diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmiko" "b/data_all_eng_slimpj/shuffled/split2/finalzzmiko" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmiko" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{intr}\n\nHyperk\\\"{a}hler manifolds first appeared within the framework of differential geometry as Riemannian manifolds with holonomy group of special restricted group. Nowadays, hyperk\\\"{a}hler geometry forms a separate research subject fusing traditional areas of mathematics such as differential and algebraic geometry of complex manifolds, holomorphic symplectic geometry, Hodge theory and many others. \n\nOne of the latest links can be found in theoretical physics: In 2009, Gaiotto, Moore and Neitzke \\cite{gaiotto} proposed a new construction of hyperk\\\"{a}hler metrics $g$ on target spaces $\\mathcal{M}$ of quantum field theories with $d = 4, \\mathcal{N} = 2$ superysmmetry. Such manifolds were already known to be hyperk\\\"{a}hler (see \\cite{seiberg}), but no known explicit hyperk\\\"{a}hler metrics have been constructed.\n\nThe manifold $\\mathcal{M}$ is a total space of a complex integrable system and it can be expressed as follows. There exists a complex manifold $\\mathcal{B}$, a divisor $D \\subset \\mathcal{B}$ and a subset $\\mathcal{M}' \\subset \\mathcal{M}$ such that $\\mathcal{M}'$ is a torus fibration over $\\mathcal{B}' := \\mathcal{B} \\backslash D$. On the divisor $D$, the torus fibers of $\\mathcal{M}$ degenerate, as Figure \\ref{nodtorus} shows.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{nodal_torus.eps}\n\t\\caption{Hyperk\\\"{a}hler manifolds realized as torus fibrations}\n\t\\label{nodtorus}\n\\end{figure}\n\nModuli spaces $\\mathcal{M}$ of Higgs bundles on Riemann surfaces with prescribed singularities at finitely many points are one of the prime examples of this construction. Hyperk\\\"{a}hler geometry is useful since we can use Hitchin's twistor space construction \\cite{hitchin} and consider all $\\mathbb{P}^1$-worth of complex structures at once. In the case of moduli spaces of Higgs bundles, this allows us to consider $\\mathcal{M}$ from three distinct viewpoints:\n\n\\begin{enumerate}\n\t\\item (Dolbeault) $\\mathcal{M}_{\\text{Dol}}$ is the moduli space of Higgs bundles, i.e. pairs $(E, \\Phi)$, $E \\to C$ a rank $n$ degree zero holomorphic vector bundle and $\\Phi \\in \\Gamma(\\text{End}(E) \\otimes \\Omega^1)$ a Higgs field.\n\t\n\t\\item (De Rham) $\\mathcal{M}_{\\text{DR}}$ is the moduli space of flat connections on rank $n$ holomorphic vector bundles, consisting of pairs $(E, \\nabla)$ with $\\nabla : E \\to \\Omega^1 \\otimes E$ a holomorphic connection and\n\t\n\t\\item (Betti) $\\mathcal{M}_{\\text{B}} = \\text{Hom}(\\pi_1(C) \\to \\text{GL}_n(\\mathbb{C}))\/\\text{GL}_n(\\mathbb{C})$ of conjugacy classes of representations of the fundamental group of $C$. \n\\end{enumerate}\n\n\\noindent All these algebraic structures form part of the family of complex structures making $\\mathcal{M}$ into a hyperk\\\"{a}hler manifold. \n\nTo prove that the manifolds $\\mathcal{M}$ from the integrable systems are indeed hyperk\\\"{a}hler, we start with the existence of a simple, explicit hyperk\\\"{a}hler metric $g^{\\text{sf}}$ on $\\mathcal{M}'$. Unfortunately, $g^{\\text{sf}}$ does not extend to $\\mathcal{M}$. To construct a complete metric $g$, it is necessary to do ``quantum corrections'' to $g^{\\text{sf}}$. These are obtained by solving a certain explicit integral equation (see \\eqref{inteq} below). The novelty is that the solutions, acting as Darboux coordinates for the hyperk\\\"{a}hler metric $g$, have discontinuities at a specific locus in $\\mathcal{B}$. Such discontinuities cancel the global monodromy around $D$ and is thus feasible to expect that $g$ extends to the entire $\\mathcal{M}$.\n\nWe start by defining a Riemann-Hilbert problem on the $\\mathbb{P}^1$-slice of the twistor space $\\mathcal{Z} = \\mathcal{M}' \\times \\mathbb{P}^1$. That is, we look for functions $\\mathcal{X}_\\gamma$ with prescribed discontinuities and asymptotics. In the language of Riemann-Hilbert theory, this is known as \\textit{monodromy data}. Rather than a single Riemann-Hilbert problem, we have a whole family of them parametrized by the $\\mathcal{M}'$ manifold. We show that this family constitutes an \\textit{isomonodromic deformation} since by the Kontsevich-Soibelman Wall-Crossing Formula, the monodromy data remains invariant. \n\nAlthough solving Riemann-Hilbert problems in general is not always possible, in this case it can be reduced to an integral equation solved by standard Banach contraction principles. We will focus on a particular case known as the ``Pentagon'' (a case of Hitchin systems with gauge group $\\text{SU}(2)$). The family of Riemann-Hilbert problems and their methods of solutions is a topic of independent study so we leave this construction to a second article that can be of interest in the study of boundary-value problems.\n\nThe extension of the manifold $\\mathcal{M}'$ is obtained by gluing a circle bundle with an appropriate gauge transformation eliminating any monodromy problems near the divisor $D$. The circle bundle constructs the degenerate tori at the discriminant locus $D$ (see Figure \\ref{pinch}).\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.15\\textwidth]{part_torus2.ps}\n\t\\caption{Construction of degenerate fibers}\n\t\\label{pinch}\n\\end{figure}\n\nOn the extended manifold $\\mathcal{M}$ we prove that the solutions $\\mathcal{X}_\\gamma$ of the Riemann-Hilbert problem on $\\mathcal{M}'$ extend and the resulting holomorphic symplectic form $\\varpi(\\zeta)$ gives the desired hyperk\\\"{a}hler metric $g$.\n\nAlthough for the most basic examples of this construction such as the moduli space of Higgs bundles it was already known that $\\mathcal{M}'$ extends to a hyperk\\\"{a}hler manifold $\\mathcal{M}$ with degenerate torus fibers, the construction here works for the general case of $\\dim_\\mathbb{C} \\mathcal{B} = 1$. Moreover, the functions $\\mathcal{X}_\\gamma$ here are special coordinates arising in moduli spaces of flat connections, Teichm\\\"{u}ller theory and Mirror Symmetry. In particular, these functions are used in \\cite{chan} for the construction of holomorphic discs with boundary on special Lagrangian torus fibers of mirror manifolds.\n\nThe organization of the paper is as follows. In Section \\ref{intsys} we introduce the complex integrable systems to be considered in this paper. These systems arose first in the study of moduli spaces of Higgs bundles and they can be written in terms of initial data and studied abstractly. This leads to a formulation of a family of Riemann-Hilbert problems, whose solutions provide Darboux coordinates for the moduli spaces $\\mathcal{M}$ considered and hence equip the latter with a hyperk\\\"{a}hler structure. In Section \\ref{ov} we fully work the simplest example of these integrable systems: the Ooguri-Vafa case. Although the existence of this hyperk\\\"{a}hler metric was already known, this is the first time it is obtained via Riemann-Hilbert methods. In Section \\ref{gmetric}, we explicitly show that this metric is a smooth deformation of the well-known Taub-NUT metric near the singular fiber of $\\mathcal{M}$ thus proving its extension to the entire manifold. In Section \\ref{pent} we introduce our main object of study, the Pentagon case. This is the first nontrivial example of the integrable systems considered and here the Wall Crossing phenomenon is present. We use the KS wall-crossing formula to apply an isomonodromic deformation of the Riemann-Hilbert problems leading to solutions continuous at the wall of marginal stability. Finally, Section \\ref{sfiber} deals with the extension of these solutions $\\mathcal{X}_\\gamma$ to singular fibers of $\\mathcal{M}$ thought as a torus fibration. What we do is to actually complete the manifold $\\mathcal{M}$ from a regular torus fibration $\\mathcal{M}'$ by gluing circle bundles near a discriminant locus $D$. This involves a change of the torus coordinates for the fibers of $\\mathcal{M}'$. In terms of the new coordinates, the $\\mathcal{X}_\\gamma$ functions extend to the new patch and parametrize the complete manifold $\\mathcal{M}$. We finish the paper by showing that, near the singular fibers of $\\mathcal{M}$, the hyperk\\\"{a}hler metric $g$ looks like the metric for the Ooguri-Vafa case plus some smooth corrections, thus proving that this metric is complete.\n\n\\textbf{Acknowledgment:} The author likes to thank Andrew Neitzke for his guidance, support and incredibly helpful conversations. \n\n\\section{Integrable Systems Data}\\label{intsys}\n\nWe start by presenting the complex integrable systems introduced in \\cite{gaiotto}. As motivation, consider the moduli space $\\mathcal{M}$ of Higgs bundles on a complex curve $C$ with Higgs field $\\Phi$ having prescribed singularities at finitely many points. In \\cite{wkb}, it is shown that the space of quadratic differentials $u$ on $C$ with fixed poles and residues is a complex affine space $\\mathcal{B}$ and the map $\\text{det} : \\mathcal{M} \\to \\mathcal{B}$ is proper with generic fiber $\\text{Jac}(\\Sigma_u)$, a compact torus obtained from the \\textit{spectral curve} $\\Sigma_u : = \\{(z, \\phi) \\in T^*C : \\phi^2 = u\\}$, a double-branched cover of $C$ over the zeroes of the quadratic differential $u$. $\\Sigma_u$ has an involution that flips $\\phi \\mapsto -\\phi$. If we take $\\Gamma_u$ to be the subgroup of $H_1(\\Sigma_u, \\mathbb{Z})$ odd under this involution, $\\Gamma$ forms a lattice of rank 2 over $\\mathcal{B}'$, the space of quadratic differentials with only simple zeroes. This lattice comes with a non-degenerate anti-symmetric pairing $\\left\\langle , \\right\\rangle$ from the intersection pairing in $H_1$. It is also proved in \\cite{wkb} that the fiber $\\text{Jac}(\\Sigma_u)$ can be identified with the set of characters $\\text{Hom}(\\Gamma_u, \\mathbb{R}\/2\\pi \\mathbb{Z})$. If $\\lambda$ denotes the tautological 1-form in $T^* C$, then for any $\\gamma \\in \\Gamma$,\n\\[ Z_\\gamma = \\frac{1}{\\pi} \\oint_\\gamma \\lambda \\]\ndefines a holomorphic function $Z_\\gamma$ in $\\mathcal{B}'$. Let $\\{\\gamma_1, \\gamma_2\\}$ be a local basis of $\\Gamma$ with $\\{\\gamma^1, \\gamma^2\\}$ the dual basis of $\\Gamma^*$. Without loss of generality, we also denote by $\\left\\langle , \\right\\rangle$ the pairing in $\\Gamma^*$. Let $\\left\\langle dZ \\wedge dZ \\right\\rangle$ be short notation for $\\left\\langle \\gamma^1, \\gamma^2 \\right\\rangle dZ_{\\gamma_1} \\wedge dZ_{\\gamma_2}$. Since $\\dim_\\mathbb{C} \\mathcal{B}' = 1$, $\\left\\langle dZ \\wedge dZ \\right\\rangle = 0$.\n\nThis type of data arises in the construction of hyperk\\\"{a}hler manifolds as in \\cite{gaiotto} and \\cite{notes}, so we summarize the conditions required:\n \n\nWe start with a complex manifold $\\mathcal{B}$ (later shown to be affine) of dimension $n$ and a divisor $D \\subset \\mathcal{B}$. Let $\\mathcal{B}' = \\mathcal{B} \\backslash D$. Over $\\mathcal{B}'$ there is a local system $\\Gamma$ with fiber a rank $2n$ lattice, equipped with a non-degenerate anti-symmetric integer valued pairing $\\left\\langle \\, , \\right\\rangle$. \n\nWe will denote by $\\Gamma^*$ the dual of $\\Gamma$ and, by abuse of notation, we'll also use $\\left\\langle \\, , \\right\\rangle$ for the dual pairing (not necessarily integer-valued) in $\\Gamma^*$. Let $u$ denote a general point of $\\mathcal{B}'$. We want to obtain a torus fibration over $\\mathcal{B}'$, so let $\\text{TChar}_u(\\Gamma)$ be the set of twisted unitary characters of $\\Gamma_u$\\footnote{Although we can also work with the set of unitary characters (no twisting involved) by shifting the $\\theta$-coordinates, we choose not to do so, as that results in more complex calculations}, i.e. maps $\\theta : \\Gamma_u \\to \\mathbb{R}\/2\\pi \\mathbb{Z}$ satisfying\n\\begin{equation*}\n\\theta_\\gamma + \\theta_{\\gamma'} = \\theta_{\\gamma + \\gamma'} + \\pi \\left\\langle \\gamma, \\gamma' \\right\\rangle.\n\\end{equation*}\nTopologically, $\\text{TChar}_u(\\Gamma)$ is a torus $(S^1)^{2n}$. Letting $u$ vary, the $\\text{TChar}_u(\\Gamma)$ form a torus bundle $\\mathcal{M}'$ over $\\mathcal{B}'$. Any local section $\\gamma$ gives a local angular coordinate of $\\mathcal{M}'$ by ``evaluation on $\\gamma$'', $\\theta_\\gamma : \\mathcal{M}' \\to \\mathbb{R}\/2\\pi \\mathbb{Z}$.\n\nWe also assume there exists a homomorphism $Z : \\Gamma \\to \\mathbb{C}$ such that the vector $Z(u) \\in \\Gamma^*_u \\otimes \\mathbb{C}$ varies holomorphically with $u$. If we pick a patch $U \\subset \\mathcal{B}'$ on which $\\Gamma$ admits a basis $\\{\\gamma_1, \\ldots, \\gamma_{2n}\\}$ of local sections in which $\\left\\langle , \\right\\rangle$ is the standard symplectic pairing, then (after possibly shrinking $U$) the functions\n\\[ f_i = \\text{Re}(Z_{\\gamma_i}) \\]\nare real local coordinates. The transition functions on overlaps $U \\cap U'$ are valued on $\\text{Sp}(2n, \\mathbb{Z})$, as different choices of basis in $\\Gamma$ must fix the symplectic pairing. This gives an affine structure on $\\mathcal{B}'$.\n\n By differentiating and evaluating in $\\gamma$, we get 1-forms $d\\theta_\\gamma, d Z_\\gamma$ on $\\mathcal{M}'$ which are linear on $\\Gamma$. For a local basis $\\{\\gamma_1, \\ldots, \\gamma_{2n}\\}$ as in the previous paragraph, let $\\{\\gamma^1, \\ldots, \\gamma^{2n}\\}$ denote its dual basis on $\\Gamma^*$. We write $\\left\\langle dZ \\wedge dZ \\right\\rangle$ as short notation for\n \\begin{equation}\\label{dzdz}\n \\left\\langle \\gamma^i , \\gamma^j \\right\\rangle dZ_{\\gamma_i} \\wedge dZ_{\\gamma_j},\n \\end{equation}\n where we sum over repeated indices. Observe that the anti-symmetric pairing $\\left\\langle \\, , \\right\\rangle$ and the anti-symmetric wedge product of 1-forms makes \\eqref{dzdz} symmetric. We require that:\n \n\\begin{equation}\\label{dz0}\n\\left\\langle dZ \\wedge dZ \\right\\rangle = 0,\n\\end{equation}\n\nBy \\eqref{dz0}, near $u$, $\\mathcal{B}'$ can be locally identified with a complex Lagrangian submanifold of $\\Gamma^* \\otimes_\\mathbb{Z} \\mathbb{C}$.\n\nIn the example of moduli spaces of Higgs bundles, as $u$ approaches a quadratic differential with non-simple zeros, one homology cycles vanishes (see Figure \\ref{nodtorus}). This cycle $\\gamma_0$ is primitive in $H_1$ and its monodromy around the critical quadratic differential is governed by the Picard-Lefschetz formula. In the general case, let $D_0$ be a component of the divisor $D \\subset \\mathcal{B}$. We also assume the following:\n\n\\begin{itemize}\n\t\\item $Z_{\\gamma_0}(u) \\to 0$ as $u \\to u_0 \\in D_0$ for some $\\gamma_0 \\in \\Gamma$.\n\t\n\n\t\n\t\\item $\\gamma_0$ is primitive (i.e. there exists some $\\gamma'$ with $\\left\\langle \\gamma_0, \\gamma'\\right\\rangle = 1$).\n\t\n\t\\item The monodromy of $\\Gamma$ around $D_0$ is of ``Picard-Lefschetz type'', i.e.\n\t\\begin{equation}\\label{piclf}\n\t\\gamma \\mapsto \\gamma + \\left\\langle \\gamma, \\gamma_0 \\right\\rangle \\gamma_0\n\t\\end{equation}\n\\end{itemize}\n\n\nWe assign a complex structure and a holomorphic symplectic form on $\\mathcal{M}'$ as follows (see \\cite{notes} and the references therein for proofs). Take a local basis $\\{\\gamma_1, \\ldots, \\gamma_{2n}\\}$ of $\\Gamma$. If $\\epsilon^{ij} = \\left\\langle \\gamma_i , \\gamma_j\\right\\rangle$ and $\\epsilon_{ij}$ is its dual, let\n\\begin{equation}\\label{ome}\n\\omega_+ = \\left\\langle dZ \\wedge d\\theta \\right\\rangle = \\epsilon_{ij} \\, dZ_{\\gamma_i} \\wedge d\\theta_{\\gamma_j}.\n\\end{equation}\nBy linearity on $\\gamma$ of the 1-forms, $\\omega_+$ is independent of the choice of basis. There is a unique complex structure $J$ on $\\mathcal{M}'$ for which $\\omega_+$ is of type (2,0). The 2-form $\\omega_+$ gives a holomorphic symplectic structure on $(\\mathcal{M}', J)$. With respect to this structure, the projection $\\pi: \\mathcal{M}' \\to \\mathcal{B}'$ is holomorphic, and the torus fibers $\\mathcal{M}'_u = \\pi^{-1}(u)$ are compact complex Lagrangian submanifolds. \n\nRecall that a positive 2-form $\\omega$ on a complex manifold is a real 2-form for which $\\omega(v,Jv) >0$ for all real tangent vectors $v$. From now on, we assume that $\\left\\langle dZ \\wedge d\\overline{Z} \\right\\rangle$ is a positive 2-form on $\\mathcal{B}'$. Now fix $R > 0$. Then we can define a 2-form on $\\mathcal{M}'$ by\n\\begin{equation*}\n\\omega_3^{\\text{sf}} = \\frac{R}{4} \\left\\langle dZ \\wedge d\\overline{Z} \\right\\rangle - \\frac{1}{8\\pi^2 R}\\left\\langle d\\theta \\wedge d\\theta \\right\\rangle.\n\\end{equation*}\nThis is a positive form of type (1,1) in the $J$ complex structure. Thus, the triple $(\\mathcal{M}', J, \\omega_3^{\\text{sf}})$ determines a K\\\"{a}hler metric $g^{\\text{sf}}$ on $\\mathcal{M}'$. This metric is in fact hyperk\\\"{a}hler (see \\cite{freed}), so we have a whole $\\mathbb{P}^1$-worth of complex structures for $\\mathcal{M}'$, parametrized by $\\zeta \\in \\mathbb{P}^1$. The above complex structure $J$ represents $J(\\zeta = 0)$, the complex structure at $\\zeta = 0$ in $\\mathbb{P}^1$. The superscript ${}^\\text{sf}$ stands for ``semiflat''. This is because $g^{\\text{sf}}$ is flat on the torus fibers $\\mathcal{M}'_u$.\n\nAlternatively, it is shown in \\cite{gaiotto} that if\n\\begin{equation}\\label{xsfr}\n\\mathcal{X}_\\gamma^{\\text{sf}}(\\zeta) = \\exp\\left( \\frac{\\pi R Z_\\gamma}{\\zeta} + i\\theta_\\gamma + \\pi R \\zeta \\overline{Z_\\gamma} \\right)\n\\end{equation}\nThen the 2-form\n\\[ \\varpi(\\zeta) = \\frac{1}{8\\pi^2 R} \\left\\langle d\\log \\mathcal{X}^{\\text{sf}}(\\zeta) \\wedge d\\log \\mathcal{X}^{\\text{sf}}(\\zeta) \\right\\rangle \\]\n(where the DeRham operator $d$ is applied to the $\\mathcal{M}'$ part only) can be expressed as\n\\[ -\\frac{i}{2\\zeta}\\omega_+ + \\omega^{\\text{sf}}_3 -\\frac{i \\zeta}{2} \\omega_-, \\]\nfor $\\omega_- = \\overline{\\omega_+} = \\left\\langle d\\overline{Z} \\wedge d\\overline{\\theta} \\right\\rangle$, that is, in the twistor space $\\mathcal{Z} = \\mathcal{M}' \\times \\mathbb{P}^1$ of \\cite{hitchin}, $\\varpi(\\zeta)$ is a holomorphic section of $\\Omega_{\\mathcal{Z}\/\\mathbb{P}^1} \\otimes \\mathcal{O}(2)$ (the twisting by $\\mathcal{O}(2)$ is due to the poles at $\\zeta = 0$ and $\\zeta = \\infty$ in $\\mathbb{P}^1$). This is the key step in Hitchin's twistor space construction. By \\cite[\\S 3]{gaiotto}, $\\mathcal{M}'$ is hyperk\\\"{a}hler.\n\nWe want to reproduce the same construction of a hyperk\\\"{a}hler metric now with corrected Darboux coordinates $\\mathcal{X}_\\gamma(\\zeta)$. For that, we need another piece of data. Namely, a function $\\Omega : \\Gamma \\to \\mathbb{Z}$ such that $\\Omega(\\gamma;u) = \\Omega(-\\gamma;u)$. Furthermore, we impose a condition on the nonzero $\\Omega(\\gamma;u)$. Introduce a positive definite norm on $\\Gamma$. Then we require the existence of $K > 0$ such that\n\\begin{equation}\\label{support}\n\\frac{|Z_\\gamma|}{\\left\\| \\gamma \\right\\|} > K\n\\end{equation}\nfor those $\\gamma$ such that $\\Omega(\\gamma; u) \\neq 0$. This is called the \\emph{Support Property}, as in \\cite{gaiotto}.\n\nFor a component of the singular locus $D_0$ and for $\\gamma_0$ the primitive element in $\\Gamma$ for which $Z_{\\gamma_0} \\to 0$ as $u \\to u_0 \\in D_0$, we also require\n\\[ \\Omega(\\gamma_0; u) = 1 \\text{ for all $u$ in a neighborhood of $D_0$} \\]\n\nTo see where these invariants arise from, consider the example of moduli spaces of Higgs bundles again. A quadratic differential $u \\in \\mathcal{B}'$ determines a metric $h$ on $C$. Namely, if $u = P(z)dz^2$, $h = |P(z)| dz d\\overline{z}$. Let $C'$ be the curve obtained after removing the poles and zeroes of $u$. Consider the finite length inextensible geodesics on $C'$ in the metric $h$. These come in two types:\n\n\\begin{enumerate}\n\t\\item \\textit{Saddle connections}: geodesics running between two zeroes of $u$. See Figure \\ref{saddlec}.\n\t\n\t\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{saddle_connection.ps}\n\t\\caption{Saddle connections on $C'$}\n\t\\label{saddlec}\n\\end{figure}\n\n\\item \\textit{Closed geodesics}: When they exist, they come in 1-parameter families sweeping out annuli in $C'$. See Figure \\ref{clgeod}.\n\n\t\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{closed_loop.ps}\n\t\\caption{Closed geodesics on $C'$ sweeping annuli}\n\t\\label{clgeod}\n\\end{figure}\n\n\\end{enumerate}\n\nOn the branched cover $\\Sigma_u \\to C$, each geodesic can be lifted to a union of closed curves in $\\Sigma_u$, representing some homology class $\\gamma \\in H_1(\\Sigma_u, \\mathbb{Z})$. See Figure \\ref{lift}.\n\n\t\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{double_cover.ps}\n\t\\caption{Lift of geodesics to $\\Sigma_u$}\n\t\\label{lift}\n\\end{figure}\n\n\n In this case, $\\Omega(\\gamma,u)$ counts these finite length geodesics: every saddle connection with lift $\\gamma$ contributes $+1$ and every closed geodesic with lift $\\gamma$ contributes $-2$.\n\n\nBack to the general case, we're ready to formulate a Riemann-Hilbert problem on the $\\mathbb{P}^1$-slice of the twistor space $\\mathcal{Z} = \\mathcal{M}' \\times \\mathbb{P}^1$. Recall that in a RH problem we have a contour $\\Sigma$ dividing a complex plane (or its compactification) and one tries to obtain functions which are analytic in the regions defined by the contour, with continuous extensions along the boundary and with prescribed discontinuities along $\\Sigma$ and fixed asymptotics at the points where $\\Sigma$ is non-smooth. In our case, the contour is a collection of rays at the origin and the discontinuities can be expressed as symplectomorphisms of a complex torus:\n\nDefine a ray associated to each $\\gamma \\in \\Gamma_u$ as:\n\\[ \\ell_\\gamma(u) = Z_\\gamma \\mathbb{R}_-. \\]\nWe also define a transformation of the functions $\\mathcal{X}_{\\gamma'}$ given by each $\\gamma \\in \\Gamma_u$:\n\\begin{equation}\\label{kjump}\n\\mathcal{K}_\\gamma \\mathcal{X}_{\\gamma'} = \\mathcal{X}_{\\gamma'} (1- \\mathcal{X}_{\\gamma})^{\\left\\langle \\gamma', \\gamma \\right\\rangle}\n\\end{equation}\nLet $T_u$ denote the space of twisted complex characters of $\\Gamma_u$, i.e. maps $\\mathcal{X} : \\Gamma_u \\to \\mathbb{C}^{\\times}$ satisfying\n\\begin{equation}\\label{xprop}\n \\mathcal{X}_\\gamma \\mathcal{X}_{\\gamma'} = (-1)^{\\left\\langle \\gamma, \\gamma'\\right\\rangle} \\mathcal{X}_{\\gamma + \\gamma'}\n\\end{equation}\n$T_u$ has a canonical Poisson structure given by\n\\[ \\{ \\mathcal{X}_\\gamma, \\mathcal{X}_{\\gamma'} \\} = \\left\\langle \\gamma, \\gamma' \\right\\rangle \\mathcal{X}_{\\gamma + \\gamma'}\\]\nThe $T_u$ glue together into a bundle over $\\mathcal{B}'$ with fiber a complex Poisson torus. Let $T$ be the pullback of this system to $\\mathcal{M}'$. We can interpret the transformations $\\mathcal{K}_\\gamma$ as birational automorphisms of $T$.\nTo each ray $\\ell$ going from 0 to $\\infty$ in the $\\zeta$-plane, we can define a transformation\n\\begin{equation}\\label{stkfac}\nS_\\ell = \\prod_{\\gamma : \\ell_\\gamma(u) = \\ell} \\mathcal{K}_\\gamma^{\\Omega(\\gamma;u)}\n\\end{equation}\nNote that all the $\\gamma$'s involved in this product are multiples of each other, so the $\\mathcal{K}_\\gamma$ commute and it is not necessary to specify an order for the product.\n\nTo obtain the corrected $\\mathcal{X}_\\gamma$, we can formulate a Riemann-Hilbert problem for which the former functions are solutions to it. We seek a map $\\mathcal{X} : \\mathcal{M}'_u \\times \\mathbb{C}^{\\times} \\to T_u$ with the following properties:\n\\begin{enumerate}[label=\\textnormal{(\\arabic*)}]\n\t\\item $\\mathcal{X}$ depends piecewise holomorphically on $\\zeta$, with discontinuities only at the rays $\\ell_\\gamma(u)$ for which $\\Omega(\\gamma;u) \\neq 0$.\n\t\\item The limits $\\mathcal{X}^{\\pm}$ as $\\zeta$ approaches any ray $\\ell$ from both sides exist and are related by\n\t\\begin{equation}\\label{invjmp}\n\t\\mathcal{X}^+ = S_\\ell^{-1} \\circ \\mathcal{X}^-\n\t\\end{equation}\n\t\\item $\\mathcal{X}$ obeys the reality condition\n\t\\begin{equation}\\label{realcond}\n\t\\overline{\\mathcal{X}_{-\\gamma}(-1\/\\overline{\\zeta})} = \\mathcal{X}_\\gamma(\\zeta)\n\t\\end{equation}\n\t\\item For any $\\gamma \\in \\Gamma_u$, $\\lim_{\\zeta \\to 0} \\mathcal{X}_\\gamma(\\zeta) \/ \\mathcal{X}^{\\text{sf}}_\\gamma(\\zeta)$ exists and is real. \\label{asymptotic}\n\\end{enumerate}\n\nIn \\cite{gaiotto}, this RH problem is formulated as an integral equation:\n\\begin{equation}\\label{inteq}\n\\mathcal{X}_\\gamma(u,\\zeta) = \\mathcal{X}^{\\text{sf}}_\\gamma(u,\\zeta)\\exp\\left[ -\\frac{1}{4\\pi i} \\sum_{\\gamma'} \\Omega(\\gamma';u) \\left\\langle \\gamma, \\gamma' \\right\\rangle \\int_{\\ell_{\\gamma'(u)}} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta}\\log\\left( 1 - \\mathcal{X}_{\\gamma'}(u,\\zeta')\\right)\\right],\n\\end{equation}\nOne can define recursively, setting $\\mathcal{X}^{(0)} = \\mathcal{X}^{\\text{sf}}$:\n\\begin{equation}\\label{recurs}\n\\mathcal{X}^{(\\nu+1)}_\\gamma(u,\\zeta) = \\mathcal{X}^{\\text{sf}}_\\gamma(u,\\zeta)\\exp\\left[ -\\frac{1}{4\\pi i} \\sum_{\\gamma'} \\Omega(\\gamma';u) \\left\\langle \\gamma, \\gamma' \\right\\rangle \\int_{\\ell_{\\gamma'(u)}} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta}\\log\\left( 1 - \\mathcal{X}^{(\\nu)}_{\\gamma'}(u,\\zeta')\\right)\\right],\n\\end{equation}\n\nMore precisely, we have a family of RH problems, parametrized by $u \\in \\mathcal{B}'$, as this defines the rays $\\ell_\\gamma(u)$, the complex torus $T_u$ where the symplectomorphisms are defined and the invariants $\\Omega(\\gamma;u)$ involved in the definition of the problem.\n\nWe still need one more piece of the puzzle, since the latter function $\\Omega$ may not be continuous. In fact, $\\Omega$ jumps along a real codimension-1 loci in $\\mathcal{B}'$ called the ``wall of marginal stability''. This is the locus where 2 or more functions $Z_\\gamma$ coincide in phase, so two or more rays $\\ell_{\\gamma}(u)$ become one. More precisely:\n\\[ W = \\{u \\in \\mathcal{B}': \\exists \\gamma_1, \\gamma_2 \\text{ with } \\Omega(\\gamma_1;u) \\neq 0, \\Omega(\\gamma_2;u) \\neq 0, \\left\\langle \\gamma_1, \\gamma_2\\right\\rangle \\neq 0, Z_{\\gamma_1}\/Z_{\\gamma_2} \\in \\mathbb{R}_+\\}\\]\n The jumps of $\\Omega$ are not arbitrary; they are governed by the Kontsevich-Soibelman wall-crossing formula.\n\nTo describe this, let $V$ be a strictly convex cone in the $\\zeta$-plane with apex at the origin. Then for any $u \\notin W$ define\n\\begin{equation}\nA_V(u) = \\prod^\\text{\\Large$\\curvearrowleft$}_{\\gamma : Z_\\gamma(u) \\in V} \\mathcal{K}_\\gamma^{\\Omega(\\gamma;u)} = \\prod^\\text{\\Large$\\curvearrowleft$}_{\\ell \\subset V} S_\\ell\\footnote{This product may be infinite. One should more precisely think of $A_V(u)$ as living in a certain prounipotent completion of the group generated by $\\{\\mathcal{K}_\\gamma\\}_{\\gamma : Z_\\gamma(u) \\in V}$ as explained in \\cite{kont}}\n\\end{equation}\n\nThe arrow indicates the order of the rational maps $\\mathcal{K}_\\gamma$. $A_V(u)$ is a birational Poisson automorphism of $T_u$. Define a $V$-\\textit{good path} to be a path $p$ in $\\mathcal{B}'$ along which there is no point $u$ with $Z_\\gamma(u) \\in \\partial V$ and $\\Omega(\\gamma;u) \\neq 0$. (So as we travel along a $V$-good path, no $\\ell_\\gamma$ rays enter or exit V.) If $u, u'$ are the endpoints of a $V$-good path $p$, the wall-crossing formula is the condition that $A_V(u), A_V(u')$ are related by parallel transport in $T$ along $p$. See Figure \\ref{partrnpt}.\n\n\n\t\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.80\\textwidth]{wall_crossing.ps}\n\t\\caption{For a good path $p$, the two automorphisms $A_V(u), A_V(u')$ are related by parallel transport}\n\t\\label{partrnpt}\n\\end{figure}\n\n\n\n\\subsection{Statement of Results}\\label{results}\n\nWe will restrict in this paper to the case $\\dim_\\mathbb{C} \\mathcal{B} = 1$, so $n = \\dim \\Gamma = 2$. We want to extend the torus fibration $\\mathcal{M}'$ to a manifold $\\mathcal{M}$ with degenerate torus fibers. To give an example, in the case of Hitchin systems, the torus bundle $\\mathcal{M}'$ is not the moduli space of Higgs bundles yet, as we have to consider quadratic differentials with non-simple zeroes too. The main results of this paper center on the extension of the manifold $\\mathcal{M}'$ to a manifold $\\mathcal{M}$ with an extended fibration $\\mathcal{M} \\to \\mathcal{B}$ such that the torus fibers $\\mathcal{M}'_u$ degenerate to nodal torus (i.e. ``singular'' or ``bad'' fibers) for $u \\in D$.\n\nWe start by fully working out the simplest example known as Ooguri-Vafa \\cite{cecotti}. Here we have a fibration over the open unit disk $\\mathcal{B} := \\{u \\in \\mathbb{C} : |u| < 1 \\}$. At the discriminant locus $D : = \\{ u = 0 \\}$, the fibers degenerate into a nodal torus. The local rank-2 lattice $\\Gamma$ has a basis $(\\gamma_m, \\gamma_e)$ and the skew-symmetric pairing is defined by $\\left\\langle \\gamma_m, \\gamma_e \\right\\rangle = 1$. The monodromy of $\\Gamma$ around $u = 0$ is $\\gamma_e \\mapsto \\gamma_e, \\gamma_m \\mapsto \\gamma_m + \\gamma_e$. We also have functions $Z_{\\gamma_e}(u) = u, Z_{\\gamma_m}(u) = \\frac{u}{2\\pi i }( \\log u - 1) + f(u)$, for $f$ holomorphic and admitting an extension to $\\mathcal{B}$. Finally, the integer-valued function $\\Omega$ in $\\Gamma$ is here: $\\Omega(\\pm \\gamma_e; u) = 1$ and $\\Omega(\\gamma; u) = 0$ for any other $\\gamma \\in \\Gamma_u$. There is no wall of marginal stability in this case. The integral equation \\eqref{inteq} can be solved after just 1 iteration.\n\nFor all other nontrivial cases, in order to give a satisfactory extension of the $\\mathcal{X}_\\gamma$ coordinates, it was necessary to develop the theory of Riemann-Hilbert-Birkhoff problems to suit these infinite-dimensional systems (as the transformations $S_\\ell$ defining the problem can be thought as operators on $C^\\infty(T_u)$, rather than matrices). It is not clear that such coordinates can be extended, since we may approach the bad fiber from two different sides of the wall of marginal stability and obtain two different extensions. To overcome this first obstacle, we have to use the theory of isomonodromic deformations as in \\cite{boalch} to reformulate the Riemann-Hilbert problem in \\cite{gaiotto} independent of the regions determined by the wall.\n\nHaving redefined the problem, we want our $\\mathcal{X}_\\gamma$ to be smooth on the parameters $\\theta_{\\gamma_1}, \\theta_{\\gamma_{2}}$ and $u$,\naway from where the prescribed jumps are. Even at $\\mathcal{M}'$, there was no mathematical proof that such condition must be true. In the companion paper \\cite{rhprob}, we combine classical Banach contraction methods and Arzela-Ascoli results on uniform convergence in compact sets to obtain:\n\\begin{theorem}\\label{smooth}\nIf the collection $J$ of nonzero $\\Omega(u; \\gamma)$ satisfies the support property \\eqref{support} and if the parameter $R$ of \\eqref{xsfr} is large enough (determined by the values $|Z_\\gamma(u)|, \\gamma \\in J$), there exists a unique collection of functions $\\mathcal{X}_\\gamma$ with the prescribed asymptotics and jumps as in \\cite{gaiotto}. These functions are smooth on $u$ and the torus coordinates $\\theta_1, \\theta_{2}$ (even for $u$ at the wall of marginal stability), and piecewise holomorphic on $\\zeta$\n\\end{theorem}\n\nSince we're considering only the case $n=1$, $\\Gamma$ is a rank-1 lattice over the Riemann surface $\\mathcal{B}'$ and the discriminant locus $D$ where the torus fibers degenerate is a discrete subset of $\\mathcal{B}'$. \n\nFrom this point on, we restrict our attention to the next nontrivial system, known as the Pentagon case \\cite{notes}. Here $\\mathcal{B} = \\mathbb{C}$ with 2 bad fibers which we can assume are at $u = -2, u = 2$ and $\\mathcal{B}'$ is the twice-punctured plane. There is a wall of marginal stability where all $Z_\\gamma$ are contained in the same line. This separates $\\mathcal{B}$ in two domains $\\mathcal{B}_\\text{out}$ and a simply-connected $\\mathcal{B}_\\text{in}$. See Figure \\ref{aplane}.\n\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{walls.ps}\n\t\\caption{The wall $W$ in $\\mathcal{B}$ for the Pentagon case}\n\t\\label{aplane}\n\\end{figure}\n\n\nOn $\\mathcal{B}_\\text{in}$ we can trivialize $\\Gamma$ and choose a basis $\\{\\gamma_1, \\gamma_2\\}$ with pairing $\\left\\langle \\gamma_1, \\gamma_2\\right\\rangle = 1$. This basis does not extend to a global basis for $\\Gamma$ since it is not invariant under monodromy. However, the set $\\{\\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2, \\gamma_1 + \\gamma_2, -\\gamma_1 - \\gamma_2\\}$ is indeed invariant so the following definition of $\\Omega$ makes global sense:\n \\begin{align*}\n \\text{For $u \\in \\mathcal{B}_\\text{in}$}, \\Omega(\\gamma; u) = & \\left\\{ \\begin{array}{ll}\n 1 & \\text{for } \\gamma \\in \\{ \\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2\\}\\\\\n 0 & \\text{otherwise}\n \\end{array} \\right. \\\\\n \\text{For $u \\in \\mathcal{B}_\\text{out}$} , \\Omega(\\gamma; u) = & \\left\\{ \\begin{array}{ll}\n 1 & \\text{for } \\gamma \\in \\{ \\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2, \\gamma_1 + \\gamma_2, -\\gamma_1 - \\gamma_2\\}\\\\\n 0 & \\text{otherwise}\n \\end{array} \\right.\n \\end{align*}\n \n \n \n \n \n The Pentagon case appears in the study of Hitchin systems with gauge group $\\text{SU}(2)$. The extension of $\\mathcal{M}'$ was previously obtained by hyperk\\\"{a}hler quotient methods in \\cite{biquard}, but no explicit hyperk\\\"{a}hler metric was constructed. \n\nOnce the $\\{ \\mathcal{X}_{\\gamma_i} \\}$ are obtained by Theorem \\ref{smooth},\nit is necessary to do an analytic continuation along $\\mathcal{B}'$ for the particular $\\mathcal{X}_{\\gamma_i}$ for which $Z_{\\gamma_i} \\to 0$ as $u \\to u_0 \\in D$. Without loss of generality, we can assume there\nis a local basis $\\{\\gamma_1, \\gamma_{2}\\}$ of $\\Gamma$ such that $Z_{\\gamma_2} \\to 0$ in $D$. After that, an analysis of the possible divergence of $\\mathcal{X}_\\gamma$ as $u \\to u_0$ shows the necessity of performing a gauge transformation on the torus coordinates of the fibers $\\mathcal{M}_u$ that allows us to define an integral equation even at $u_0 \\in D$. This series of transformations are defined in \\eqref{newmp}, \\eqref{outmp}, \\eqref{outmp2} and \\eqref{fingauge}, and constitute a new result that was not expected in \\cite{gaiotto}. We basically deal with a family of boundary value problems for which the jump function vanishes at certain points and\nsingularities of certain kind appear as $u \\to u_0$. As this is of independent interest, we leave the relevant results to \\cite{rhprob} and we show that our solutions contain at worst branch singularities at 0 or $\\infty$ in the $\\zeta$-plane. As in the case of normal fibers, we can run a contraction argument to obtain Darboux coordinates even at the singular fibers and conclude\n\n\\begin{theorem}\\label{extbf}\nLet $\\{\\gamma_{1}, \\gamma_2\\}$ be a local basis for $\\Gamma$ in a small sector centered at $u_0 \\in D$ such that $Z_{\\gamma_2} \\to 0$ as $u \\to u_0 \\in D$. For the Pentagon integrable system, the local function $\\mathcal{X}_{\\gamma_1}$ admits an analytic continuation $\\widetilde{\\mathcal{X}}_{\\gamma_1}$ to a punctured disk centered at $u_0$ in $\\mathcal{B}$. There exists a gauge transformation $\\theta_1 \\mapsto \\widetilde{\\theta}_1$ that extends the torus fibration $\\mathcal{M}'$ to a manifold $\\mathcal{M}$ that is locally, for each point in $D$, a (trivial) fibration over $\\mathcal{B} \\times S^{1}$ with fiber $S^1$ coordinatized by $\\theta_1$ and with one fiber collapsed into a point. For $R > 0$ big enough, it is possible to extend $\\widetilde{\\mathcal{X}}_{\\gamma_1}$ and $\\mathcal{X}_{\\gamma_2}$ to $\\mathcal{M}$, still preserving the smooth properties as in Theorem \\ref{smooth}.\n\\end{theorem}\n\n\n\n\nAfter we have the smooth extension of the $\\{ \\mathcal{X}_{\\gamma_i} \\}$ by Theorem \\ref{extbf}, we can extend the holomorphic symplectic form $\\varpi(\\zeta)$ labeled by $\\zeta \\in \\mathbb{P}^1$ as in \\cite{hitchin} for all points except possibly one at the singular fiber. From $\\varpi(\\zeta)$ we can obtain the hyperk\\\"{a}hler metric $g$ and, in the case of the Pentagon, after a change of coordinates, we realize $g$ locally as the Taub-NUT metric plus smooth corrections, finishing the construction of $\\mathcal{M}$ and its hyperk\\\"{a}hler metric. The following is the main theorem of the paper.\n\n\\begin{theorem}\\label{smfrm}\nFor the Pentagon case, the extension $\\mathcal{M}$ of the manifold $\\mathcal{M}'$ constructed in Theorem \\ref{extbf} admits, for $R$ large enough, a hyperk\\\"{a}hler metric $g$ obtained by extending the hyperk\\\"{a}hler metric on $\\mathcal{M}'$ determined by the Darboux coordinates $\\{ \\mathcal{X}_{\\gamma_i} \\}$.\n\\end{theorem}\n\n\n\n \n\\section{The Ooguri-Vafa Case}\\label{ov}\n\n\\subsection{Classical Case}\\label{clasov}\n\nWe start with one of the simplest cases, known as the Ooguri-Vafa case, first treated in \\cite{cecotti}. To see where this case comes from, recall that by the SYZ picture of K3 surfaces \\cite{gross}, any K3 surface $\\mathcal{M}$ is a hyperk\\\"{a}hler manifold. In one of its complex structures (say $J^{(\\zeta = 0)}$) is elliptically fibered, with base manifold $\\mathcal{B} = \\mathbb{P}^1$ and generic fiber a compact complex torus. There are a total of 24 singular fibers, although the total space is smooth. See Figure \\ref{k3}.\n\n\t\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{k3.ps}\n\t\\caption{A K3 surface $\\mathcal{M}$ as an elliptic fibration}\n\t\\label{k3}\n\\end{figure}\n\nGross and Wilson \\cite{gross2} constructed a hyperk\\\"{a}hler metric $g$ on a K3 surface by gluing in the Ooguri-Vafa metric constructed in \\cite{oovf} with a standard metric $g^{\\text{sf}}$ away from the degenerate fiber. Thus, this simple case can be regarded as a local model for K3 surfaces.\n \nWe have a fibration over the open unit disk $\\mathcal{B} := \\{a \\in \\mathbb{C} : |a| < 1 \\}$. At the locus $D : = \\{ a = 0 \\}$ (in the literature this is also called the \\textit{discriminant locus}), the fibers degenerate into a nodal torus. Define $\\mathcal{B}'$ as $\\mathcal{B} \\backslash D$, the punctured unit disk. On $\\mathcal{B}'$ there exists a local system $\\Gamma$ of rank-2 lattices with basis $(\\gamma_m, \\gamma_e)$ and skew-symmetric pairing defined by $\\left\\langle \\gamma_m, \\gamma_e \\right\\rangle = 1$. The monodromy of $\\Gamma$ around $a = 0$ is $\\gamma_e \\mapsto \\gamma_e, \\gamma_m \\mapsto \\gamma_m + \\gamma_e$. We also have functions $Z_{\\gamma_e}(a) = a, Z_{\\gamma_m}(a) = \\frac{a}{2\\pi i }( \\log a - 1)$. On $\\mathcal{B}'$ we have local coordinates $(\\theta_m, \\theta_e)$ for the torus fibers with monodromy $\\theta_e \\mapsto \\theta_e, \\theta_m \\mapsto \\theta_m + \\theta_e - \\pi$. Finally, the integer-valued function $\\Omega$ in $\\Gamma$ is here: $\\Omega(\\pm \\gamma_e, a) = 1$ and $\\Omega(\\gamma, a) = 0$ for any other $\\gamma \\in \\Gamma_a$. There is no wall of marginal stability in this case.\n\n We call this the ``classical Ooguri-Vafa'' case as it is the one appearing in \\cite{oovf} already mentioned at the beginning of this section. In the next section, we'll generalize this case by adding a function $f(a)$ to the definition of $Z_{\\gamma_m}$.\n\nLet\n\\begin{equation}\\label{xesf}\n \\mathcal{X}^{\\text{sf}}_\\gamma(\\zeta, a) := \\exp\\left( \\pi R \\zeta^{-1} Z_\\gamma(a) + i\\theta_\\gamma + \\pi R \\zeta \\overline{Z_\\gamma(a)}\\right)\n\\end{equation}\nThese functions receive corrections defined as in \\cite{gaiotto}. We are only interested in the pair $(\\mathcal{X}_m, \\mathcal{X}_e)$ which will constitute our desired Darboux coordinates for the holomorphic symplectic form $\\varpi$. The fact that $\\Omega(\\gamma_m, a) = 0$ gives that $\\mathcal{X}_e = \\mathcal{X}^{\\text{sf}}_e$. As $a \\to 0$, $Z_{\\gamma_e}$ and $Z_{\\gamma_m}$ approach 0. Thus $\\mathcal{X}_e|_{a = 0} = e^{i\\theta_e}$. Since $\\mathcal{X}_e = \\mathcal{X}^{\\text{sf}}_e$ the actual $\\mathcal{X}_m$ is obtained after only 1 iteration of \\eqref{recurs}. For each $a \\in \\mathcal{B}'$, let $\\ell_+$ be the ray in the $\\zeta$-plane defined by $\\{\\zeta : a\/\\zeta \\in \\mathbb{R}_- \\}$. Similarly, $\\ell_- : = \\{\\zeta : a\/\\zeta \\in \\mathbb{R}_+\\}$.\n\nLet\n\\begin{equation}\\label{defxm}\n\\mathcal{X}_m = \\mathcal{X}^{\\text{sf}}_m \\exp \\left[ \\frac{i}{4\\pi} \\int_{\\ell_+} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' +\n \\zeta}{\\zeta' - \\zeta} \\log[1 - \\mathcal{X}_e(\\zeta')] - \\frac{i}{4\\pi} \\int_{\\ell_-} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' +\n \\zeta}{\\zeta' - \\zeta} \\log[1 - \\mathcal{X}_e(\\zeta')^{-1}] \\right].\n\\end{equation}\nFor convenience, from this point on we assume $a$ is of the form $sb$, where $s$ is a positive number, $b$ is fixed and $|b| = 1$. Moreover, in $\\ell_+$, $\\zeta' = -tb$, for $t \\in (0, \\infty)$, and a similar parametrization holds in $\\ell_-$.\n\\begin{lemma}\nFor fixed $b$, $\\mathcal{X}_m$ as in \\eqref{defxm} has a limit as $|a| \\to 0$.\n\\end{lemma}\n\\begin{proof}\nWriting $\\dfrac{\\zeta' + \\zeta}{\\zeta'(\\zeta' - \\zeta)} = \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta}$, we want to find the limit as $a \\to 0$ of\n\\begin{align}\n& \\int_{\\ell_+} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& - \\int_{\\ell_-} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(-\\pi Ra\/\\zeta' - i\\theta_e - \\pi R\\zeta' \\bar{a})] d\\zeta' \\label{integs}.\n\\end{align}\nFor simplicity, we'll focus in the first integral only, the second one can be handled similarly. Rewrite:\n\\begin{align}\n& \\int_{\\ell_+} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& = \\int_{0}^{-b} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{-b}^{-b\\infty} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& = \\int_{0}^{-b} \\left\\{ \\dfrac{-1}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{-b}^{-b\\infty} \\left\\{ \\dfrac{-1}{\\zeta'} + \\frac{2}{\\zeta'} + \\dfrac{2}{\\zeta' - \\zeta} - \\frac{2}{\\zeta'} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& = \\int_{0}^{-b} \\dfrac{-1}{\\zeta'} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{-b}^{-b\\infty} \\dfrac{1}{\\zeta'} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{0}^{-b} \\dfrac{2}{\\zeta' - \\zeta} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{-b}^{-b\\infty} \\left\\{ \\dfrac{2}{\\zeta' - \\zeta} - \\frac{2}{\\zeta'} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\label{4sums}\n\\end{align}\n\n\\noindent Observe that\n\\begin{align*}\n& \\int_{0}^{-b} \\dfrac{-1}{\\zeta'} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta'\\\\\n& = -\\int_0^1 \\frac{1}{t} \\log[1 - \\exp(-\\pi Rs(t + 1\/t))] dt\\\\\n\\intertext{and after a change of variables $\\tilde{t} = 1\/t$, we get}\n& = -\\int_1^\\infty \\frac{1}{\\tilde{t}} \\log[1 - \\exp(-\\pi Rs(\\tilde{t} + 1\/\\tilde{t}))] d\\tilde{t}\\\\\n& = -\\int_{-b}^{-b\\infty} \\dfrac{1}{\\zeta'} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta'.\n\\end{align*}\nThus, (\\ref{4sums}) reduces to\n\\begin{align}\n& \\int_{0}^{-b} \\dfrac{2}{\\zeta' - \\zeta} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\notag\\\\\n& + \\int_{-b}^{-b\\infty} \\left\\{ \\dfrac{2}{\\zeta' - \\zeta} - \\frac{2}{\\zeta'} \\right\\} \\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})] d\\zeta' \\label{2sums}.\n\\end{align}\nIf $\\theta_e = 0$, (\\ref{integs}) diverges to $-\\infty$, in which case $\\mathcal{X}_m = 0$. Otherwise, $\\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})]$ is bounded away from 0. Consequently,\n$|\\log[1 - \\exp(\\pi Ra\/\\zeta' + i\\theta_e + \\pi R\\zeta' \\bar{a})]| < C < \\infty$ in $\\ell_+$.\nAs $a \\to 0$, the integrals are dominated by\n\\[ \\int_0^{-b} \\dfrac{2C}{|\\zeta' - \\zeta|} |d\\zeta'| + \\int_{-b}^{-b\\infty} \\frac{C|\\zeta\/b|}{|\\zeta'(\\zeta' - \\zeta)|} |d\\zeta'| < \\infty \\]\nif $\\theta_e \\neq 0$. Hence we can interchange the limit and the integral in (\\ref{2sums}) and obtain that, as $a \\to 0$, this reduces to\n\n\\begin{align}\n & 2\\log(1 - e^{i \\theta_e})\\left[\\int_{0}^{-b} \\frac{d\\zeta'}{\\zeta' - \\zeta} + \\int_{-b}^{-b\\infty} d\\zeta' \\left\\{ \\frac{1}{\\zeta' - \\zeta} - \\frac{1}{\\zeta'}\\right\\} \\right] \\notag\\\\\n & = 2\\log(1 - e^{i \\theta_e})[F(-b) + G(-b)], \\label{odes}\n \\end{align}\nwhere\n\\[ F(z) := \\log\\left( 1 - \\dfrac{z}{\\zeta}\\right), G(z) := \\log\\left( 1 - \\dfrac{\\zeta}{z}\\right) \\]\n are the (unique) holomorphic solutions in the simply connected domain $U := \\mathbb{C} - \\{z : z\/\\zeta \\in \\mathbb{R}_+\\}$ to the ODEs\n\\[ F'(z) = \\frac{1}{z - \\zeta}, F(0) = 0 \\hspace{10 mm} G'(z) = \\frac{1}{z - \\zeta} - \\frac{1}{z}, \\lim_{z \\to \\infty} G(z) = 0. \\]\nThis forces us to rewrite (\\ref{odes}) uniquely as\n\\begin{equation}\\label{prinbran}\n2\\log(1 - e^{i \\theta_e})\\left[\\log\\left(1 + \\frac{b}{\\zeta}\\right) - \\log\\left(1 + \\frac{\\zeta}{b}\\right)\\right]\n\\end{equation}\nHere $\\log$ denotes the principal branch of the log in both cases, and the equation makes sense for $\\{b \\in \\mathbb{C} : b \\notin \\ell_+ \\}$ (recall that by construction, we have the additional datum $|b| = 1$). We want to conclude that\n\\begin{equation}\\label{logfusion}\n\\log(1 + b\/\\zeta) - \\log(1 + \\zeta\/b) = \\log(b\/\\zeta),\n\\end{equation}\nstill using the principal branch of the log. To see this, define $H(z)$ as $F(z) - G(z) - \\log(-z\/\\zeta)$. This is an analytic function on $U$ and clearly $H'(z) \\equiv 0$. Thus $H$ is constant in $U$. It is easy to show that the identity holds for a suitable choice of $z$ (for example, if $\\zeta$ is not real, choose $z = 1$) and by the above, it holds on all of $U$; in particular, for $z = -b$.\n\nAll the arguments so far can be repeated to the ray $\\ell_-$ to get the final form of (\\ref{integs}):\n \\begin{equation}\\label{fextov}\n 2\\left\\{\\log\\left[\\frac{b}{\\zeta}\\right]\\log(1 - e^{i\\theta_e})\n -\\log\\left[\\frac{- b}{\\zeta}\\right]\\log(1 - e^{-i\\theta_e}) \\right\\}, \\hspace{3 mm} \\theta_e \\neq 0.\n \\end{equation}\n This yields that (\\ref{defxm}) simplifies to:\n\\begin{align}\n\\mathcal{X}_m & = \\mathcal{X}^{\\text{sf}}_m \\exp\\left( \\frac{i}{2\\pi} \\left\\{ \\log\\left[\\frac{b}{\\zeta}\\right]\\log(1 - e^{i\\theta_e})\n -\\log\\left[\\frac{- b}{\\zeta}\\right]\\log(1 - e^{-i\\theta_e})\\right\\} \\right) \\notag\\\\\n & = \\mathcal{X}^{\\text{sf}}_m \\exp\\left( \\frac{i}{2\\pi} \\left\\{ \\log\\left[\\frac{a}{|a|\\zeta}\\right]\\log(1 - e^{i\\theta_e})\n -\\log\\left[\\frac{- a}{|a|\\zeta}\\right]\\log(1 - e^{-i\\theta_e})\\right\\} \\right) \\label{xmnice}\n\\end{align}\nin the limiting case $a \\to 0$.\n\\end{proof}\n\nTo obtain a function that is continuous everywhere and independent of $\\arg a$, define regions I, II and III in the $a$-plane as follows: $\\mathcal{X}^{\\text{sf}}_m$ has a fixed cut in the negative real axis, both in the $\\zeta$-plane and the $a$-plane. Assuming for the moment that $\\arg \\zeta \\in (0,\\pi)$, define region I as the half plane $\\{a \\in \\mathbb{C} : \\text{Im}\\left( a\/\\zeta\\right) < 0 \\}$. Region II is that enclosed by the $\\ell_-$ ray and the cut in the negative real axis, and region III is the remaining domain so that as we travel counterclockwise we traverse regions I, II and III in this order (see Figure \\ref{3reg}).\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{regions.ps}\n\t\\caption{The three regions in the $a$-plane, as we traverse them counterclockwise}\n\t\\label{3reg}\n\\end{figure}\n\n\nFor $a \\neq 0$, Gaiotto, Moore and Neitzke \\cite{gaiotto} proved that $\\mathcal{X}_m$ has a continuous extension to the punctured disk of the form:\n\n\\begin{equation}\\label{regmod}\n \\widetilde{\\mathcal{X}}_m = \\left\\{ \\begin{array}{ll}\n\t \\mathcal{X}_m & \\text{in region I}\\\\\n\t (1 - \\mathcal{X}^{-1}_e) \\mathcal{X}_m & \\text{in region II}\\\\\n\t - \\mathcal{X}_e (1 - \\mathcal{X}^{-1}_e) \\mathcal{X}_m = (1 - \\mathcal{X}_e)\\mathcal{X}_m & \\text{in region III}\n\t \\end{array} \\right. \n\\end{equation}\n\nIf we regard $\\mathcal{M}'$ as a $S^1$-bundle over $\\mathcal{B}' \\times S^1$, with the fiber parametrized by $\\theta_m$, then we seek to extend $\\mathcal{M}'$ to a manifold $\\mathcal{M}$ by gluing to $\\mathcal{M}'$ another $S^1$-bundle over $D \\times (0,2\\pi)$, for $D$ a small open disk around $a = 0$, and $\\theta_e \\in (0,2\\pi)$. The $S^1$-fiber is parametrized by a different coordinate $\\theta'_m$ where the Darboux coordinate $\\widetilde{\\mathcal{X}}_m$ can be extended to $\\mathcal{M}$. This is the content of the next theorem. \n\n\\begin{theorem}\\label{mprtom}\n$\\mathcal{M}'$ can be extended to a manifold $\\mathcal{M}$ where the torus fibers over $\\mathcal{B}'$ degenerate at $D = \\{a = 0\\}$ and $\\widetilde{\\mathcal{X}}_m $ can be extended to $D$, independent of the value of $\\arg a$.\n\\end{theorem}\n\\begin{proof}\nWe'll use the following identities:\n\\begin{align}\n\\log(1 - e^{i\\theta_e}) & = \\log(1 - e^{-i\\theta_e}) +i(\\theta_e - \\pi), \\hspace{5mm} \\text{for } \\theta_e \\in (0, 2\\pi) \\label{logs}\\\\\n \\log\\left[\\frac{-a}{|a|\\zeta}\\right] & = \\left\\{ \\begin{array}{ll}\n \\log\\left[\\frac{a}{|a|\\zeta}\\right] + i\\pi & \\text{in region I}\\\\\n\t \\log\\left[\\frac{a}{|a|\\zeta}\\right] - i\\pi & \\text{in regions II and III}\n\t \\end{array} \\right. \\label{regions}\\\\\n\t \\log [a\/\\zeta] & = \\left\\{ \\begin{array}{ll}\n \\log a - \\log \\zeta & \\text{in regions I and II}\\\\\n\t \\log a - \\log \\zeta + 2\\pi i & \\text{in region III}\n\t \\end{array} \\right. \\label{breaklog}\n\\end{align}\nto obtain a formula for $\\widetilde{\\mathcal{X}}_m$ at $a = 0$ independent of the region. Formula \\eqref{breaklog} can be proved with an argument analogous to that used for the proof of \\eqref{logfusion}. Starting with region I, by \\eqref{xmnice}, (\\ref{regmod}), (\\ref{logs}) and (\\ref{regions}):\n\\begin{align*}\n\\widetilde{\\mathcal{X}}_m & = \\exp\\left[ i\\theta_m - \\frac{1}{2\\pi} (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|\\zeta}\\right] + \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right) \\right] \\hspace{5 mm} \\text{in region I.}\\\\\n\\intertext{By \\eqref{breaklog},}\n& = \\exp\\left[ i\\theta_m - \\frac{1}{2\\pi} (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|}\\right] + \\frac{\\theta_e - \\pi}{2\\pi}\\log \\zeta + \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right) \\right]\n\\end{align*}\nIn region II, by our formulas above, we get\n\\begin{align*}\n\\widetilde{\\mathcal{X}}_m & = \\exp\\left[i\\theta_m - \\frac{1}{2\\pi} (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|\\zeta}\\right] - \\frac{1}{2} \\log \\left(1 - e^{-i\\theta_e}\\right) \\right]\\left(1 - e^{-i\\theta_e}\\right) \\\\\n& = \\exp\\left[i\\theta_m - \\frac{1}{2\\pi} (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|\\zeta}\\right] - \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right) + \\log\\left(1 - e^{-i\\theta_e}\\right) \\right]\\\\\n & = \\exp\\left[ i\\theta_m - \\frac{1}{2\\pi} (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|}\\right] + \\frac{\\theta_e - \\pi}{2\\pi}\\log \\zeta + \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right) \\right] \\text{in region II.}\n\\end{align*}\nFinally, in region III, and making use of \\eqref{logs}, \\eqref{regions}, \\eqref{breaklog}:\n\\begin{align}\n\\widetilde{\\mathcal{X}}_m & = \\exp\\left[i\\theta_m - \\frac{1}{2\\pi} (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|\\zeta}\\right] - \\frac{1}{2} \\log \\left(1 - e^{-i\\theta_e}\\right) \\right]\\left(1 - e^{i\\theta_e}\\right) \\notag\\\\\n& = \\exp\\left[ i\\theta_m - \\frac{1}{2\\pi} (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|}\\right] + \\frac{\\theta_e - \\pi}{2\\pi}\\log \\zeta - i(\\theta_e - \\pi) \\notag \\right. \\\\\n& \\left. \\hspace{14 mm} - \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right) + \\log\\left(1 - e^{-i\\theta_e}\\right) + i(\\theta_e - \\pi) \\right] \\notag \\\\\n& = \\exp\\left[ i\\theta_m - \\frac{1}{2\\pi} (\\theta_e - \\pi) \\log\\left[\\frac{a}{|a|}\\right] + \\frac{\\theta_e - \\pi}{2\\pi}\\log\n \\zeta + \\frac{1}{2} \\log\\left(1 - e^{-i\\theta_e}\\right) \\right] \\label{xmany}.\n\\end{align}\n\n Observe that, throughout all these calculations, we only had to use the natural branch of the complex logarithm. In summary, (\\ref{xmany}) works for any region in the $a$-plane, with a cut in the negative real axis.\n\nThis also suggest the following coordinate transformation\n\\begin{equation}\\label{thetapr}\n \\theta'_m = \\theta_m + \\frac{i(\\theta_e - \\pi)}{4\\pi} \\left( \\log\\frac{a}{\\Lambda} - \\log\\frac{\\bar{a}}{\\overline{\\Lambda}} \\right) \n\\end{equation}\nHere $\\Lambda$ is the same cutoff constant as in \\cite{gaiotto}. Let $\\varphi$ parametrize the phase of $a\/|a|$. Then \\eqref{thetapr} simplifies to\n\\begin{equation}\\label{nicethm}\n\\theta'_m = \\theta_m - \\frac{(\\theta_e - \\pi)\\varphi}{2\\pi}\n\\end{equation}\n\nOn a coordinate patch around the singular fiber, $\\theta'_m$ is single-valued.\nThus, the above shows that we can glue to $\\mathcal{M}'$ another $S^1$-bundle over $D \\times (0,2\\pi)$, for $D$ a small open disk around $a = 0$, and $\\theta_e \\in (0,2\\pi)$. The $S^1$-fiber is parametrized by $\\theta'_m$ and the transition function is given by \\eqref{nicethm}, yielding a manifold $\\mathcal{M}$. In this patch, we can extend $\\widetilde{\\mathcal{X}}_m$ to $a = 0$ as:\n\n\\begin{equation}\\label{reblow}\n\\left. \\widetilde{\\mathcal{X}}_m\\right|_{a = 0} = e^{i\\theta'_m} \\zeta^{\\frac{\\theta_e - \\pi}{2\\pi}} (1 - e^{-i\\theta_e})^{\\frac{1}{2}}\n\\end{equation}\nwhere the branch of $\\zeta^{\\frac{\\theta_e - \\pi}{2\\pi}}$ is determined by the natural branch of the logarithm in the $\\zeta$ plane. Note that when $\\theta_e = 0$, $\\widetilde{\\mathcal{X}}_m \\equiv 0$ in \\eqref{reblow} and by definition, $\\mathcal{X}_e \\equiv 1$. Since these two functions are Darboux coordinates for $\\mathcal{M}$, the $S^1$ fibration over $D \\times (0, 2\\pi)$ we glued to $\\mathcal{M}'$ to get $\\mathcal{M}$ degenerates into a point when $\\theta_e = 0$. \n\nNow consider the case that $\\arg \\zeta \\in (-\\pi, 0)$. Label the regions as one travels counterclockwise, starting with the region bounded by the cut and the $\\ell_-$ (See Figure \\ref{negarg}). We can do an analytic continuation similar to \\eqref{regmod} starting in region I, but formulas \\eqref{regions}, \\eqref{breaklog} become now:\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{regions_negative_argument.ps}\n\t\\caption{The three regions in the case $\\arg \\zeta < 0$.}\n\t\\label{negarg}\n\\end{figure}\n\n\\begin{align*}\n \\log\\left[\\frac{-a}{|a|\\zeta}\\right] & = \\left\\{ \\begin{array}{ll}\n \\log\\left[\\frac{a}{|a|\\zeta}\\right] - i\\pi & \\text{in region II}\\\\\n\t \\log\\left[\\frac{a}{|a|\\zeta}\\right] + i\\pi & \\text{in regions I and III}\n\t \\end{array} \\right. \\\\\n\t \\log [a\/\\zeta] & = \\left\\{ \\begin{array}{ll}\n \\log a - \\log \\zeta & \\text{in regions I and II}\\\\\n\t \\log a - \\log \\zeta - 2\\pi i & \\text{in region III}\n\t \\end{array} \\right.\n\\end{align*}\n\nBy an argument entirely analogous to the case $\\arg \\zeta > 0$, we get again:\n\\begin{equation*}\n\\left. \\widetilde{\\mathcal{X}}_m\\right|_{a = 0} = e^{i\\theta'_m} \\zeta^{\\frac{\\theta_e - \\pi}{2\\pi}} (1 - e^{-i\\theta_e})^{\\frac{1}{2}}\n\\end{equation*}\n\nThe case $\\zeta$ real and positive is even simpler, as Figure \\ref{zeroarg} shows. Here we have only two regions, and the jumps at the cut and the $\\ell_+$ ray are combined, since these two lines are the same. Label the lower half-plane as region I and the upper half-plane as region II. Start an analytic continuation of $\\mathcal{X}_m$ in region I as before, using the formulas:\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{regions_zero_argument.ps}\n\t\\caption{Only two regions in the case $\\arg \\zeta = 0$.}\n\t\\label{zeroarg}\n\\end{figure}\n\n\\begin{align*}\n \\log\\left[\\frac{-a}{|a|\\zeta}\\right] & = \\left\\{ \\begin{array}{ll}\n \\log\\left[\\frac{a}{|a|\\zeta}\\right] - i\\pi & \\text{in region II}\\\\\n\t \\log\\left[\\frac{a}{|a|\\zeta}\\right] + i\\pi & \\text{in region I}\n\t \\end{array} \\right. \\\\\n\t \\log [a\/\\zeta] & = \\log a - \\log \\zeta \\hspace{4 mm} \\text{in both regions}\n\\end{align*}\n\nThe result is equation \\eqref{reblow} again. The case $\\arg \\zeta = \\pi$ is entirely analogous to this and it yields the same formula, thus proving that \\eqref{reblow} holds for all $\\zeta$ and is independent of $\\arg a$.\n\\end{proof}\n\n\\subsection{Alternative Riemann-Hilbert problem}\\label{altrh}\n\nWe may obtain the function $\\mathcal{X}_m$ (and consequently, the analytic extension $\\widetilde{\\mathcal{X}}_m$) at $a = 0$ through a slightly different formulation of the Riemann-Hilbert problem stated in \\eqref{defxm}. Namely, instead of defining a jump of $\\mathcal{X}_m$ at two opposite rays $\\ell_+, \\ell_-$, we combine these into a single jump at the line $\\ell$ defined by $\\ell_+$ and $\\ell_-$, as in Figure \\ref{onejump}. Note that because of the orientation of $\\ell$ one of the previous jumps has to be reversed.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{singlejump.ps}\n\t\\caption{The reversed orientation on $\\ell_+$ inverts the jump.}\n\t\\label{onejump}\n\\end{figure}\n\nFor all values $a \\neq 0$, $\\mathcal{X}_e = \\mathcal{X}_e^{\\text{sf}}$ approaches 0 as $\\zeta \\to 0$ or $\\zeta \\to \\infty$ along the $\\ell$ ray due to the exponential decay in formula \\eqref{xesf}. Thus, the jump function\n\\[ G(\\zeta) := \\left\\{ \\begin{array}{ll} 1-\\mathcal{X}^{-1}_e & \\text{ for $\\zeta = t a, 0 \\leq t \\leq \\infty$} \\\\ \n 1- \\mathcal{X}_e & \\text{ for $\\zeta = t a, -\\infty \\leq t \\leq 0$} \\end{array} \\right. \\] \nis continuous on $\\ell$ regarded as a closed contour on $\\mathbb{P}^1$, and it approaches the identity transformation exponentially fast at the points $0$ and $\\infty$.\n\n\n\n\nThe advantage of this reformulation of the Riemann-Hilbert problem is that it can be extended to the case $a = 0$ and we can obtain estimates on the solutions $\\mathcal{X}_m$ even without an explicit formulation. If we fix $\\arg a$ and let $|a| \\to 0$ as before, the jump function $G(\\zeta)$ approaches the constant jumps\n\\begin{equation}\\label{jumpd}\n\\left. G(\\zeta) \\right|_{|a|=0} := \\left\\{ \\begin{array}{ll} 1-e^{-i\\theta_e} & \\text{ for $\\zeta = t a, 0 < t < \\infty$} \\\\ \n 1- e^{i\\theta_e} & \\text{ for $\\zeta = t a, -\\infty < t < 0$} \\end{array} \\right.\n\\end{equation}\nThus, $\\left. G(\\zeta) \\right|_{|a|=0}$ has two discontinuities at $0$ and $\\infty$. If we denote by \n\\[ \\Delta_0 = \\lim_{t \\to 0^+} G(\\zeta) - \\lim_{t \\to 0^-} G(\\zeta), \\qquad \\Delta_\\infty = \\lim_{t \\to \\infty^+} G(\\zeta) - \\lim_{t \\to \\infty^-} G(\\zeta), \\]\nthen, by \\eqref{jumpd},\n\\begin{equation*}\n\\Delta_0 = -\\Delta_\\infty\n\\end{equation*}\n\n\n\n\nLet $D^+$ be the region in $\\mathbb{P}^1$ bounded by $\\ell$ with the positive, counterclockwise orientation. Denote by $D^-$ the region where $\\ell$ as a boundary has the negative orientation. We look for solutions of the homogeneous boundary problem\n\\begin{equation}\\label{bcond}\nX_m^+(\\zeta) = G(\\zeta) X_m^-(\\zeta)\n\\end{equation}\nwith $G(\\zeta)$ as in \\eqref{jumpd}. This is Lemma 4.1 in \\cite{rhprob}.\n\n \n \n The solutions $X_m^\\pm$ obtained therein are related to $\\mathcal{X}_m$ via $\\mathcal{X}_m (\\zeta) = \\mathcal{X}^{\\text{sf}}_m (\\zeta) X_m (\\zeta)$. Uniqueness of solutions of the homogeneous Riemann-Hilbert problem shows that these are the same functions (up to a constant factor) constructed in the previous section. Observe that the term $\\zeta^{\\frac{\\theta_e - \\pi}{2\\pi}}$ appears naturally due to the nature of the discontinuity of the jump function at 0 and $\\infty$. The analytic continuation around the point $a = 0$ and the gauge transformation $\\theta_m \\mapsto \\theta'_m$ are still performed as before.\n \n\n\n\n\\subsection{Generalized Ooguri-Vafa coordinates}\\label{genvafa}\n\n\nWe can generalize the previous extension to the case $Z_{\\gamma_m} := \\frac{1}{2\\pi i}a \\log a + f(a)$, where $f : \\mathcal{B}' \\to \\mathbb{C}$ is holomorphic and admits a holomorphic extension into $\\mathcal{B}$. In particular,\n\\begin{equation}\\label{newxmsf}\n\\mathcal{X}_m^\\text{sf} = \\exp \\left( \\frac{-iR}{2\\zeta}a \\log a + \\frac{\\pi R f(a)}{\\zeta} + i \\theta_m + \\frac{i \\zeta R}{2} \\overline{a} \\log \\overline{a} + \\pi R \\zeta \\overline{f(a)}\\right)\n\\end{equation}\nThe value at the singular locus $f(0)$ does not have to be 0. All the other data remains the same. \n\nThe first thing we observe is that $\\mathcal{X}_e$ remains the same. Consequently, the corrections for the generalized $\\mathcal{X}_m$ are as before. Using the change of coordinates as in \\eqref{nicethm}, we can thus write\n\\begin{equation}\\label{genov}\n\\left. \\widetilde{\\mathcal{X}}_m\\right|_{a = 0} = \\exp\\left[ \\frac{\\pi R f(0)}{\\zeta} + i\\theta'_m + \\pi R \\zeta f(0) \\right] \\zeta^{\\frac{\\theta_e - \\pi}{2\\pi}} (1 - e^{-i\\theta_e})^{\\frac{1}{2}}\n\\end{equation}\n\n\\section{Extension of the Ooguri-Vafa metric}\\label{gmetric}\n\n\\subsection{Classical Case}\\label{clasmet}\n\n\\subsubsection{A $C^1$ extension of the coordinates}\\label{c1ext}\n\nIn section \\ref{clasov} we extended the fibered manifold $\\mathcal{M}'$ to a manifold $\\mathcal{M}$ with a degenerate fiber at $a = 0$ in $\\mathcal{B}$. We also extended $\\widetilde{\\mathcal{X}}_m$ continuously to this bad fiber. Now we extend the metric by enlarging the holomorphic symplectic form $\\varpi(\\zeta)$. Recall that this is of the form\n\\[ \\varpi(\\zeta) = -\\frac{1}{4\\pi^2 R} \\frac{d \\mathcal{X}_e}{\\mathcal{X}_e} \\wedge \\frac{d\\widetilde{\\mathcal{X}}_m}{\\widetilde{\\mathcal{X}}_m} \\]\nClearly there are no problems extending $d \\log \\mathcal{X}_e$, so it remains only to extend $d \\log \\widetilde{\\mathcal{X}}_m$.\n\n\\begin{lemma}\nLet $\\widetilde{\\mathcal{X}}_m$ denote the analytic continuation around $a = 0$ of the magnetic function, as in the last section. The 1-form\n\\begin{equation}\\label{c1dlog}\nd \\log \\widetilde{\\mathcal{X}}_m = \\frac{d \\widetilde{\\mathcal{X}}_m}{\\widetilde{\\mathcal{X}}_m},\n\\end{equation}\n(where $d$ denotes the differential of a function on the torus fibration $\\mathcal{M}'$ only) has an extension to $\\mathcal{M}$\n\\end{lemma}\n\\begin{proof}\nWe proceed as in section \\ref{clasov} and work in different regions in the $a$-plane (see Figure \\ref{3reg}), starting with region I, where $\\widetilde{\\mathcal{X}}_m = \\mathcal{X}_m$. Then observe that we can write the corrections on $\\mathcal{X}_m$ as a complex number $\\Upsilon_m(\\zeta) \\in (\\mathcal{M}'_a)^{\\mathbb{C}}$ such that\n\\[ \\mathcal{X}_m = \\exp\\left( \\frac{-i R }{2\\zeta}(a\\log a - a) + i \\Upsilon_m + \\frac{i\\zeta R}{2} (\\overline{a} \\log\n\\overline{a} - \\overline{a} )\\right). \\]\nThus, by \\eqref{c1dlog} and ignoring the $i$ factor, it suffices to obtain an extension of \n\\begin{align}\n& d\\left[ \\frac{- R }{2\\zeta}(a\\log a - a) + \\Upsilon_m + \\frac{\\zeta R}{2} (\\overline{a} \\log\n\\overline{a} - \\overline{a} ) \\right] \\notag \\\\\n& = \\frac{ -R}{2\\zeta} \\log a \\, da + d \\Upsilon_m + \\frac{ \\zeta R}{2} \\log \\overline{a} \\, d\\overline{a}. \\label{3ext}\n\\end{align}\nUsing \\eqref{defxm},\n\\begin{align*}\nd \\Upsilon_m = d\\theta_m & - \\frac{1}{4\\pi}\\int_{\\ell_+} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e}\\left( \\frac{\\pi R}{\\zeta'} da +id\\theta_e+ \\pi R \\zeta' d\\overline{a}\\right) \\notag\\\\\n& +\\frac{1}{4\\pi}\\int_{\\ell_-} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}\\left( -\\frac{\\pi R}{\\zeta'} da -id\\theta_e - \\pi R \\zeta' d\\overline{a}\\right).\n\\end{align*}\nWe have to change our $\\theta_m$ coordinate into $\\theta'_m$ according to \\eqref{nicethm} and differentiate to obtain:\n\\begin{align}\nd \\Upsilon_m & = \nd\\theta'_m - \\frac{i(\\theta_e - \\pi)}{4\\pi} \\left( \\frac{da}{a} - \\frac{d\\overline{a}}{\\overline{a}}\\right)+\\frac{\\arg a}{2\\pi}d\\theta_e \\notag \\\\\n& - \\frac{1}{4\\pi}\\int_{\\ell_+} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e}\\left( \\frac{\\pi R}{\\zeta'} da +id\\theta_e+ \\pi R \\zeta' d\\overline{a}\\right) \\notag\\\\\n& +\\frac{1}{4\\pi}\\int_{\\ell_-} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}\\left( -\\frac{\\pi R}{\\zeta'} da -id\\theta_e - \\pi R \\zeta' d\\overline{a}\\right) \\label{dups1}\n\\end{align}\nRecall that, since we have introduced the change of coordinates $\\theta_m \\mapsto \\theta'_m$, we are working on a patch on $\\mathcal{M}$ that contains $a = 0$ with a degenerate fiber here. It then makes sense to ask if \\eqref{3ext} extends to $a =0$. If this is true, then every independent 1-form extends individually. Let's consider the form involving $d\\theta_e$ first. By \\eqref{dups1}, this part consists of:\n\\begin{equation}\\label{dthpart}\n\\frac{\\arg a}{2\\pi}d\\theta_e - \\frac{i}{4\\pi}\\int_{\\ell_+} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e} d\\theta_e - \\frac{i}{4\\pi}\\int_{\\ell_-} \\dfrac{d\\zeta'}{\\zeta'} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e} d\\theta_e.\n\\end{equation}\nWe can use the exact same technique in section \\ref{clasov} to find the limit of \\eqref{dthpart} as $a \\to 0$. Namely, split each integral into four parts, use the symmetry of $\\dfrac{\\mathcal{X}_e}{1- \\mathcal{X}_e}$ between $0$ and $\\infty$ to cancel two of these integrals and take the limit in the remaining ones. The result is:\n\\begin{align}\n& \\frac{\\arg a}{2\\pi} - \\frac{ie^{i\\theta_e}}{2\\pi(1-e^{i\\theta_e})}\\log\\left[ \\frac{e^{i \\arg a}}{\\zeta} \\right] - \n\\frac{ie^{-i\\theta_e}}{2\\pi(1-e^{-i\\theta_e})}\\log\\left[ \\frac{-e^{i \\arg a}}{\\zeta} \\right] \\notag \\\\\n& = \\frac{\\arg a}{2\\pi} - \\frac{ie^{i\\theta_e}}{2\\pi(1-e^{i\\theta_e})}\\log\\left[ \\frac{e^{i \\arg a}}{\\zeta} \\right] + \n\\frac{i}{2\\pi(1-e^{i\\theta_e})}\\log\\left[ \\frac{-e^{i \\arg a}}{\\zeta} \\right] \\label{dthe}\n\\end{align}\nin region I (we omitted the $d\\theta_e$ factor for simplicity). Making use of formulas \\eqref{regions} and \\eqref{breaklog}, we can simplify the above expression and get rid of the apparent dependence on $\\arg a$ until finally getting:\n\\[ -\\frac{i\\log \\zeta}{2\\pi} - \\frac{1}{2(1-e^{i\\theta_e})}, \\hspace{7 mm} \\theta_e \\neq 0. \\]\nIn other regions of the $a$-plane we have to modify $\\widetilde{\\mathcal{X}}_m$ as in \\eqref{regmod}. Nonetheless, by \\eqref{regions} and \\eqref{breaklog}, the result is the same and we conclude that at least the terms involving $d\\theta_e$ have an extension to $a=0$ for $\\theta_e \\neq 0$.\n\nNext we extend the terms involving $da$. By \\eqref{3ext} and \\eqref{dups1}, these are:\n\\begin{equation*}\n\\frac{ -R}{2\\zeta} \\log a \\, da - \\frac{i(\\theta_e - \\pi)}{4\\pi a} da - \\frac{R}{4}\\int_{\\ell_+} \\dfrac{d\\zeta'}{(\\zeta')^2} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e} da - \\frac{R}{4}\\int_{\\ell_-} \\dfrac{d\\zeta'}{(\\zeta')^2} \\frac{\\zeta'+\\zeta}{\\zeta'-\\zeta} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e} da \n\\end{equation*}\nIn what follows, we ignore the $da$ part and focus on the coefficients for the extension. The partial fraction decomposition\n\\begin{equation}\\label{parfrac}\n\\frac{\\zeta'+\\zeta}{(\\zeta')^2(\\zeta'-\\zeta)} = \\frac{2}{\\zeta'(\\zeta'-\\zeta)} - \\frac{1}{(\\zeta')^2}\n\\end{equation}\nsplits each integral above into two parts. We will consider first the terms\n\\begin{equation}\\label{dap1}\n- \\frac{i(\\theta_e - \\pi)}{4\\pi a} + \\frac{R}{4}\\int_{\\ell_+} \\dfrac{d\\zeta'}{(\\zeta')^2}\\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e} + \\frac{R}{4}\\int_{\\ell_-} \\dfrac{d\\zeta'}{(\\zeta')^2} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}.\n\\end{equation}\nUse the fact that $\\mathcal{X}_e$ (resp. $\\mathcal{X}^{-1}_e$) has norm less than 1 on $\\ell_+$ (resp. $\\ell_-$) and the uniform convergence of the geometric series on $\\zeta'$ to write \\eqref{dap1} as:\n\\begin{align*}\n- \\frac{i(\\theta_e - \\pi)}{4\\pi a} + \\frac{R}{4}\\sum_{n=1}^\\infty \\left\\{ \\vphantom{\\int_{\\ell_+}} \\right. & \\int_{\\ell_+}\\frac{d\\zeta'}{(\\zeta')^2}\\exp\\left(\n\\frac{\\pi R n a}{\\zeta'} +i n \\theta_e +\\pi R n \\zeta' \\overline{a}\\right) + \\\\\n& \\left.\n\\int_{\\ell_-}\\frac{d\\zeta'}{(\\zeta')^2}\\exp\\left(\n\\frac{-\\pi R n a}{\\zeta'} -i n \\theta_e -\\pi R n \\zeta' \\overline{a}\\right)\\right\\},\n\\end{align*}\n\\begin{align*}\n& = - \\frac{i(\\theta_e - \\pi)}{4\\pi a} + \\left(\\frac{R}{4}\\right) \\left( \\frac{-2|a|}{a}\\right)\\sum_{n=1}^\\infty \\left( e^{in\\theta_e} - e^{-in\\theta_e}\\right)K_1(2\\pi R n |a|)\\\\\n& = - \\frac{i(\\theta_e - \\pi)}{4\\pi a} - \\frac{R|a|}{2a}\\sum_{n=1}^\\infty \\left( e^{in\\theta_e} - e^{-in\\theta_e}\\right)K_1(2\\pi R n |a|).\n\\end{align*}\nSince $K_1(x) \\thicksim 1\/x$, for $x$ real and $x \\to 0$, we obtain, letting $a \\to 0$:\n\\begin{align*}\n& - \\frac{i(\\theta_e - \\pi)}{4\\pi a} - \\frac{R|a|}{2a\\cdot 2\\pi R |a|} \\sum_{n=1}^\\infty \\frac{\\left( e^{in\\theta_e} - e^{-in\\theta_e}\\right)}{n}\\\\\n& = - \\frac{i(\\theta_e - \\pi)}{4\\pi a} + \\frac{1}{4\\pi a}[\\log(1-e^{i\\theta_e})-\\log(1-e^{-i\\theta_e})]\\\\\n\\intertext{and by \\eqref{logs},}\n& = - \\frac{i(\\theta_e - \\pi)}{4\\pi a} +\\frac{i(\\theta_e -\\pi)}{4\\pi a} = 0.\n\\end{align*}\nTherefore this part of the $da$ terms extends trivially to 0 in the singular fiber.\n\nIt remains to extend the other terms involving $da$. Recall that by \\eqref{parfrac}, these terms are (after getting rid of a factor of $-R\/2$):\n\\begin{equation}\\label{last3}\n\\frac{ \\log a}{\\zeta} + \\int_{\\ell_+} \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e} + \\int_{\\ell_-} \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}.\n\\end{equation}\n\nWe'll focus in the first integral in \\eqref{last3}. As a starting point, we'll prove that as $a \\to 0$, the limiting value of this integral is the same as the limit of\n\\begin{equation}\\label{simpler}\n \\int_{\\ell_+} \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} \\frac{\\exp\\left( \\frac{\\pi R a}{\\zeta'} +i\\theta_e \\right)}{1-\\exp\\left( \\frac{\\pi R a}{\\zeta'} +i\\theta_e +\\pi R \\zeta' \\overline{a}\\right)}.\n\\end{equation}\nIt suffices to show that\n\\begin{equation}\\label{lebes}\n \\int_{\\ell_+} \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} \\frac{\\exp\\left( \\frac{\\pi R a}{\\zeta'} \\right)}{1-\\exp\\left( \\frac{\\pi R a}{\\zeta'} +i\\theta_e +\\pi R \\zeta' \\overline{a}\\right)} [1-\\exp(\\pi R \\zeta' \\overline{a})] \\to 0, \\hspace{5 mm} \\text{as $a \\to 0$, $\\theta_e \\neq 0$}\n\\end{equation}\nTo see this, we can assume $|a| < 1$. Let $b = a\/|a|$. Observe that in the $\\ell_+$ ray, $|\\exp(\\pi Ra\/\\zeta')| < 1$, and since $\\theta_e \\neq 0$, we can bound \\eqref{lebes} by\n\\[ \\text{const} \\int_{\\ell_+} \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} [1-\\exp(\\pi R \\zeta' \\overline{b})] < \\infty. \\]\nEquation \\eqref{lebes} now follows from Lebesgue Dominated Convergence and the fact that $1-\\exp(\\pi R \\zeta' \\overline{a}) \\to 0$ as $a \\to 0$. A similar application of Dominated Convergence allows us to reduce the problem to the extension of\n\\begin{equation}\\label{secsimp}\n\\int_{\\ell_+} \\frac{d\\zeta'}{\\zeta'(\\zeta'-\\zeta)} \\frac{\\exp\\left( \\frac{\\pi R a}{\\zeta'} +i\\theta_e \\right)}{1-\\exp\\left( \\frac{\\pi R a}{\\zeta'} +i\\theta_e \\right)}.\n\\end{equation}\nIntroduce the real variable $s = -\\pi R a \/ \\zeta'$. We can write \\eqref{secsimp} as:\n\\begin{align}\n& e^{i\\theta_e}\\int_0^\\infty \\frac{ds}{s\\left[ \\frac{-\\pi R a}{s} - \\zeta \\right]} \\frac{e^{-s}}{1-e^{i\\theta_e-s}} \\notag\\\\\n& = -\\frac{1}{\\zeta}\\int_0^\\infty \\frac{ds}{s+\\frac{\\pi R a}{\\zeta}} \\cdot \\frac{e^{-s}}{e^{-i\\theta_e}-e^{-s}} \\notag\\\\\n& = \\frac{1}{\\zeta}\\int_0^\\infty \\frac{ds}{s+\\frac{\\pi R a}{\\zeta}} \\cdot \\frac{1}{1-e^{s-i\\theta_e}} \\label{doublez}\n\\end{align}\nThe integrand of \\eqref{doublez} has a double zero at $\\infty$, when $a \\to 0$, so the only possible non-convergent part in the limit $a=0$ is the integral\n\\[ \\frac{1}{\\zeta}\\int_0^1 \\frac{ds}{s+\\frac{\\pi R a}{\\zeta}} \\cdot \\frac{1}{1-e^{s-i\\theta_e}}. \\]\nSince\n\\[ \\int_0^1 \\frac{ds}{s} \\left[ \\frac{1}{1-e^{s-i\\theta_e}} - \\frac{1}{1-e^{-i\\theta_e}}\\right] < \\infty, \\]\nwe can simplify this analysis even further and focus only on\n\\begin{align}\n& \\frac{1}{\\zeta(1-e^{-i\\theta_e})} \\int_0^1 \\frac{ds}{s+\\frac{\\pi R a}{\\zeta}} \\\\\n& = -\\frac{\\log (\\pi R a \/\\zeta)}{\\zeta(1-e^{-i\\theta_e})}.\n\\end{align}\nWe can apply the same technique to obtain a limit for the second integral in \\eqref{last3}. The result is\n\\[ -\\frac{\\log (-\\pi R a \/\\zeta)}{\\zeta(1-e^{i\\theta_e})}, \\]\nwhich means that the possibly non-convergent terms in \\eqref{last3} are:\n\\begin{equation}\\label{cancel}\n\\frac{\\log a}{\\zeta} - \\frac{\\log a}{\\zeta(1-e^{-i\\theta_e})} - \\frac{\\log a}{\\zeta(1-e^{i\\theta_e})} = 0.\n\\end{equation}\nNote that the corrections of $\\mathcal{X}_m$ in other regions of the $a$-plane as in \\eqref{regmod} depend only on $\\mathcal{X}_e$, which clearly has a smooth extension to the singular fiber.\n\n\nThe extension of the $d\\overline{a}$ part is performed in exactly the same way as with the $da$ forms. We conclude that the 1-form\n\\[ \\frac{d\\widetilde{\\mathcal{X}}_m}{\\widetilde{\\mathcal{X}}_m} \\]\nhas an extension to $\\mathcal{M}$; more explicitly, to the fiber at $a=0$ in the classical Ooguri-Vafa case. This holds true also in the generalized Ooguri-Vafa case since here we simply add factors of the form $f'(a)da$ and it is assumed that $f(a)$ has a smooth extension to the singular fiber.\n\\end{proof}\n\nIn section \\ref{sfiber}, we will reinterpret these extension of the derivatives of $\\mathcal{X}_m$ if we regard the gauge transformation \\eqref{nicethm} as a contour integral between symmetric contours. It will be then easier to see that the extension can be made smooth.\n\n\\subsubsection{Extension of the metric}\\label{extmetric}\n\n\nThe results of the previous section already show the continuous extension of the holomorphic symplectic form\n\\[ \\varpi(\\zeta) = -\\frac{1}{4\\pi^2 R} \\frac{d \\mathcal{X}_e}{\\mathcal{X}_e} \\wedge \\frac{d\\widetilde{\\mathcal{X}}_m}{\\widetilde{\\mathcal{X}}_m} \\]\nto the limiting case $a = 0$, but we excluded the special case $\\theta_e = 0$. Here we obtain $\\varpi(\\zeta)$ at the singular fiber with a different approach that will allow us to see that such an extension is smooth without testing the extension for each derivative. Although it was already known that $\\mathcal{M}'$ extends to the hyperk\\\"{a}hler manifold $\\mathcal{M}$ constructed here, this approach is new, as it gives an explicit construction of the metric as we will see. Furthermore, the Ooguri-Vafa model can be thought as an elementary model for which more complex integrable systems are modeled locally (see \\S \\ref{sfiber}).\n\\begin{theorem}\nThe holomorphic symplectic form $\\varpi(\\zeta)$ extends smoothly to $\\mathcal{M}$. Near $a = 0$ and $\\theta_e = 0$, the hyperk\\\"{a}hler metric $g$ looks like a constant multiple of the Taub-NUT metric $g_{\\text{Taub-NUT}}$ plus some smooth corrections.\n\\end{theorem}\n\\begin{proof}\n By \\cite{gaiotto}, near $a = 0$,\n\\[ \\varpi(\\zeta) = -\\frac{1}{4\\pi^2 R} \\frac{d \\mathcal{X}_e}{\\mathcal{X}_e} \\wedge \\left[ id\\theta_m + 2\\pi i A + \\pi i V \n\\left(\\frac{1}{\\zeta}da - \\zeta d\\bar{a}\\right)\\right], \\]\nwhere\n\\[ A = \\frac{1}{8\\pi^2}\\left( \\log \\frac{a}{\\Lambda} - \\log\\frac{\\bar{a}}{\\overline{\\Lambda}} \\right)d\\theta_e - \\frac{R}{4\\pi} \\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right)\\sum_{n \\neq 0} (\\text{sgn} \\,n) e^{in\\theta_e} |a|\n K_1(2\\pi R|na|) \\]\nshould be understood as a $U(1)$ connection over the open subset of $\\mathbb{C} \\times S^1$ parametrized by $(a,\\theta_e)$ and $V$ is given by Poisson re-summation as\n\\begin{equation}\\label{potent}\nV = \\frac {R}{4\\pi}\\left[ \\frac{1}{\\sqrt{R^2|a|^2 + \\frac{\\theta_e^2}{4\\pi^2}}} + \\sum_{\\substack{n = -\\infty \\\\ n \\neq 0}}^\\infty \\left( \\frac{1}{\\sqrt{R^2 |a|^2 + (\\frac{\\theta_e}{2\\pi} + n)^2}} - \\kappa_n \\right) \\right].\n\\end{equation}\nHere $\\kappa_n$ is a regularization constant introduced to make the sum convergent, even at $a = 0, \\theta_e \\neq 0$. The curvature $F$ of the unitary connection satisfies\n\\begin{equation}\\label{curv}\ndA = *dV.\n\\end{equation}\nConsider now a gauge transformation $\\theta_m \\mapsto \\theta_m + \\alpha$ and its induced change in the connection $A \\mapsto A' = A - d\\alpha\/2\\pi$ (see \\cite{gaiotto}). We have $id\\theta'_m + 2\\pi i A' = id\\theta_m + id\\alpha + 2\\pi i A - id\\alpha = id\\theta_m + 2\\pi i A$. Furthermore, for the particular gauge transformation in (\\ref{thetapr}), at $a = 0$ and for $\\theta_e \\neq 0$:\n\\begin{align*}\nA' & = A - \\frac{d\\alpha}{2\\pi}\\\\\n& = \\frac{1}{8\\pi^2}\\left( \\log \\frac{a}{\\Lambda} - \\log\\frac{\\bar{a}}{\\overline{\\Lambda}} \\right)d\\theta_e - \\frac{1}{8\\pi^2} \\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right) \\left[ \\sum_{n = 1}^\\infty \\frac{e^{in\\theta_e}}{n} - \\sum_{n = 1}^\\infty \\frac{e^{-in\\theta_e}}{n} \\right]\\\\\n& - \\frac{1}{8\\pi^2}\\left( \\log \\frac{a}{\\Lambda} - \\log\\frac{\\bar{a}}{\\overline{\\Lambda}} \\right)d\\theta_e - \\frac{i(\\theta_e - \\pi)}{8\\pi^2}\\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right),\\\\\n\\intertext{(here we're using the fact that $K_1(x) \\to 1\/x$ as $x \\to 0$)}\n& = \\frac{i(\\theta_e - \\pi)}{8\\pi^2}\\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right) - \\frac{i(\\theta_e - \\pi)}{8\\pi^2}\\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right) = 0.\\\\\n\\intertext{since the above sums converge to $-\\log(1 - e^{i\\theta_e}) + \\log(1 - e^{-i\\theta_e}) = -i(\\theta_e - \\pi)$ for $\\theta_e \\neq 0$.}\n\\end{align*}\n\nWriting $V_0$ (observe that this only depends on $\\theta_e$) for the limit of $V$ as $a \\to 0$, we get at $a = 0$\n\\begin{align*}\n\\varpi(\\zeta) & = -\\frac{1}{4\\pi^2 R} \\left( \\frac{\\pi R}{\\zeta}da + id\\theta_e + \\pi R \\zeta d\\bar{a} \\right) \\wedge \\left(\n id\\theta'_m + \\pi i V_0 \\left( \\frac{da}{\\zeta} - \\zeta d\\bar{a}\\right) \\right) \\\\\n & = \\frac{1}{4\\pi^2 R} d\\theta_e \\wedge d\\theta'_m + \\frac{iV_0}{2}da \\wedge d\\bar{a} -\\frac{i}{4\\pi \\zeta}da \\wedge d\\theta'_m - \\frac{V_0}{4\\pi R\\zeta}da \\wedge d\\theta_e \\\\\n & - \\frac{i\\zeta}{4\\pi} d\\bar{a} \\wedge d\\theta'_m + \\frac{V_0 \\zeta}{4\\pi R} d\\bar{a} \\wedge d\\theta_e.\n\\end{align*}\n\nThis yields that, at the singular fiber,\n\\begin{align}\n\\omega_3 & = \\frac{1}{4\\pi^2 R} d\\theta_e \\wedge d\\theta'_m + \\frac{iV_0}{2}da \\wedge d\\bar{a} \\label{symp3}\\\\\n\\omega_+ & = \\frac{1}{2\\pi} da \\wedge \\left( d\\theta'_m - \\frac{iV_0}{R}d\\theta_e \\right)\\label{symp+}\\\\\n\\omega_- & = \\frac{1}{2\\pi} d\\bar{a} \\wedge \\left( d\\theta'_m + \\frac{iV_0}{R}d\\theta_e \\right)\\label{symp-}\n\\end{align}\n\nFrom the last two equations we obtain that $d\\theta'_m - iV_0\/R d\\theta_e$ and $d\\theta'_m + iV_0\/R d\\theta_e$ are respectively (1,0) and (0,1) forms under the complex structure $J_3$. A $(1,0)$ vector field dual to the $(1,0)$ form above is then $\\dfrac{1}{2}\\left(\\partial_{\\theta'_m} + iR\/V_0 \\partial_{\\theta_e}\\right)$. In particular,\n\\[ J_3(\\partial_{\\theta'_m}) = -\\frac{R}{V_0} \\partial_{\\theta_e}, \\hspace{5 mm} J_3 \\left(-\\frac{R}{V_0}\\partial_{\\theta_e} \\right) = -\\partial_{\\theta'_m}. \\]\nWith this and (\\ref{symp3}) we can reconstruct the metric at $a = 0$. Observe that\n\\begin{align*}\ng(\\partial_{\\theta_e}, \\partial_{\\theta_e}) & = \\omega_3(\\partial_{\\theta_e}, J_3(\\partial_{\\theta_e})) = \\omega_3\\left(\\partial_{\\theta_e}, \\frac{V_0}{R}\\partial_{\\theta'_m}\\right) = \\frac{V_0}{4\\pi^2 R^2} \\\\\ng(\\partial_{\\theta'_m}, \\partial_{\\theta'_m}) & = \\omega_3(\\partial_{\\theta'_m}, J_3(\\partial_{\\theta'_m})) = \\omega_3\\left(\\partial_{\\theta'_m}, -\\frac{R}{V_0}\\partial_{\\theta_e}\\right) = \\frac{1}{4\\pi^2 V_0}\n\\end{align*}\n\nConsequently,\n\\[ g = \\frac{1}{V_0} \\left( \\frac{d\\theta'_m}{2\\pi}\\right)^2 + V_0 d\\vec{x}^2, \\]\nwhere $a = x^1 + ix^2, \\theta_e = 2\\pi R x^3$. Since $V_0(\\theta_e)$ is undefined for $\\theta_e = 0$, we have to check that $g$ extends to this point. Let $(r,\\vartheta, \\phi)$ denote spherical coordinates for $\\vec{x}$. The formula above is the natural extension of the metric given in \\cite{gaiotto} for nonzero $a$:\n\\[ g = \\frac{1}{V(\\vec{x})} \\left( \\frac{d\\theta'_m}{2\\pi} + A'(\\vec{x})\\right)^2 + V(\\vec{x}) d\\vec{x}^2 \\]\nTo see that this extends to $r =0$, we rewrite\n\\begin{align}\nV & = \\frac{R}{4\\pi}\\left[ \\frac{1}{\\sqrt{R^2 |a|^2 + \\frac{\\theta_e^2}{4\\pi^2}}} + \\sum_{n \\neq 0} \\left(\n \\frac{1}{\\sqrt{R^2 |a|^2 + (\\frac{\\theta_e}{2\\pi} + n)^2}} - \\kappa_n \\right)\\right] \\notag\\\\\n & = \\frac{1}{4\\pi}\\left[ \\frac{1}{\\sqrt{ |a|^2 + \\frac{\\theta_e^2}{4R^2 \\pi^2}}} + R\\sum_{n \\neq 0} \\left(\n \\frac{1}{\\sqrt{R^2 |a|^2 + (\\frac{\\theta_e}{2\\pi} + n)^2}} - \\kappa_n \\right) \\right] \\notag\\\\\n & = \\frac{1}{4\\pi} \\left( \\frac{1}{r} + C(\\vec{x}) \\right), \\label{vtaub}\n\\end{align}\nwhere $C(\\vec{x})$ is smooth and bounded in a neighborhood of the origin.\n\nSimilarly, we do Poisson re-summation for the unitary connection\n\n\\[ A' = - \\frac{1}{4\\pi} \\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right) \\left[ \\frac{i(\\theta_e - \\pi)}{2\\pi} + R \\sum_{n \\neq 0} (\\text{sgn} \\,n) e^{in\\theta_e} |a| K_1(2\\pi R|na|) \\right]. \\]\n\nUsing the fact that the inverse Fourier transform of $(\\text{sgn }\\xi)e^{i\\theta_e \\xi}|a|K_1(2\\pi R|a\\xi|)$ is\n\\[ \\frac{i(\\frac{\\theta_e}{2\\pi} + t)}{2R\\sqrt{R^2|a|^2 + ( \\frac{\\theta_e}{2\\pi} + t)^2}}, \\]\nwe obtain\n\\begin{align}\nA' & = - \\frac{i}{8\\pi} \\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right)\\sum_{n = -\\infty}^\\infty \\left( \\dfrac{ \\frac{\\theta_e}{2\\pi} + n}{\\sqrt{R^2 |a|^2 + (\\frac{\\theta_e}{2\\pi} + n)^2}} - \\kappa_n \\right) \\notag\\\\\n& = \\frac{1}{4\\pi}\\left( \\frac{da}{a} - \\frac{d\\bar{a}}{\\bar{a}}\\right)\\left[-\\frac{i\\theta_e}{4\\pi \\sqrt{R^2 |a|^2 + \\left( \n \\frac{\\theta_e}{2\\pi}\\right)^2}} - \\frac{i}{2}\\sum_{n \\neq 0} \\left( \\dfrac{ \\frac{\\theta_e}{2\\pi} + n}{\\sqrt{R^2 |a|^2 + (\\frac{\\theta_e}{2\\pi} + n)^2}} - \\kappa_n \\right)\\right] \\notag\\\\\n \\intertext{since $d\\phi = d\\arg a = -id\\log \\dfrac{a}{|a|} = -\\dfrac{i}{2}\\left(\\dfrac{da}{a} - \\dfrac{d\\bar{a}}{\\bar{a}} \\right)$ and $\\cos \\vartheta = \\dfrac{x^3}{r}$, this simplifies to:}\n & = \\frac{1}{4\\pi}(\\cos \\vartheta + D(\\vec{x}))d\\phi. \\label{ataub}\n\\end{align}\nHere $\\kappa_n$ is a regularization constant that makes the sum converge, and $D(\\vec{x})$ is smooth and bounded in a neighborhood of $r = 0$. By (\\ref{vtaub}) and (\\ref{ataub}), it follows that near $r = 0$\n\\begin{align*}\ng & = V^{-1}\\left( \\frac{d\\theta'_m}{2\\pi} + A' \\right)^2 + Vd\\vec{x}^2\\\\\n& = 4\\pi \\left( \\frac{1}{r} + C \\right)^{-1} \\left( \\frac{d\\theta'_m}{2\\pi} + \\frac{1}{4\\pi}\\cos \\vartheta d\\phi + D d\\phi \\right)^2 + \\frac{1}{4\\pi} \\left( \\frac{1}{r} + C \\right) d\\vec{x}^2\\\\\n& = \\frac{1}{4\\pi}\\left[ \\left( \\frac{1}{r} + C \\right)^{-1} \\left( 2d\\theta'_m + \\cos \\vartheta d\\phi + \\tilde{D} d\\phi \\right)^2\n + \\left( \\frac{1}{r} + C \\right) d\\vec{x}^2 \\right]\\\\\n& = \\frac{1}{4\\pi} g_{\\text{Taub-NUT}} + \\text{smooth corrections}.\n\\end{align*}\nThis shows that our metric extends to $r = 0$ and finishes the construction of the singular fiber.\n\\end{proof}\n\n\\subsection{General case}\\label{genextmtr}\n\nHere we work with the assumption in subsection \\ref{genvafa}. To distinguish this case to the previous one, we will denote by $\\varpi_\\text{old}, g_\\text{old}$, etc. the forms obtained in the classical case.\n\nLet $C := -i\/2 + \\pi f'(0)$ and let\n\\[ B_0 = V_0 + \\frac{R \\, \\text{Im }C}{\\pi}. \\]\nWe will see that, to extend the holomorphic symplectic form $\\varpi(\\zeta)$ and consequently the hyperk\\\"{a}hler metric $g$ to $\\mathcal{M}$, it is necessary to impose a restriction on the class of functions $f(a)$ on $\\mathcal{B}$ for the generalized Ooguri-Vafa case.\n\n\\begin{theorem}\nIn the General Ooguri-Vafa case, the holomorphic symplectic form $\\varpi(\\zeta)$ and the hyperk\\\"{a}hler metric $g$ extend to $\\mathcal{M}$, at least for the set of functions $f(a)$ as in \\S \\ref{genvafa} with $f'(0) > B_0$.\n\\end{theorem}\n\\begin{proof}\n By formula \\eqref{newxmsf},\n\\begin{equation}\\label{dlogxm}\nd \\log \\mathcal{X}_m^\\text{sf} = d \\log \\mathcal{X}_{m, \\text{old}}^\\text{sf} + \\frac{R}{\\zeta}\\left( -\\frac{i}{2} + \\pi f'(a) \\right)da + R\\zeta \\left( \\frac{i}{2} + \\pi \\overline{f'(a)} \\right)d\\overline{a}\n\\end{equation}\n\nRecall that the corrections of $\\mathcal{X}_m$ are the same as the classical Ooguri-Vafa case. Thus, using \\eqref{dlogxm}, at $a = 0$\n\\begin{equation*}\n\\varpi(\\zeta) = \\varpi_\\text{old}(\\zeta) + \\frac{iR }{2\\pi} \\text{Im } C da \\wedge d\\overline{a} + \\frac{i C}{4\\pi^2 \\zeta}\n da \\wedge d\\theta_e + \\frac{i \\zeta \\overline{C}}{4\\pi^2} d\\overline{a} \\wedge d\\theta_e.\n\\end{equation*}\n\nDecomposing $\\varpi(\\zeta) = -i\/2\\zeta \\omega_+ + \\omega_3 -i\\zeta \/2 \\omega_-$, we obtain:\n\\begin{align}\n\\omega_3 & = \\omega_{3, \\text{old}} + \\frac{i R}{2\\pi} \\text{Im } C da \\wedge d\\overline{a}, \\label{omeg3}\\\\\n\\omega_+ & = \\omega_{+, \\text{old}} - \\frac{C}{2\\pi^2} da \\wedge d\\theta_e \\label{newomp}\\\\\n\\omega_- & = \\omega_{-, \\text{old}} - \\frac{\\overline{C}}{2\\pi^2} d\\overline{a} \\wedge d\\theta_e \\label{newomm}\n\\end{align}\n\nBy \\eqref{newomp} and \\eqref{newomm},\n\\[ d\\theta'_m - \\frac{i}{R}\\left( V_0 - \\frac{iRC}{\\pi} \\right)d\\theta_e \\hspace{6 mm} \\text{and} \\hspace{6 mm} d\\theta'_m + \\frac{i}{R}\\left( V_0 + \\frac{iR\\overline{C}}{\\pi} \\right)d\\theta_e \\]\nare, respectively, (1,0) and (0,1) forms. It's not hard to see that\n\\begin{align*}\n\\frac{-V_0 \\pi -iR \\overline{C}}{R\\pi}\\partial_{\\theta'_m} & - i\\partial_{\\theta_e} \\\\\n\\intertext{or, rearranging real parts,}\n\\left( -\\frac{V_0}{R} -\\frac{\\text{Im }C}{\\pi} \\right) \\partial_{\\theta'_m} & -i \\left( \\frac{\\text{Re }C}{\\pi} \\partial_{\\theta'_m} + \\partial_{\\theta_e}\\right)\n\\end{align*}\nis a $(1,0)$ vector field. This allow us to obtain\n\\begin{align*}\nJ_3\\left[ \\left( -\\frac{V_0}{R} -\\frac{\\text{Im }C}{\\pi} \\right) \\partial_{\\theta'_m} \\right] & = \\frac{\\text{Re }C}{\\pi} \\partial_{\\theta'_m} + \\partial_{\\theta_e}\\\\\nJ_3\\left[\\frac{\\text{Re }C}{\\pi} \\partial_{\\theta'_m} + \\partial_{\\theta_e} \\right] & = \\left( \\frac{V_0}{R} +\\frac{\\text{Im }C}{\\pi} \\right) \\partial_{\\theta'_m}.\n\\end{align*}\nBy linearity,\n\\begin{align*}\nJ_3(\\partial_{\\theta'_m}) & = \\text{const} \\cdot \\partial_{\\theta'_m} - \\frac{R\\pi}{V_0 \\pi + R\\text{Im }C} \\partial_{\\theta_e}\\\\\nJ_3(\\partial_{\\theta_e}) & = \\left( \\frac{V_0 \\pi + R\\text{Im }C }{\\pi R} + \\frac{(\\text{Re }C)^2 R}{\\pi(V_0 \\pi \n + R\\text{Im }C)} \\right)\\partial_{\\theta'_m} + \\text{const} \\cdot \\partial_{\\theta_e}.\n\\end{align*}\nWith this we can compute\n\\begin{align*}\ng(\\partial_{\\theta'_m}, \\partial_{\\theta'_m}) & = \\omega_3(\\partial_{\\theta'_m}, J_3(\\partial_{\\theta'_m}))\\\\\n& = \\frac{1}{4\\pi(V_0 \\pi + R\\text{Im }C)}\\\\\ng(\\partial_{\\theta_e}, \\partial_{\\theta_e}) & = \\omega_3(\\partial_{\\theta_e}, J_3(\\partial_{\\theta_e}))\\\\\n& = \\frac{V_0 \\pi + R\\text{Im }C}{4\\pi^3 R^2} + \\frac{(\\text{Re }C)^2}{4\\pi^3(V_0 \\pi + R\\text{Im }C)}\\\\\n& = \\frac{B_0}{4\\pi^3 R^2} + \\frac{(\\text{Re }C)^2}{4\\pi^3 B_0}\n\\end{align*}\n\nWe can see that, if $B_0 > 0$, the metric at $a = 0$ is\n\\begin{equation}\ng = \\frac{1}{B_0} \\left( \\frac{d\\theta'_m}{2\\pi}\\right)^2 + B_0 d\\vec{x}^2 + \\left(\\frac{R\\cdot\\text{Re }C}{\\pi}\\right)^2 \\frac{dx_3^2}{B_0}.\n\\end{equation}\nThis metric can be extended to the point $\\theta_e = 0$ ($r = 0$ in \\S \\ref{clasmet}) exactly as before, by writing $g$ as the Taub-NUT metric plus smooth corrections and observing that, since $\\lim_{\\theta_e \\to 0} B_0 = \\infty$,\n\\[ \\lim_{\\theta_e \\to 0} \\left(\\frac{R\\cdot\\text{Re }C}{\\pi}\\right)^2 \\frac{dx_3^2}{B_0} = 0. \\]\n\\end{proof}\n\n\n\\section{The Pentagon case}\\label{pent}\n\n\n\n\\subsection{Monodromy Data}\\label{solut}\n\nNow we will extend the results of the Ooguri-Vafa case to the general problem. We will start with the Pentagon example. This example is presented in detail in \\cite{notes}. By \\cite{wkb}, this example represents the moduli space of Higgs bundles with gauge group $\\text{SU}(2)$ over $\\mathbb{P}^1$ with 1 irregular singularity at $z = \\infty$.\n\nHere $\\mathcal{B} = \\mathbb{C}$ with discriminant locus a 2-point set, which we can assume is $\\{-2,2\\}$ in the complex plane. Thus $\\mathcal{B}'$ is the twice-punctured plane. $\\mathcal{B}$ is divided into two domains $\\mathcal{B}_{\\text{in}}$ and $\\mathcal{B}_{\\text{out}}$ by the locus\n\\[ W = \\{u : Z(\\Gamma_u) \\text{ is contained in a line in } \\mathbb{C} \\} \\subset \\mathcal{B} \\]\nSee Figure \\ref{walls}. Since $\\mathcal{B}_{\\text{in}}$ is simply connected $\\Gamma$ can be trivialized over $\\mathcal{B}_{\\text{in}}$ by primitive cycles $\\gamma_1, \\gamma_2$, with $Z_{\\gamma_1} = 0$ at $u = -2$, $Z_{\\gamma_2} = 0$ at $u = 2$. We can choose them also so that $\\left\\langle \\gamma_1, \\gamma_2 \\right\\rangle = 1$. \n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{walls.ps}\n\t\\caption{The wall $W$ in $\\mathcal{B}$ for the Pentagon case}\n\t\\label{walls}\n\\end{figure}\n\nTake the set $\\{\\gamma_1, \\gamma_2\\}$. To compute its monodromy around infinity, take cuts at each point of $D = \\{-2,2\\}$ (see Figure \\ref{moninf}) and move counterclockwise. By \\eqref{piclf}, the jump of $\\gamma_2$ when you cross the cut at $-2$ is of the form $\\gamma_2 \\mapsto \\gamma_1 + \\gamma_2$. As you return to the original place and cross the cut at $2$, the jump of $\\gamma_1$ is of the type $\\gamma_1 \\mapsto \\gamma_1 - \\gamma_2$.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{monodromy_infinity.ps}\n\t\\caption{The monodromy around infinity of $\\Gamma$}\n\t\\label{moninf}\n\\end{figure}\n\n\nThus, around infinity, $\\{\\gamma_1, \\gamma_2\\}$ transforms into $\\{-\\gamma_2, \\gamma_1 + \\gamma_2\\}$. The set $\\{\\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2, \\gamma_1 + \\gamma_2, -\\gamma_1 - \\gamma_2\\}$ is therefore invariant under monodromy at infinity and it makes global sense to define\n\n \\begin{align}\n \\text{For $u \\in \\mathcal{B}_\\text{in}$}, \\hspace{5 mm} \\Omega(\\gamma; u) = & \\left\\{ \\begin{array}{ll}\n 1 & \\text{for } \\gamma \\in \\{ \\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2\\}\\\\\n 0 & \\text{otherwise}\n \\end{array} \\right. \\notag\\\\\n \\text{For $u \\in \\mathcal{B}_\\text{out}$} , \\hspace{5 mm} \\Omega(\\gamma; u) = & \\left\\{ \\begin{array}{ll}\n 1 & \\text{for } \\gamma \\in \\{ \\gamma_1, \\gamma_2, -\\gamma_1, -\\gamma_2, \\gamma_1 + \\gamma_2, -\\gamma_1 - \\gamma_2\\}\\\\\n 0 & \\text{otherwise}\n \\end{array} \\right. \\label{omgpar}\n \\end{align}\n\n\n\n\nLet $\\mathcal{M}'$ denote the torus fibration over $\\mathcal{B}'$ constructed in \\cite{notes}. Near $u=2$, we'll denote $\\gamma_1$ by $\\gamma_m$ and $\\gamma_2$ by $\\gamma_e$ (the labels will change for $u = - 2$). To shorten notation, we'll write $\\ell_{e}, Z_e$, etc. instead of $\\ell_{\\gamma_e}, Z_{\\gamma_e}$, etc. Let $\\theta$ denote the vector of torus coordinates $(\\theta_e, \\theta_m)$. With the change of variables $a := Z_e(u)$ we can assume, without loss of generality, that the bad fiber is at $a = 0$ and\n\\begin{equation}\\label{zmnotz}\n\\lim_{a \\to 0} Z_m(a) = c \\neq 0.\n\\end{equation}\nLet $T$ denote the complex torus fibration over $\\mathcal{M}'$ constructed in \\cite{gaiotto}. By the definition of $\\Omega(\\gamma; a)$, the functions $(\\mathcal{X}_e, \\mathcal{X}_m)$ both receive corrections. Recall that by \\eqref{recurs}, for each $\\nu \\in \\mathbb{N}$, we get a function $\\mathcal{X}_\\gamma^{(\\nu)}$, which is the $\\nu$-th iteration of the function $\\mathcal{X}_\\gamma$. We can write\n\\[ \\mathcal{X}_\\gamma^{(\\nu)}(a, \\zeta, \\theta) = \\mathcal{X}_\\gamma^{\\text{sf}}(a, \\zeta, \\theta)C_{\\gamma}^{(\\nu)}(a, \\zeta, \\theta). \\]\nIt will be convenient to rewrite the above equation as in \\cite[C.17]{gaiotto}. For that, let $\\Upsilon^{(\\nu)}$ be the map from $\\mathcal{M}_a$ to its complexification $\\mathcal{M}_a^{\\mathbb{C}}$ such that\n\\begin{equation}\\label{upsi}\n \\mathcal{X}_\\gamma^{(\\nu)}(a, \\zeta, \\theta) = \\mathcal{X}_\\gamma^{\\text{sf}}(a, \\zeta, \\Upsilon^{(\\nu)}).\n\\end{equation}\n\nWe'll do a modification in the construction of \\cite{gaiotto} as follows: We'll use the term ``BPS ray'' for each ray $\\{\\ell_\\gamma : \\Omega(\\gamma,a) \\neq 0 \\}$ as in \\cite{gaiotto}. This terminology comes from Physics. In the language of Riemann-Hilbert problems, these are known as ``anti-Stokes'' rays. That is, they represent the contour $\\Sigma$ where a function has prescribed discontinuities.\n\nThe problem is local on $\\mathcal{B}$, so instead of defining a Riemann-Hilbert problem using the BPS rays $\\ell_\\gamma$, we will cover $\\mathcal{B}'$ with open sets $\\{U_\\alpha : \\alpha \\in \\Delta \\}$ such that for each $\\alpha$, $\\overline{U_\\alpha}$ is compact, $\\overline{U_\\alpha} \\subset V_\\alpha$, with $V_\\alpha$ open and $\\left. \\mathcal{M}' \\right|_{V_\\alpha}$ a trivial fibration. For any ray $r$ in the $\\zeta$-plane, define $\\mathbb{H}_r$ as the half-plane of vectors making an acute angle with $r$. Assume that there is a pair of rays $r, -r$ such that for all $a \\in U_\\alpha$, half of the rays lie inside $\\mathbb{H}_r$ and the other half lie in $\\mathbb{H}_{-r}$. We call such rays \\textit{admissible rays}. If $U_\\alpha$ is small enough, there exists admissible rays for such a neighborhood. We are allowing the case that $r$ is a BPS ray $\\ell_\\gamma$, as long as it satisfies the above condition. As $a$ varies in $U_\\alpha$, some BPS rays (or anti-Stokes rays, in RH terminology) converge into a single ray (wall-crossing phenomenon) (see Figures \\ref{3rays} and \\ref{2rays}). \n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{3raysbeforewall.ps}\n\t\\caption{3 anti-Stokes rays before hitting the wall}\n\t\\label{3rays}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{2raysafterwall.ps}\n\t\\caption{At the other side of the wall there are only 2 anti-Stokes rays}\n\t\\label{2rays}\n\\end{figure}\n\n\n For $\\gamma \\in \\Gamma$, we define $\\gamma > 0$ (resp. $\\gamma < 0$) as $\\ell_\\gamma \\in \\mathbb{H}_r$ (resp. $\\ell_\\gamma \\in \\mathbb{H}_{-r}$). Our Riemann-Hilbert problem will have only two anti-Stokes rays, namely $r$ and $-r$. The specific discontinuities at the anti-Stokes rays for the function we're trying to obtain are called \\textit{Stokes factors} (see \\cite{boalch}). In \\eqref{invjmp}, the Stokes factor was given by $S^{-1}_\\ell$.\n \n In this case, the Stokes factors are the concatenation of all the Stokes factors $S^{-1}_\\ell$ in \\eqref{stkfac} in the counterclockwise direction:\n\\begin{align*}\nS_+ & = \\prod^\\text{\\Large$\\curvearrowleft$}_{\\gamma > 0}{\\mathcal{K}^{\\Omega(\\gamma; a)}_\\gamma}\\\\\nS_- & = \\prod^\\text{\\Large$\\curvearrowleft$}_{\\gamma < 0}{\\mathcal{K}^{\\Omega(\\gamma; a)}_\\gamma}\n\\end{align*}\n\nWe will denote the solutions of this Riemann-Hilbert problem by $\\mathcal{Y}$. As in \\eqref{upsi}, we can write $\\mathcal{Y}$ as\n\\begin{equation}\\label{thet}\n \\mathcal{Y}_\\gamma(a, \\zeta, \\theta) = \\mathcal{X}_\\gamma^{\\text{sf}}(a, \\zeta, \\Theta),\n\\end{equation}\nfor $\\Theta : \\mathcal{M}_a \\to \\mathcal{M}_a^\\mathbb{C}$.\n\nA different choice of admissible pairs $r', -r'$ gives an equivalent Riemann-Hilbert problem, where the two solutions $\\mathcal{Y}, \\mathcal{Y}'$ differ only for $\\zeta$ in the sector defined by the rays $r,r'$, and one can be obtained from the other by analytic continuation.\n\nIn the case of the Pentagon, we have two types of wall-crossing phenomenon. Namely, as $a$ varies, $\\ell_e$ moves in the $\\zeta$-plane until it coincides with the $\\ell_m$ ray for some value of $a$ in the wall of marginal stability (Fig. \\ref{3rays} and \\ref{2rays}). We'll call this type I of wall-crossing. In this case we have the Pentagon identity\n\\begin{equation}\\label{pentid}\n \\mathcal{K}_e \\mathcal{K}_m = \\mathcal{K}_m \\mathcal{K}_{e+m} \\mathcal{K}_e,\n\\end{equation}\nAs $a$ goes around 0, the $\\ell_e$ ray will then intersect with the $\\ell_{-m}$ ray now. Because of the monodromy $\\gamma_m \\mapsto \\gamma_{-e+m}$ around 0, $\\ell_m$ becomes $\\ell_{-e+m}$. This second type (type II) of wall-crossing is illustrated in Fig. \\ref{2rays2} and \\ref{3rays2}.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{2raysafterwalltoo.ps}\n\t\\caption{2 anti-Stokes rays before hitting the wall}\n\t\\label{2rays2}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.40\\textwidth]{3raysbeforewalltoo.ps}\n\t\\caption{At the other side of the wall there are now 3 anti-Stokes rays}\n\t\\label{3rays2}\n\\end{figure}\n\nThis gives a second Pentagon identity\n\\[ \\mathcal{K}_{-e} \\mathcal{K}_m = \\mathcal{K}_m \\mathcal{K}_{-e+m} \\mathcal{K}_{-e} \\]\n\n\nIn any case, the Stokes factors above remain the same even if $a$ is in the wall of marginal stability. The way we defined $S_+, S_-$ makes this true for the general case also.\n\nSpecifically, in the Pentagon the two Stokes factors for the first type of wall-crossing are given by the maps:\n\\begin{align}\n\\left. \\begin{array}{ll}\n \\mathcal{Y}_m & \\mapsto \\mathcal{Y}_m(1-\\mathcal{Y}_e(1-\\mathcal{Y}_m))^{-1} \\\\\n \\mathcal{Y}_e & \\mapsto \\mathcal{Y}_e(1-\\mathcal{Y}_m)\n \\end{array} \\right\\} & S_+ \\label{newj1}\\\\\n\\intertext{and, similarly}\n\\left. \\begin{array}{ll}\n \\mathcal{Y}_m & \\mapsto \\mathcal{Y}_m(1-\\mathcal{Y}^{-1}_e(1-\\mathcal{Y}^{-1}_m))\\\\\n \\mathcal{Y}_e & \\mapsto \\mathcal{Y}_e(1-\\mathcal{Y}^{-1}_m)^{-1}\n \\end{array} \\right\\} & S_- \\label{newj2}\n\\end{align}\n\n\\noindent For the second type:\n\n\\begin{align}\n\\left. \\begin{array}{ll}\n \\mathcal{Y}_m & \\mapsto \\mathcal{Y}_m(1-\\mathcal{Y}^{-1}_e) \\\\\n \\mathcal{Y}_e & \\mapsto \\mathcal{Y}_e(1-\\mathcal{Y}_m(1-\\mathcal{Y}^{-1}_e))\n \\end{array} \\right\\} & S_+ \\label{2newj1}\\\\\n\\left. \\begin{array}{ll}\n \\mathcal{Y}_m & \\mapsto \\mathcal{Y}_m(1-\\mathcal{Y}_e)^{-1}\\\\\n \\mathcal{Y}_e & \\mapsto \\mathcal{Y}_e(1-\\mathcal{Y}^{-1}_m(1-\\mathcal{Y}_e))^{-1}\n \\end{array} \\right\\} & S_- \\label{2newj2}\n\\end{align}\n\n\\subsection{Solutions}\n\nIn \\cite{rhprob} we prove the following theorem (in fact, a more general version is proven).\n\n\\begin{theorem}\\label{yfunctions}\nThere exist functions $ \\mathcal{Y}_m(a, \\zeta, \\theta_e, \\theta_m), \\mathcal{Y}_e(a, \\zeta, \\theta_e, \\theta_m)$ defined for $a \\neq 0$, smooth on $a$, $\\theta_e$ and $\\theta_m$. The functions are sectionally analytic on $\\zeta$ and obey the jump condition\n\\[ \\begin{array}{rll}\n\t\\mathcal{Y}^+ & = S_+ \\mathcal{Y}^-, & \\qquad \\text{along $r$} \\\\\n\t\\mathcal{Y}^+ & = S_{-} \\mathcal{Y}^-, & \\qquad \\text{along $-r$}\n\t\\end{array} \\]\nMoreover, $\\mathcal{Y}_m, \\mathcal{Y}_e$ obey the reality condition \\eqref{realcond} and the asymptotic condition \\ref{asymptotic}.\n\\end{theorem}\n\n\\begin{remark}\nOur construction used integrals along a fixed admissible pair $r,-r$ and our Stokes factors are concatenation of the Stokes factors in \\cite{gaiotto}. Thus, the coefficients $f^{\\gamma'}$ are different here, but they are still obtained by power series expansion of the explicit Stokes factor. In particular, it may not be possible to express\n\\[ f^{\\gamma'} = c_{\\gamma'} \\gamma' \\]\nfor some constant $c_{\\gamma'}$. For instance, in the pentagon, wall-crossing type I, we have, for $0\\leq j\\leq i$ and $\\gamma' = \\gamma_{ie +jm}$:\n\\[ f^{\\gamma'} = \\frac{(-1)^{j}\\binom{i}{j}}{i^2} \\gamma_{ie}. \\]\nBecause of this, we didn't use the Cauchy-Schwarz property of the norm in $\\Gamma$ in the estimates above as in \\cite{gaiotto}. Nevertheless, the tameness condition on the $\\Omega(\\gamma',a)$ invariants still give us the desired contraction.\n\\end{remark}\n\nObserve that, since we used admissible rays, the Stokes matrices don't change at the walls of marginal stability and we were able to treat both sides of the wall indistinctly. Thus, the functions $\\mathcal{Y}$ in Theorem \\ref{yfunctions} are smooth across the wall.\n\n\n\nLet's reintroduce the solutions in \\cite{gaiotto}. Denote by $\\mathcal{X}_e, \\mathcal{X}_m$ the solutions to the Riemann-Hilbert problem with jumps of the form $S_\\ell^{-1}$ at each BPS ray $\\ell$ with the same asymptotics and reality condition as $\\mathcal{Y}_e, \\mathcal{Y}_m$. In fact, we can see that the functions $\\mathcal{Y}$ are the analytic continuation of $\\mathcal{X}$ up until the admissible rays $r, -r$. \n\nIn a patch $U_\\alpha \\subset \\mathcal{B}'$ containing the wall of marginal stability, define the admissible ray $r$ as the ray where $\\ell_e, \\ell_m$ (or $\\ell_e, \\ell_{-m}$) collide. Since one is the analytic continuation of the other, $\\mathcal{X}$ and $\\mathcal{Y}$ differ only in a small sector in the $\\zeta$-plane bounded by the $\\ell_e, \\ell_m$ ($\\ell_e, \\ell_{-m}$) rays, for $a$ not in the wall. As $a$ approaches the wall, such a sector converges to the single admissible ray $r$. Thus, away from the ray where the two BPS rays collide, the solutions $\\mathcal{X}$ in \\cite{gaiotto} are continuous in $a$.\n\n\n\\section{Extension to the singular fibers}\\label{sfiber}\n\nIn this paper we will only consider the Pentagon example and in this section we will extend the Darboux coordinates $\\mathcal{X}_e, \\mathcal{X}_m$ obtained above to the singular locus $D \\subset \\mathcal{B}$ where one of the charges $Z_\\gamma$ approaches zero.\n\nLet $u$ be a coordinate for $\\mathcal{B} = \\mathbb{C}$. We can assume that the two bad fibers of $\\mathcal{M}$ are at $-2,2$ in the complex $u$-plane. For almost all $\\zeta \\in \\mathbb{P}^1$, the BPS rays converge in a point of the wall of marginal stability away from any bad fiber:\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.50\\textwidth]{raysinB.ps}\n\t\\caption{For general $\\zeta$, there is only 1 pair of rays at each fiber}\n\t\\label{raysinb}\n\\end{figure}\n\n\n\n\nIt is assumed that $\\lim_{u \\to 2} Z_{\\gamma_1}$ exists and it is nonzero. If we denote this limit by $c = |c|e^{i\\phi}$, then for $\\zeta$ such that $\\arg \\zeta \\to \\phi + \\pi$, the ray $\\ell_{\\gamma_1}$ emerging from -2 approaches the other singular point $u = 2$ (see Figure \\ref{oneside}).\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{raysatoneside.ps}\n\t\\caption{The BPS rays in $\\mathcal{B}$ nearly coalesce at the singular locus}\n\t\\label{oneside}\n\\end{figure}\n\n\nWhen $\\arg \\zeta = \\phi + \\pi$, the locus $\\{ u : Z_{\\gamma}(u)\/\\zeta \\in \\mathbb{R}_-\\}$, for some $\\gamma$ such that $\\Omega(\\gamma;u) \\neq 0$ crosses $u = 2$. See Figure \\ref{otherside}.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{raysatotherside.ps}\n\t\\caption{For $\\zeta$ in a special ray, the rays intersect $u = 2$}\n\t\\label{otherside}\n\\end{figure}\n\nAs $\\zeta$ keeps changing, the rays leave the singular locus, but near $u = 2$, the tags change due to the monodromy of $\\gamma_1$ around $u=2$. Despite this change of labels, near $u = 2$ only the rays $\\ell_{\\gamma_2}, \\ell_{-\\gamma_2}$ pass through this singular point. See Figure \\ref{finalside}\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{raysatrightside.ps}\n\t\\caption{After the critical value of $\\zeta$, the rays leave $u = 2$ and their tags change}\n\t\\label{finalside}\n\\end{figure}\n\nIn the general case of Figures \\ref{raysinb}, \\ref{oneside} or \\ref{finalside}, the picture near $u = 2$ is like in the Ooguri-Vafa case, Figure \\ref{3reg}.\n\nIn any case, because of the specific values of the invariants $\\Omega$, it is possible to analytically extend the function $\\mathcal{X}_{\\gamma_1}$ around $u = 2$. The global jump coming from the rays $\\ell_{\\gamma_2}, \\ell_{-\\gamma_2}$ is the opposite of the global monodromy coming from the Picard-Lefschetz monodromy of $\\gamma_1 \\mapsto \\gamma_1 - \\gamma_2$ (see \\eqref{piclf}). Thus, it is possible to obtain a function $\\widetilde{\\mathcal{X}}_{\\gamma_1}$ analytic on a punctured disk on $\\mathcal{B}'$ near $u = 2$ extending $\\mathcal{X}_{\\gamma_1}$.\n\nFrom this point on, we use the original formulation of the Riemann-Hilbert problem using BPS rays as in \\cite{gaiotto}. We also use $a = Z_{\\gamma_2}(u)$ to coordinatize a disk near $u = 2$, and we label $\\{\\gamma_1, \\gamma_2\\}$ as $\\{\\gamma_m, \\gamma_e\\}$ as in the Ooguri-Vafa case. Recall that, to shorten notation, we write $\\ell_e, \\mathcal{X}_e$, etc. instead of $\\ell_{\\gamma_e}, \\mathcal{X}_{\\gamma_e}$, etc.\n\nBy our work in the previous section, solutions $\\mathcal{X}_\\gamma$ (or, taking logs, $\\Upsilon_\\gamma$) to the Riemann-Hilbert problem are continuous at the wall of marginal stability for all $\\zeta$ except those in the ray $\\ell_m = Z_{m}\/\\zeta \\in \\mathbb{R}_- = \\ell_e$ (to be expected by the definition of the RH problem). We want to extend our solutions to the bad fiber located at $a=0$. We'll see that to achieve this, it is necessary to introduce new $\\theta$ coordinates.\n\n\n\nFor convenience, we rewrite the integral formulas for the Pentagon in terms of $\\Upsilon$ as in \\cite{notes}. We will only write the part in $\\mathcal{B}_\\text{in}$, the $\\mathcal{B}_{\\text{out}}$ part is similar.\n\\begin{align}\n\\Upsilon_e(a,\\zeta) & = \\theta_e -\n\\frac{1}{4\\pi}\\left\\{ \\int_{\\ell_m} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta}\\log\\left[ 1 - \\mathcal{X}_m^{\\text{sf}}(a,\\zeta', \\Upsilon_m) \\right] - \\int_{\\ell_{-m}} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta}\\log\\left[ 1 - \\mathcal{X}_{-m}^{\\text{sf}}(a,\\zeta', \\Upsilon_{-m})\\right] \\right\\}, \\label{upsefor}\\\\\n\\Upsilon_m(a,\\zeta) & = \\theta_m +\n\\frac{1}{4\\pi}\\left\\{ \\int_{\\ell_e} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta}\\log\\left[ 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right] - \\int_{\\ell_{-e}} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta}\\log\\left[ 1 - \\mathcal{X}_{-e}^{\\text{sf}}(a,\\zeta', \\Upsilon_{-e}) \\right] \\right\\} \\label{upsmfor}\n\\end{align}\n\n\nWe can focus only on the integrals above, so write $\\Upsilon_\\gamma(a,\\zeta) = \\theta_\\gamma + \\dfrac{1}{4\\pi} \\Phi_\\gamma(a,\\zeta)$, for $\\gamma \\in \\{\\gamma_m, \\gamma_e\\}$. To obtain the right gauge transformation of the torus coordinates $\\theta$, we'll split the integrals above into four parts and then we'll show that two of them define the right change of coordinates (in $\\mathcal{B}_{\\text{in}}$, and a similar transformation for $\\mathcal{B}_{\\text{out}}$) that simplify the integrals and allow an extension to the singular fiber.\n\nBy Theorem \\ref{yfunctions}, both $\\Upsilon_m, \\Upsilon_e$ satisfy the ``reality condition'', which expresses a symmetry in the behavior of the complexified coordinates $\\Upsilon$:\n \\begin{equation}\\label{corresp}\n \\overline{\\Upsilon_\\gamma(a, \\zeta)} = \\Upsilon_\\gamma \\left(a, -1\/\\overline{\\zeta}\\right), \\qquad a \\neq 0\n \\end{equation} \n If we write as $\\Upsilon_0$ (resp. $\\Upsilon_\\infty$) the asymptotic of this function as $\\zeta \\to 0$ (resp. $\\zeta \\to \\infty$) so that\n \\[ \\Upsilon_0 = \\theta + \\frac{1}{4\\pi} \\Phi_0, \\]\n for a suitable correction $\\Phi_0$. A similar equation holds for the asymptotic as $\\zeta \\to \\infty$. By the asymptotic condition \\ref{asymptotic}, $\\Phi_0$ is imaginary. \n\nCondition \\eqref{corresp} also shows that $\\Phi_0 = - \\Phi_\\infty$. This and the fact that $\\Phi_0$ is imaginary give the reality condition\n \\begin{equation}\\label{reali}\n \\Upsilon_0 = \\overline{\\Upsilon_\\infty}\n \\end{equation}\n \nSplit the integrals in \\eqref{upsmfor} into four parts as in \\eqref{integs}. For example, if we denote by $\\zeta_e := -a\/|a|$, the intersection of the unit circle with the $\\ell_e$ ray, then\n\\begin{align}\n& \\int_{\\ell_e} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta}\\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) = \\notag\\\\\n& -\\int_0^{\\zeta_e} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) + \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) \\notag\\\\\n& + \\int_0^{\\zeta_e} \\frac{2 d\\zeta'}{\\zeta'-\\zeta} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) + \\int_{\\zeta_e}^{\\zeta_e \\infty} 2d\\zeta' \\left\\{\\frac{1}{\\zeta'-\\zeta} -\\frac{1}{\\zeta'}\\right\\} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) \\label{4ints}\n\\end{align}\n\nWe consider the first two integrals apart from the rest. If we take the limit $a \\to 0$ the exponential decay in $\\mathcal{X}_e^{\\text{sf}}$:\n\\[ \\exp\\left( \\frac{\\pi R a}{\\zeta'} + \\pi R \\zeta' \\overline{a} \\right)\\]\nvanishes and the integrals are no longer convergent. \n \n By combining the two integrals with their analogues in the $\\ell_{-e}$ ray we obtain:\n\\begin{align}\n & -\\int_0^{\\zeta_e} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) + \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - \\mathcal{X}_e^{\\text{sf}}(a,\\zeta', \\Upsilon_e) \\right) \\notag\\\\\n & \\int_0^{-\\zeta_e} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - {\\mathcal{X}_e^{\\text{sf}}}^{-1}(a,\\zeta', -\\Upsilon_e) \\right) - \\int_{-\\zeta_e}^{-\\zeta_e \\infty} \\frac{d\\zeta'}{\\zeta'} \\log\\left( 1 - {\\mathcal{X}_e^{\\text{sf}}}^{-1}(a,\\zeta', -\\Upsilon_e) \\right) \\label{gen4int}\n \\end{align}\nThe parametrization in the first pair of integrals is of the form $\\zeta' = t\\zeta_e$, and in the second pair $\\zeta' = -t\\zeta_e$. Making the change of variables $\\zeta' \\mapsto 1\/\\zeta'$, we can pair up these integrals in a more explicit way as:\n\\begin{align}\n & -\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a}) \\right) \\right] \\right. \\notag\\\\\n & \\left. + \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) - i\\Upsilon_e(a,\\frac{1}{t} e^{i\\arg a}) \\right) \\right] \\right\\} \\notag \\\\\n & + \\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) +i\\Upsilon_e(a,-\\frac{1}{t} e^{i\\arg a}) \\right) \\right] \\right. \\notag\\\\\n & \\left. + \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a}) \\right) \\right] \\right\\} \\label{4intreal}\n \\end{align}\nBy \\eqref{corresp}, the integrands come in conjugate pairs. Therefore, we can rewrite \\eqref{4intreal} as:\n \\begin{align*}\n -2\\int_0^1 \\frac{dt}{t} \\text{Re } \\left\\{\\vphantom{\\int_0^1}\\right. & \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a}) \\right) \\right] - \\\\\n & \\left. \\log\\left[ 1 - \\exp\\left( -\\pi R |a| \\left(\\frac{1}{t} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a}) \\right) \\right] \\right\\}\n \\end{align*}\n \\begin{equation}\\label{simpl4}\n = -2\\int_0^1 \\frac{dt}{t} \\log \\left| \\frac{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a}) \\right)}{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a}) \\right)} \\right|\n \\end{equation}\n \n\nObserve that \\eqref{simpl4} itself suggest the correct transformation of the $\\theta$ coordinates that fixes this. Indeed, for a fixed $a \\neq 0$ and $\\theta_e$, let $Q$ be the map\n\\begin{equation*}\nQ(\\theta_m) = \\theta_m + \\psi(a,\\theta),\n\\end{equation*}\nwhere\n \\begin{align}\n \\psi_{\\text{in}}(a,\\theta)& = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\log\\left| \\frac{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a}) \\right)}{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a}) \\right)} \\right| \\notag\\\\\n & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}\\right](te^{i\\arg a})} \\right| \\label{newmp}\n \\end{align}\n for $a \\in \\mathcal{B}_{\\text{in}}$. For $a \\in \\mathcal{B}_{\\text{out}}$ where the wall-crossing is of type I, let $\\varphi = \\arg (Z_{\\gamma_e + \\gamma_m}(a))$, with $\\zeta'= -t e^{i\\varphi}$ parametrizing the $\\ell_{e + m}$ ray:\n \\begin{align}\n \\psi_{\\text{out}}(a,\\theta) & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left| \\frac{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a}) \\right)}{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a}) \\right)} \\right| \\right. \\notag\\\\\n & + \\left. \\log\\left| \\frac{1 - \\exp\\left( -\\pi R |Z_{\\gamma_e + \\gamma_m}| \\left(t^{-1} + t \\right) +i\\Upsilon_{e +m}(a,-te^{i\\arg \\varphi}) \\right)}{1 - \\exp\\left( -\\pi R |Z_{\\gamma_e + \\gamma_m}| \\left(t^{-1} + t \\right) -i\\Upsilon_{e+m}(a,te^{i\\arg \\varphi}) \\right)} \\right| \\right\\} \\notag\\\\\n & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}\\right](te^{i\\arg a})} \\right| + \\log\\left| \\frac{1 - \\left[\\mathcal{X}_{e+m}\\right](-te^{i\\varphi})}{1 - \\left[\\mathcal{X}_{-e-m}\\right](te^{i\\varphi})} \\right| \\right\\} \\label{outmp}\n \\end{align}\n \n Similarly, for wall-crossing of type II, $\\varphi = \\arg (Z_{\\gamma_{-e} + \\gamma_m}(a))$, with $\\zeta'= -t e^{i\\varphi}$ for the $\\ell_{-e + m}$ ray:\n \\begin{align}\n \\psi_{\\text{out}}(a,\\theta) & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left| \\frac{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) +i\\Upsilon_e(a,-te^{i\\arg a}) \\right)}{1 - \\exp\\left( -\\pi R |a| \\left(t^{-1} + t \\right) -i\\Upsilon_e(a,te^{i\\arg a}) \\right)} \\right| \\right. \\notag\\\\\n & + \\left. \\log\\left| \\frac{1 - \\exp\\left( -\\pi R |Z_{\\gamma_{-e} + \\gamma_m}| \\left(t^{-1} + t \\right) +i\\Upsilon_{-e +m}(a,-te^{i\\arg \\varphi}) \\right)}{1 - \\exp\\left( -\\pi R |Z_{\\gamma_{-e} + \\gamma_m}| \\left(t^{-1} + t \\right) -i\\Upsilon_{-e+m}(a,te^{i\\arg \\varphi}) \\right)} \\right| \\right\\} \\notag\\\\\n & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}\\right](te^{i\\arg a})} \\right| + \\log\\left| \\frac{1 - \\left[\\mathcal{X}_{-e+m}\\right](-te^{i\\varphi})}{1 - \\left[\\mathcal{X}_{e-m}\\right](te^{i\\varphi})} \\right| \\right\\} \\label{outmp2}\n \\end{align}\n \n As $a$ approaches the wall of marginal stability $W$, $\\arg a \\to \\varphi$. We need to show the following\n\\begin{lemma}\nThe two definitions $\\psi_{\\text{in}}$ and $\\psi_{\\text{out}}$ coincide at the wall of marginal stability.\n\\end{lemma} \n\\begin{proof}\n First let $a$ approach $W$ from the ``in'' region, so we're using definition \\eqref{newmp}. Start with the pair of functions $(\\mathcal{X}_e, \\mathcal{X}_m)$ in the $\\zeta$-plane and let $\\widetilde{\\mathcal{X}}_e$ denote the analytic continuation of $\\mathcal{X}_e$. See Figure \\ref{jumpxe}. When they reach the $\\ell_e$ ray, $\\mathcal{X}_e$ jumped to $\\mathcal{X}_e(1-\\mathcal{X}_m)$ by \\eqref{kjump} and \\eqref{invjmp}. Thus $\\mathcal{X}_e = \\widetilde{\\mathcal{X}}_e(1-\\mathcal{X}_m)$ along the $\\ell_e$ ray.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{2raysafterwalljumps2.eps}\n\t\\caption{Jump of $\\mathcal{X}_e$}\n\t\\label{jumpxe}\n\\end{figure}\n\nTherefore, \n\\begin{equation*}\n\\psi_{\\text{in}}(a,\\theta) = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e(1-\\mathcal{X}_m)\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}(1-\\mathcal{X}_m)^{-1}\\right](te^{i\\arg a})} \\right|\n\\end{equation*}\n\nNow starting from the ``out'' region, and focusing on the wall-crossing of type I for the moment, we start with the pair $(\\mathcal{X}_e, \\mathcal{X}_m)$ as before. This time, $\\mathcal{X}_e$ at the $\\ell_e$ ray has not gone to any jump yet. See Figure \\ref{jumpxem}. Only $\\mathcal{X}_{e+m}$ undergoes a jump at the $\\ell_{e+m}$ ray and it is of the form $\\mathcal{X}_{e+m} \\mapsto \\mathcal{X}_{e+m}(1-\\mathcal{X}_{e})^{-1}$.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.60\\textwidth]{3raysbeforewalljumps2.eps}\n\t\\caption{Only $\\mathcal{X}_{e+m}$ has a jump}\n\t\\label{jumpxem}\n\\end{figure}\n\nWhen $a$ hits the wall $W$, $\\varphi = \\arg a$ and the integrals are taken over the same ray. Thus, we can combine the logs and obtain:\n\n \\begin{align}\n \\psi_{\\text{out}}(a,\\theta) & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\left\\{ \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}\\right](te^{i\\arg a})} \\right| + \\log\\left| \\frac{1 - \\left[\\mathcal{X}_{e+m}(1-\\mathcal{X}_{e})^{-1}\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e-m}(1-\\mathcal{X}_{e})\\right](te^{i\\arg a})} \\right| \\right\\} \\notag\\\\\n & = \\frac{1}{2\\pi}\\int_0^1 \\frac{dt}{t} \\log\\left| \\frac{1 - \\left[\\mathcal{X}_e(1-\\mathcal{X}_m)\\right](-te^{i\\arg a})}{1 - \\left[\\mathcal{X}_{-e}(1-\\mathcal{X}_m)^{-1}\\right](te^{i\\arg a})} \\right|\n \\end{align}\n \n\\noindent and the two definitions coincide. For the wall-crossing of type II the proof is entirely analogous.\n\n \n\n\\end{proof}\n \n \\begin{theorem}\\label{homeps}\n $Q$ is a reparametrization in $\\theta_m$; that is, a diffeomorphism of $\\mathbb{R}\/2\\pi \\mathbb{Z}$.\n \\end{theorem}\n \\begin{proof}\n To show that $Q$ is injective, it suffices to show that $\\left|\\frac{\\partial \\psi}{\\partial \\theta_m}\\right| < 1$. We will show this in the $\\mathcal{B}_{\\text{in}}$ region. The proof for the $\\mathcal{B}_{\\text{out}}$ region is similar.\n \n To simplify the calculations, write\n \\begin{equation}\\label{psii}\n \\psi(a,\\theta) = 2\\int_0^1 \\frac{dt}{t} \\log\\left| \\frac{1-Cf(\\theta_m)}{1-Cg(\\theta_m)}\n \\right|\n \\end{equation}\n for functions $f, g$ of the form $e^{i\\Upsilon_\\gamma}$ for different choices of $\\gamma$ (they both depend on other parameters, but they're fixed here) and a factor $C$ of the form\n \\[ C = \\exp\\left( -\\pi R |a| (t^{-1} + t)\\right) \\]\n Now take partials in both sides of \\eqref{psii} and bring the derivative inside the integral. After an application of the chain rule we get the estimate\n \\[ \\left| \\frac{\\partial \\psi}{\\partial \\theta_m} \\right| \\leq 2\\int_0^1 \\frac{dt}{t} |C| \\left\\{ \\frac{|f||\\frac{\\partial \\Upsilon_e(t)}{\\partial \\theta_m}|}{|1-Cf|} + \\frac{|g||\\frac{\\partial \\Theta_e(-t)}{\\partial \\theta_m}|}{|1-Cg|} \\right\\} \\] \n By the estimates in \\cite[\\S 3.2]{rhprob}, $\\left|\\frac{\\partial \\Upsilon_e}{\\partial \\theta_m}\\right| < 1$. In \\cite[Lemma 3.2]{rhprob}, we show that $|f|, |g|$ can be bounded by 2. The part $C$ has exponential decay so if $R$ is big enough we can bound the above by 1 and injectivity is proved. For surjectivity, just observe that $\\psi(\\theta_m + 2\\pi) = \\psi(\\theta_m)$, so $Q(\\theta_m + 2\\pi) = \\theta_m + 2\\pi$. \n \\end{proof}\n \nWith respect to the new coordinate $\\theta'_m$, the functions $\\Upsilon_e, \\Upsilon_m$ satisfy the equation:\n\n\\begin{align}\n\\Upsilon_e(a,\\zeta) = \\theta_e +\n\\frac{1}{4\\pi}\\sum_{\\gamma'} \\Omega(\\gamma';a) \\left\\langle \\gamma_e, \\gamma' \\right\\rangle & \\int_{\\gamma'} \\frac{d\\zeta'}{\\zeta'} \\frac{\\zeta' + \\zeta}{\\zeta' -\\zeta} \\log\\left[ 1 - \\mathcal{X}_{\\gamma'}^{\\text{sf}}(a,\\zeta', \\Upsilon_{\\gamma'}) \\right] \\label{inteq1} \\\\\n\\Upsilon_m(a,\\zeta) = \\theta'_m +\n\\frac{1}{2\\pi}\\sum_{\\gamma'} \\Omega(\\gamma';a) \\left\\langle \\gamma_m, \\gamma' \\right\\rangle \\left\\{ \\vphantom{\\int_0^b} \\right.\n& \\int_{0}^{b'} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - \\mathcal{X}_{\\gamma'}^{\\text{sf}}(a,\\zeta', \\Upsilon_{\\gamma'}) \\right] + \\notag \\\\\n& \\left. \\int_{b'}^{b' \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - \\mathcal{X}_{\\gamma'}^{\\text{sf}}(a,\\zeta', \\Upsilon_{\\gamma'}) \\right] \\right\\} \\label{inteq2} ,\n\\end{align}\nfor $b'$ the intersection of the unit circle with the $\\ell_{\\gamma'}$ ray. The $\\Omega(\\gamma';a)$ jump at the wall, but in the Pentagon case, the sum is finite.\n\nIn order to show that $\\Upsilon$ converges to some function, even at $a = 0$, observe that the integral equations in \\eqref{inteq1} and \\eqref{inteq2} still make sense at the singular fiber, since in the case of \\eqref{inteq1}, $\\lim_{a \\to 0} Z_m = c \\neq 0$ and the exponential decay is still present, making the integrals convergent. In the case of \\eqref{inteq2}, the exponential decay is gone, but the different kernel makes the integral convergent, at least for $\\zeta \\in \\mathbb{C}^\\times$. The limit function $\\lim_{a \\to 0} \\Upsilon$ should be then a solution to the integral equations obtained by recursive iteration, as in \\cite[\\S 3]{rhprob}.\n\nWe have to be specially careful with the Cauchy integral in \\eqref{inteq2}. It will be better to obtain each iteration $\\Upsilon^{(\\nu)}_m$ when $|a| \\to 0$ by combining the pair of rays $\\ell_{\\gamma'}, \\ell_{-\\gamma'}$ into a single line $L_{\\gamma'}$, where in the case of the Pentagon, $\\gamma'$ can be either $\\gamma_e$ or $\\gamma_{e+m}$, depending on the side of the wall we're at. We formulate a boundary problem over each infinite curve $L_{\\gamma'}$ as in \\S \\ref{altrh}. As in the Ooguri-Vafa case, the jump function\\footnote{Since we do iterations of boundary problems, we abuse notation and use simply $G(\\zeta)$ where it should be $G^{(\\nu)}(\\zeta)$. This shouldn't cause any confusion, as our main focus in this section is how to obtain \\emph{any} iteration of $\\mathcal{X}_m$} $G(\\zeta)$ has discontinuities of the first kind at 0 and $\\infty$, but we also have a new difficulty: For $\\theta_e$ close to 0, the jump function $G(\\zeta) = 1-e^{i\\Upsilon^{(\\nu - 1)}_{\\gamma'}}(\\zeta)$ may be 0 for some values of $\\zeta$.\n\nSince the asymptotics of $\\Upsilon^{(\\nu)}_{e}$ as $\\zeta \\to 0$ or $\\zeta \\to \\infty$ are $\\theta_e \\pm i\\phi_e \\neq 0$, the jump function $G(\\zeta)$ can only attain the 0 value inside a compact interval away from 0 or $\\infty$, hence these points are isolated in $L_{\\gamma'}$. By the symmetry relation expressed in Lemma \\ref{corresp}, the zeroes of $G(\\zeta)$ come in pairs in $L_{\\gamma'}$ and are of the form $\\zeta_k, -1\/\\overline{\\zeta_k}$. By our choice of orientation for $L_{\\gamma'}$, one of the jumps is inverted so that $G(\\zeta)$ has only zeroes along $L_{\\gamma'}$ and no poles.\n\nThus, as in \\S \\ref{altrh}, we have a Riemann-Hilbert problem of the form\\footnote{To simplify notation, we omit the iteration index $\\nu$ in the Riemann-Hilbert problem expressed. By definition, $\\mathcal{X}_m = \\mathcal{X}^{\\text{sf}}_m X_m$, for \\emph{any} iteration $\\nu$}\n\\begin{equation}\\label{rhpent}\nX_m^+(\\zeta) = G(\\zeta) X_m^-(\\zeta)\n\\end{equation}\n\nIn \\cite[Lemma 4.2]{rhprob}, we show that the solutions of \\eqref{rhpent} exist and are unique, given our choice of kernel in \\eqref{inteq2}. We thus obtain each iteration $\\Upsilon_m^{(\\nu)}$ of \\eqref{inteq2}. Moreover, since by \\cite{rhprob}, $\\mathcal{X}_m^+ = 0$ at points $\\zeta$ in the $L_e$ ray where $G(\\zeta) = 0$, $\\Upsilon_m^{(\\nu)+}$ has a logarithmic singularity at such points.\n\n\n\n\n\\subsection{Estimates and a new gauge transformation}\n\n\n\n\n\n\n\n\n\n\n\n\n\nAs we've seen in the Ooguri-Vafa case, we expect our solutions $\\lim_{a \\to 0} \\Upsilon$ to be unbounded in the $\\zeta$ variable.\nDefine a Banach space $\\mathrsfs{X}$ as the completion under the sup norm of the space of functions $\\Phi: \\mathbb{C}^\\times \\times \\mathbb{T} \\times U \\to \\mathbb{C}^{2n}$ that are piecewise holomorphic on $\\mathbb{C}^\\times$, smooth on $\\mathbb{T} \\times U$, for $U$ an open subset of $\\mathcal{B}$ containing $0$ and such that \\eqref{inteq1}, \\eqref{inteq2} hold.\n\nLike in the Ooguri-Vafa case, let $a \\to 0$ fixing $\\arg a$. We will later get rid of this dependence on $\\arg a$ with another gauge transformation of $\\theta_m$. The following estimates on $\\Upsilon^{(\\nu)}$ will clearly give us that the sequence converges to some limit $\\Upsilon^{(\\nu)}$.\n\n\\begin{lemma}\\label{estbadfib}\nIn the Pentagon case, at the bad fiber $a = 0$:\n\\begin{align}\n\\Upsilon_e^{(\\nu + 1)} & = \\Upsilon_e^{(\\nu)} + O\\left( e^{-2\\pi \\nu R |Z_m|} \\right), \\hspace{5 mm} \\nu \\geq 2 \\label{elstm}\\\\\n\\Upsilon_m^{(\\nu + 1)} & = \\Upsilon_m^{(\\nu)} + O\\left( e^{-2\\pi \\nu R |Z_m|} \\right), \\hspace{5 mm} \\nu \\geq 1 \\label{mgstm}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nAs before, we prove this by induction. Note that $\\Upsilon^{(1)}_m = \\Upsilon^{\\text{OV}}$, the extension of the Ooguri-Vafa case obtained in \\eqref{xmnice}, and $\\Upsilon^{(1)}_m$ differs considerably from $\\theta_m$ because of the $\\log \\zeta$ term. Hence the estimates cannot start at $\\nu = 0$. Because of this reason, $\\Upsilon^{(2)}_e$ differs considerably from $\\Upsilon^{(1)}_e$ since this is the first iteration where $\\Upsilon^{(1)}_m$ is considered.\n\nLet $\\nu = 1$. The integral equations for $\\Upsilon_e$ didn't change in this special case. By Lemma 3.3 in \\cite{rhprob}, we have for the general case:\n\n\\begin{equation}\\label{frappx}\n\\Upsilon^{(1)}_e = \\theta_e + \\sum_{\\gamma'} \\Omega(\\gamma',a) \\left\\langle \\gamma_e, \\gamma' \\right\\rangle \\frac{e^{-2\\pi R |Z_{\\gamma'}|}}{4\\pi i \\sqrt{R |Z_{\\gamma'}|}}\\frac{\\zeta_{\\gamma'} + \\zeta}{\\zeta_{\\gamma'} - \\zeta} e^{i\\theta_{\\gamma'}} + O\\left( \\frac{e^{-2\\pi R |Z_{\\gamma'}|}}{R}\\right)\n\\end{equation}\nwhere $\\zeta_{\\gamma'} = -\\frac{Z_{\\gamma'}}{|Z_{\\gamma'}|}$ is the saddle point for the integrals in \\eqref{inteq1}, and $\\zeta$ is not $\\zeta_{\\gamma'}$. Note that there is no divergence if $\\zeta \\to 0$ or $\\zeta \\to \\infty$. If $\\zeta = \\zeta_{\\gamma'}$, again by Lemma 3.3 in \\cite{rhprob}, we obtain estimates as in \\eqref{frappx} except for the $\\sqrt{R}$ terms in the denominator.\n\nIn any case, for the Pentagon, the $\\gamma'$ in \\eqref{frappx} are only $ \\gamma_{\\pm m}, \\gamma_{\\pm (e+m)}$, depending on the side of the wall of marginal stability. At $a = 0$, $Z_{e+m} = Z_m$, so \\eqref{frappx} gives that $\\log[1 - e^{i \\Upsilon^{(1)}_e}] = \\log[1 - e^{i\\theta_e}] + O(e^{-2\\pi R |Z_m|})$ along the $\\ell_e$ ray, and a similar estimate holds for $\\log[1 - e^{-i \\Upsilon^{(1)}_e}]$ along the $\\ell_{-e}$ ray. Plugging in this in \\eqref{inteq2}, we get \\eqref{mgstm} for $\\nu = 1$.\n\nFor general $\\nu$, a saddle point analysis on $\\Upsilon^{(\\nu)}_e$ can still be performed and obtain as in \\eqref{frappx}:\n\\begin{equation}\\label{appgnu}\n\\Upsilon^{(\\nu+1)}_e = \\theta_e + \\frac{e^{-2\\pi R |Z_m|}}{4\\pi i \\sqrt{R |Z_m|}} \\left\\{ \\frac{\\zeta_m + \\zeta}{\\zeta_m - \\zeta} e^{i\\Upsilon^{(\\nu)}_m(\\zeta_m)} - \\frac{\\zeta_m - \\zeta}{\\zeta_m + \\zeta} e^{-i\\Upsilon^{(\\nu)}_m(-\\zeta_m)} \\right\\} + O\\left( \\frac{e^{-2\\pi R |Z_{\\gamma'}|}}{R}\\right),\n\\end{equation}\nfrom one side of the wall. On the other side (for type I) it will contain the extra terms\n\\begin{equation}\n\\frac{e^{-2\\pi R |Z_m|}}{4\\pi i \\sqrt{R |Z_m|}} \\left\\{ \\frac{\\zeta_m + \\zeta}{\\zeta_m - \\zeta} e^{i(\\Upsilon^{(\\nu)}_m(\\zeta_m) + \\Upsilon^{(\\nu)}_e(\\zeta_m))} - \\frac{\\zeta_m - \\zeta}{\\zeta_m + \\zeta} e^{-i(\\Upsilon^{(\\nu)}_m(-\\zeta_m) - \\Upsilon^{(\\nu)}_e(-\\zeta_m))} \\right\\}.\n\\end{equation}\n\\noindent Observe that for this approximation we only need $\\Upsilon^{(\\nu)}$ at the point $\\zeta_m$. By the previous part, for $\\nu = 2$,\n\\[ e^{i\\Upsilon^{(2)}_m(\\zeta_m)} = e^{i\\Upsilon^{(1)}_m(\\zeta_m)} \\left(1 + O\\left( e^{-2\\pi R |Z_m|}\\right) \\right) \\]\nThus, for $\\nu = 2$,\n\\begin{align}\n\\Upsilon^{(3)}_e & = \\theta_e + \\frac{e^{-2\\pi R |Z_m|}}{4\\pi i \\sqrt{R |Z_m|}} \\left\\{ \\frac{\\zeta_m + \\zeta}{\\zeta_m - \\zeta} e^{i\\Upsilon^{(1)}_m(\\zeta_m)} \\left(1 + O\\left( e^{-2\\pi R |Z_m|}\\right) \\right) \\right. \\notag\\\\\n& - \\left. \\frac{\\zeta_m - \\zeta}{\\zeta_m + \\zeta} e^{-i\\Upsilon^{(1)}_m(-\\zeta_m)} \\left(1 + O\\left( e^{-2\\pi R |Z_m|}\\right) \\right) \\right\\} + O\\left( R^{1\/2}\\right)\\notag\\\\\n& = \\Upsilon^{(2)}_e + O\\left( e^{-4\\pi R|Z_m|}\\right) \\label{appgnu2}\n\\end{align}\nand similarly in the other side of the wall. For general $\\nu$, the same arguments show that \\eqref{elstm}, \\eqref{mgstm} hold after the appropriate $\\nu$.\n\\end{proof}\n\n\n\n\nThere is still one problem: the limit of $\\widetilde{\\mathcal{X}}_m$ we obtained as $a \\to 0$ for the analytic continuation of $\\mathcal{X}_m$ was only along a fixed ray $\\arg a = $ constant. To get rid of this dependence, it is necessary to perform another gauge transformation on the torus coordinates $\\theta$. Recall that we are restricted to the Pentagon case. Let $a \\to 0$ fixing $\\arg a$. Let $\\zeta_{\\gamma}$ denote $Z_{\\gamma}\/|Z_{\\gamma}|$. In particular, $\\zeta_e = a\/|a|$ and this remains constant since we're fixing $\\arg a$. Also, $\\zeta_m = Z_m\/|Z_m|$ and this is independent of $\\arg a$ since $Z_m$ has a limit as $a \\to 0$. The following lemma will allow us to obtain the correct gauge transformation.\n \n\\begin{lemma}\nFor the limit $\\left. \\widetilde{\\mathcal{X}}_m\\right|_{a=0}$ obtained above, its imaginary part is independent of the chosen ray $\\arg a = c$ along which $a \\to 0$.\n\\end{lemma}\n\\begin{proof}\nLet $\\widetilde{\\Upsilon}_m$ denote the analytic continuation of $\\Upsilon_m$ yielding $\\widetilde{\\mathcal{X}}_m$. Start with a fixed value $\\arg a \\equiv \\rho_0$, for $\\rho_0$ different from $\\arg Z_m(0), \\arg (-Z_m(0))$. For another ray $\\arg a \\equiv \\rho$, we compute $\\left. \\Upsilon_m\\right|_{\\substack{a=0\\\\ \\arg a = \\rho}} - \\left. \\Upsilon_m\\right|_{\\substack{a=0\\\\ \\arg a = \\rho_0}}$ (without analytic continuation for the moment).\n\nThe integrals in \\eqref{inteq2} are of two types. One type is of the form\n\\begin{equation}\\label{1sttype}\n\\int_0^{\\zeta_{\\pm e}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - e^{i\\Upsilon_{\\pm e}(\\zeta')} \\right] + \\int_{\\zeta_{\\pm e}}^{\\zeta_{\\pm e}\\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - e^{i\\Upsilon_{\\pm e}(\\zeta')} \\right]\n\\end{equation}\nThe other type appears only in the outside part of the wall of marginal stability. Since $Z : \\Gamma \\to \\mathbb{C}$ is a homomorphism, $Z_{\\gamma_e + \\gamma_m} = Z_{\\gamma_e} + Z_{\\gamma_m}$. At $a = 0$, $Z_e = a = 0$, so $Z_{e+m} = Z_m$. Hence, $\\ell_m = \\ell_{e+m}$ at the singular fiber. This second type of integral is thus of the form\n\\begin{equation}\\label{2ndtype}\n\\int_0^{\\zeta_{\\pm m}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - e^{i\\Upsilon_{\\pm (e+m)}(\\zeta')} \\right] + \\int_{\\zeta_{\\pm m}}^{\\zeta_{\\pm m}\\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - e^{i\\Upsilon_{\\pm (e+m)}(\\zeta')} \\right]\n\\end{equation}\nSince the $\\ell_m$ stays fixed at $a = 0$ independently of $\\arg a$, \\eqref{2ndtype} does not depend on $\\arg a$, so this has a well-defined limit as $a \\to 0$. We should focus then only on integrals of the type \\eqref{1sttype}. For a different $\\arg a$, $\\zeta_e$ changes to another point $\\widetilde{\\zeta}_e$ in the unit circle. See Figure \\ref{zetilde}. The paths of integration change accordingly. We have two possible outcomes: either $\\zeta$ lies outside the sector determined by the two paths, or $\\zeta$ lies inside the region.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{argadependence.ps}\n\t\\caption{As $\\arg a$ changes, the paths of integration change}\n\t\\label{zetilde}\n\\end{figure}\n\nIn the first case ($\\zeta_1$ on Figure \\ref{zetilde}), the integrands\n\\begin{equation}\\label{2kernl}\n \\frac{\\log[1-e^{i\\Upsilon_{\\pm e}(\\zeta')}]}{\\zeta'-\\zeta}, \\hspace{5 mm} \\frac{\\zeta\\log[1-e^{i\\Upsilon_{\\pm e}(\\zeta')}]}{\\zeta'(\\zeta'-\\zeta)}\n\\end{equation}\nare holomorphic on $\\zeta'$ in the sector between the two paths. By Cauchy's formula, the difference between the two integrals is just the integration along a path $C_{\\pm e}$ between the two endpoints $\\zeta_{\\pm e}, \\widetilde{\\zeta}_{\\pm e}$. If $f(s)$ parametrizes the path $C_e$, let $C_{-e} = -1\/\\overline{f(s)}$. The orientation of $C_{e}$ in the contour containing $\\infty$ is opposite to the contour containing 0. Similarly for $C_{-e}$. Thus, the difference of $\\Upsilon_m$ for these two values of $\\arg a$ is the integral along $C_{e}, C_{-e}$ of the difference of kernels \\eqref{2kernl}, namely:\n\\begin{equation}\\label{realdiff}\n\\int_{C_e} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{i\\Upsilon_e(\\zeta')}] - \\int_{C_{-e}} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{-i\\Upsilon_e(\\zeta')}]\n\\end{equation} \n\nEven if $e^{i\\Upsilon_e(\\zeta')} = 1$ for $\\zeta'$ in the contour, the integrals in \\eqref{realdiff} are convergent, so this is well-defined for any values of $\\theta_e \\neq 0$. By symmetry of $C_e, C_{-e}$ and the reality condition \\eqref{corresp}, the second integral is the conjugate of the first one. Thus \\eqref{realdiff} is only real.\n\nWhen $\\zeta$ hits one of the contours, $\\zeta$ coincides with one of the $\\ell_e$ or $\\ell_{-e}$ rays, for some value of $\\arg a$. The contour integrals jump since $\\zeta$ lies now inside the contour ($\\zeta_2$ in Figure \\ref{zetilde}). The jump is by the residue of the integrands \\eqref{2kernl}. This gives the jump of $\\mathcal{X}_m$ that the analytic continuation around $a = 0$ cancels. Therefore, only the real part of $\\Upsilon_m$ depends on $\\arg a$.\n\\end{proof}\n\nBy the previous lemma, $\\left. \\widetilde{\\Upsilon}_m\\right|_{\\substack{a=0\\\\ \\arg a = \\rho}} - \\left. \\widetilde{\\Upsilon}_m\\right|_{\\substack{a=0\\\\ \\arg a = \\rho_0}}$ is real and is given by \\eqref{realdiff}. Define then a new gauge transformation:\n\\begin{equation}\\label{fingauge}\n\\widetilde{\\theta}_m = \\theta'_m - \\frac{1}{2\\pi} \\left\\{\\int_{C_e} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{i\\Upsilon_e(\\zeta')}] + \\int_{C_{-e}} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{-i\\Upsilon_e(\\zeta')}] \\right\\}\n\\end{equation}\n\nThis eliminates the dependence on $\\arg a$ for the limit $\\left. \\widetilde{\\mathcal{X}}_m \\right|_{a=0}$. As we did in \\S \\ref{clasov} in Theorem \\ref{mprtom}, we can extend the torus fibration $\\mathcal{M}'$ by gluing a $S^1$-fiber bundle of the form $D \\times (0, 2\\pi) \\times S^1$ for $D$ a disk around $a = 0$, $\\theta_e \\in (0,2\\pi)$ and $\\widetilde{\\theta}_m$ the new coordinate of the $S^1$ fibers. Using Taub-NUT space as a local model for this patch, the trivial $S^1$ bundle can be extended to $\\theta_e = 0$ where the fiber degenerates into a point (nevertheless, in Taub-NUT coordinates the space is still locally isomorphic to $\\mathbb{C}^2$). Since $\\widetilde{\\mathcal{X}}_m \\equiv 0$ if $\\theta_e = 0$ as in \\S \\ref{clasov}, in this new manifold $\\mathcal{M}$ we thus obtain a well defined function $\\widetilde{\\mathcal{X}}_m$.\n\n\n\n\\subsection{Extension of the derivatives}\\label{exderv}\n\\index{Extension of the derivatives@\\emph{Extension of the derivatives}}%\n\nSo far we were able to extend the functions $\\mathcal{X}_e, \\widetilde{\\mathcal{X}}_m$ to $\\mathcal{M}$. Unfortunately, we can no longer bound uniformly on $\\nu$ the derivatives of $\\widetilde{\\mathcal{X}}_m$ near $a = 0$, so the Arzela-Ascoli arguments no longer work here. Since there's no difference on the definition of $\\mathcal{X}_e$ at $a = 0$ from that of the regular fibers, this function extends smoothly to $a = 0$.\n\nWe have to obtain the extension of all derivatives of $\\widetilde{\\mathcal{X}}_m$ directly from its definition. It suffices to extend the derivatives of $\\mathcal{X}_m$ only, as the analytic continuation doesn't affect the symplectic form $\\varpi(\\zeta)$ (see below).\n\n\\begin{lemma}\n$\\log \\mathcal{X}_m$ extends smoothly to $\\mathcal{M}$, for $\\theta_e \\neq 0$.\n\\end{lemma}\n\\begin{proof}\nFor convenience, we rewrite $\\Upsilon_m$ with the final magnetic coordinate $\\widetilde{\\theta_m}$:\n\\begin{align*}\n\\Upsilon_m & = \\widetilde{\\theta_m} + \\frac{1}{2\\pi} \\left\\{\\int_{C_e} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{i \\Upsilon_e(\\zeta')}] -\n\\int_{C_{-e}} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{-i \\Upsilon_e(\\zeta')}] \\right\\}\\\\\n& + \\frac{1}{2\\pi}\\sum_{\\gamma'} \\Omega(\\gamma';a) \\left\\langle \\gamma_m, \\gamma' \\right\\rangle \\left\\{ \\vphantom{\\int_0^b} \\int_{0}^{\\zeta_{\\gamma'}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - \\mathcal{X}_{\\gamma'}^{\\text{sf}}(a,\\zeta', \\Upsilon_{\\gamma'}) \\right] \\right. + \\\\\n& \\left. \\int_{\\zeta_{\\gamma'}}^{\\zeta_{\\gamma'} \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - \\mathcal{X}_{\\gamma'}^{\\text{sf}}(a,\\zeta', \\Upsilon_{\\gamma'}) \\right] \\right\\}\n\\end{align*}\nwhere $e^{i \\Upsilon_e(\\zeta')}$ is evaluated only at $a = 0$. For $\\gamma'$ of the type $\\pm \\gamma_e \\pm \\gamma_m$, $\\mathcal{X}_{\\gamma'}$ and its derivatives still have exponential decay along the $\\ell_{\\gamma'}$ ray, so these parts in $\\Upsilon_m$ extend to $a =0$ smoothly. It thus suffices to extend only\n\n\\begin{align}\n\\Upsilon_m & = \\widetilde{\\theta_m} + \\frac{1}{2\\pi} \\left\\{\\int_{C_e} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{i \\Upsilon_e(\\zeta')}] -\n\\int_{C_{-e}} \\frac{d\\zeta'}{\\zeta'} \\log[1-e^{-i \\Upsilon_e(\\zeta')}] \\right. \\notag\\\\\n& + \\int_{0}^{\\zeta_{e}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - \\mathcal{X}_{e}^{\\text{sf}}(a,\\zeta', \\Upsilon_{e}) \\right] + \\int_{\\zeta_{e}}^{\\zeta_{e} \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - \\mathcal{X}_{e}^{\\text{sf}}(a,\\zeta', \\Upsilon_{e}) \\right] \\notag\\\\\n& -\\left. \\int_{0}^{-\\zeta_{e}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\log\\left[ 1 - {\\mathcal{X}_{e}^{\\text{sf}}}^{-1}(a,\\zeta', -\\Upsilon_{e}) \\right] - \\int_{-\\zeta_{e}}^{-\\zeta_{e} \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\log\\left[ 1 - {\\mathcal{X}_{e}^{\\text{sf}}}^{-1}(a,\\zeta', -\\Upsilon_{e}) \\right]\\right\\}\\label{upsmcompl}\n\\end{align}\ntogether with the semiflat part $\\pi R \\frac{Z_m}{\\zeta} + \\pi R \\zeta \\overline{Z_m}$, which we assume is as in the Generalized Ooguri-Vafa case, namely:\n\\begin{equation}\\label{xmcomplet}\n \\mathcal{X}_m = \\exp\\left( \\frac{-i R }{2\\zeta}(a\\log a - a + f(a)) + i \\Upsilon_m + \\frac{i\\zeta R}{2} (\\overline{a} \\log\n\\overline{a} - \\overline{a} + \\overline{f(a)} )\\right)\n\\end{equation}\nfor a holomorphic function $f$ near $a = 0$ and such that $f(0) \\neq 0$. The derivatives of the terms involving $f(a)$ clearly extend to $a = 0$, so we focus on the rest, as in \\S \\ref{c1ext}.\n\nWe show first that $\\dfrac{\\partial \\log \\mathcal{X}_m}{\\partial_{\\theta_e}}, \\dfrac{\\partial \\log \\mathcal{X}_m}{\\partial_{\\theta_m}}$ extend to $a = 0$. Since there is no difference in the proof between electric or magnetic coordinates, we'll denote by $\\partial_\\theta$ a derivative with respect to any of these two variables.\n\nWe have:\n\\begin{align*}\n\\frac{\\partial}{\\partial \\theta}\\log \\mathcal{X}_m & = \\frac{-i}{2\\pi} \\left\\{ \\int_{C_e} \\frac{d\\zeta'}{\\zeta'} \\frac{e^{i \\Upsilon_e(\\zeta')}}{1-e^{i \\Upsilon_e(\\zeta')}} \\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} - \\int_{C_{-e}} \\frac{d\\zeta'}{\\zeta'} \\frac{e^{-i \\Upsilon_e(\\zeta')}}{1-e^{-i \\Upsilon_e(\\zeta')}} \\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} \\right.\\\\\n& + \\int_{0}^{\\zeta_{e}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} + \\int_{\\zeta_{e}}^{\\zeta_{e} \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta}\\\\\n& \\left. + \\int_{0}^{-\\zeta_{e}} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} + \\int_{-\\zeta_{e}}^{\\zeta_{e} \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} \\right\\}\n\\end{align*}\n\n\nwhen $a \\to 0$, $ \\dfrac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')} \\to \\dfrac{e^{i \\Upsilon_e(\\zeta')}}{1-e^{i \\Upsilon_e(\\zeta')}}$. The integrals along $C_e$ and $C_{-e}$ represent a difference of integrals along the contour in the last integrals and a fixed contour, as in Figure \\ref{zetilde}. Thus, when $a = 0$,\n\n\\begin{align}\n\\left. 2\\pi i \\frac{\\partial}{\\partial \\theta}\\log \\Upsilon_m\\right|_{a =0} & = \\int_{0}^{b} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} + \\int_{b}^{b \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} \\notag\\\\\n& \\left. + \\int_{0}^{-b} \\frac{d\\zeta'}{\\zeta' - \\zeta} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} + \\int_{-b}^{-b \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\frac{\\partial \\Upsilon_e(\\zeta')}{\\partial \\theta} \\right\\} \\label{polec}\n\\end{align}\nfor a fixed point $b$ in the unit circle, independent of $a$. If $\\Upsilon_e(\\zeta') = 1$ for a point $c$ in the line $L$ passing through the origin and $b$, then as seen in \\cite[Lemma 4.2]{rhprob}, the function $\\mathcal{X}_m$ develops a zero on the right side of such line. Nevertheless, the analytic continuation $\\widetilde{\\mathcal{X}}_m$ around $a = 0$ introduces a factor of the form $(1 - \\mathcal{X}_e)^{-1}$ when $a$ changes from region III to region I in Figure \\ref{3reg}, so the pole at $c$ on the right side of $L$ for the derivative $\\dfrac{\\partial}{\\partial \\theta}\\log \\Upsilon_m$ coming from the integrand in \\eqref{polec} is canceled by analytic continuation. Hence, the integrals are well defined and thus the left side has an extension to $a = 0$.\n\nNow, for the partials with respect to $a, \\overline{a}$, there are two different types of dependence: one is the dependence of the contours, the other is the dependence of the integrands. The former dependence is only present in \\eqref{upsmcompl}, as the contours in Figure \\ref{zetilde} change with $\\arg a$. A simple application of the Fundamental Theorem of Calculus in each integral in \\eqref{upsmcompl} gives that this change is:\n\n\\begin{align*}\n\\left. -2\\pi i \\frac{\\partial}{\\partial \\arg a}\\log \\Upsilon_m\\right|_{a =0} & = \\log[1-e^{-i \\Upsilon_e(\\zeta_e)}] - \\log[1-e^{-i \\Upsilon_e(\\zeta_e)}]\\\\\n& - \\log[1-e^{-i \\Upsilon_e(\\zeta_e)}] + \\log[1-e^{-i \\Upsilon_e(\\zeta_e)}] = 0,\n\\end{align*}\nwhere we again used the fact that the integrals along $C_e$ and $C_{-e}$ represent the difference between the integrals in the other pairs with respect to two different rays, one fixed. By continuity on parameters, the terms are still 0 if $\\Upsilon_e(\\zeta_e) = 0$. Compare this with \\eqref{dap1}, where we obtained this explicitly.\n\nThen there is the dependence on $a, \\overline{a}$ on the integrands and the semiflat part. Focusing on $a$ only, we take partials on $\\log \\mathcal{X}_m$ in \\eqref{xmcomplet} (ignoring constants and parts that clearly extend to $a = 0$). This is:\n\n\\begin{equation}\\label{logadiv}\n\\frac{\\log a}{\\zeta} + \\int_0^{\\zeta_e} \\frac{d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}_e}{1-\\mathcal{X}_e} + \\int_0^{-\\zeta_e} \\frac{d\\zeta'}{\\zeta'(\\zeta' - \\zeta)} \\frac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}\n\\end{equation}\n\nThis is the equivalent of \\eqref{last3} in the general case. In the limit $a \\to 0$, we can do an asymptotic expansion of $\\dfrac{e^{i \\Upsilon_e(\\zeta')}}{1-e^{i \\Upsilon_e(\\zeta')}} = \\dfrac{e^{i \\Upsilon_e(0)}}{1-e^{i \\Upsilon_e(0)}} + O(\\zeta')$. Clearly when we write this expansion in \\eqref{logadiv}, the only divergent term at $a = 0$ is the first degree approximation in the integral. Thus, we can focus on that and assume that the $\\dfrac{\\mathcal{X}_e}{1-\\mathcal{X}_e}$ (resp. $\\dfrac{\\mathcal{X}^{-1}_e}{1-\\mathcal{X}^{-1}_e}$) factor is constant. If we do the partial fraction decomposition, we can run the same argument as in Eqs. \\eqref{simpler} up to \\eqref{cancel} and obtain that \\eqref{logadiv} is actually 0 at $a = 0$. The only identity needed is\n\\[ \\frac{1}{1-e^{i\\Upsilon_e(0)}} + \\frac{1}{1-e^{-i\\Upsilon_e(0)}} = 1 \\]\n\nThe argument also works for the derivative with respect to $\\overline{a}$, now with an asymptotic expansion around $\\infty$ of $\\Upsilon_e$.\n\nThis shows that $\\widetilde{\\mathcal{X}_m}$ extends in a $C^1$ way to $a = 0$. For the $C^\\infty$ extension, derivatives with respect to any $\\theta$ coordinate work in the same way, all that was used was the specific form of the contours $C_e, C_{-e}$. The same thing applies to the dependence on the contours $C_e, C_{-e}$. For derivatives with respect to $a, \\overline{a}$ in the integrands, we can again do an asymptotic expansion of $\\Upsilon_e$ at 0 or $\\infty$ and compare it to the asymptotic of the corresponding derivative of $a \\log a - a$ as $a \\to 0$.\n\n\\end{proof}\n\nNothing we have done in this section is particular of the Pentagon example. We only needed the specific values of $\\Omega(\\gamma;u)$ given in \\eqref{omgpar} to obtain the Pentagon identities at the wall and to perform the analytic continuation of $\\mathcal{X}_m$ around $u = 2$. For any integrable systems data as in section \\ref{intsys} with suitable invariants $\\Omega(\\gamma;u)$ allowing the wall-crossing formulas and analytic continuation, we can do the same isomonodromic deformation of putting all the jumps at a single admissible ray, perform saddle-point analysis and obtain the same extensions of the Darboux coordinates $\\mathcal{X}_\\gamma$. This finishes the proof of Theorem \\ref{extbf}.\n\nWhat is exclusive of the Pentagon case is that we have a well-defined hyperk\\\"{a}hler metric $g_\\text{OV}$ that we can use as a local model of the metric to be constructed here.\n\nThe extension of the holomorphic symplectic form $\\varpi(\\zeta)$ is now straightforward. We proceed as in \\cite{gaiotto} by first writing:\n\\begin{equation*}\n\\varpi(\\zeta) = -\\frac{1}{4\\pi^2 R} \\frac{d\\mathcal{X}_e}{\\mathcal{X}_e} \\wedge \\frac{d\\mathcal{X}_m}{\\mathcal{X}_m}\n\\end{equation*} \n\nWhere we used the fact that the jumps of the functions $\\mathcal{X}_\\gamma$ are via the symplectomorphisms $\\mathcal{K}_{\\gamma'}$ of the complex torus $T_a$ (see \\eqref{kjump}) so $\\varpi(\\zeta)$ remains the same if we take $\\mathcal{X}_m$ or its analytic continuation $\\widetilde{\\mathcal{X}_m}$.\n\nWe need to show that $\\varpi(\\zeta)$ is of the form\n\\begin{equation}\n-\\frac{i}{2\\zeta}\\omega_+ + \\omega_3 -\\frac{i \\zeta}{2} \\varpi_-\n\\end{equation}\nthat is, $\\varpi(\\zeta)$ must have simple poles at $\\zeta = 0$ and $\\zeta = \\infty$, even at the singular fiber where $a = 0$.\n\nBy definition, $\\mathcal{X}_e = \\exp(\\frac{\\pi R a}{\\zeta} + i\\Upsilon_e + \\pi R \\zeta \\overline{a})$. Thus\n\n\\begin{equation*}\n\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} = \\frac{\\pi R da}{\\zeta} + i d\\Upsilon_e(\\zeta) +\\pi R \\zeta d\\overline{a}\n\\end{equation*}\n\nBy \\eqref{inteq1}, and since $\\lim_{a \\to 0} Z_m \\neq 0$, $\\mathcal{X}_m$ (resp. $\\mathcal{X}_{-m}$) of the form $\\exp(\\frac{\\pi R Z_m(a)}{\\zeta} + i\\Upsilon_m + \\pi R \\zeta \\overline{Z_m(a)})$ still has exponential decay when $\\zeta$ lies in the $\\ell_m$ ray (resp. $\\ell_{-m}$), even if $a = 0$. The differential $d \\Upsilon_e(\\zeta)$ thus exists for any $\\zeta \\in \\mathbb{P}^1$ since the integrals defining it converge for any $\\zeta$.\n\nAs in \\cite{gaiotto}, we can write\n\\[ \\frac{d\\mathcal{X}_e}{\\mathcal{X}_e} \\wedge \\frac{d\\mathcal{X}_m}{\\mathcal{X}_m} = \\frac{d\\mathcal{X}_e}{\\mathcal{X}_e} \\wedge \\left( \\frac{d\\mathcal{X}^{\\text{sf}}_m}{\\mathcal{X}^{\\text{sf}}_m} + \\mathcal{I_{\\pm}} \\right), \\]\nfor $\\mathcal{I}_\\pm$ denoting the corrections to the semiflat function. By the form of $\\mathcal{X}^{\\text{sf}} = \\exp(\\frac{\\pi R Z_m(a)}{\\zeta} + i\\theta_m + \\pi R \\zeta \\overline{Z_m(a)})$, the wedge involving only the semiflat part has only simple poles at $\\zeta = 0$ and $\\zeta = \\infty$, so we can focus on the corrections. These are of the form\n\\begin{align*}\n\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\mathcal{I_{\\pm}} & = \\frac{-i}{2\\pi} \\left\\{ \\int_0^{\\zeta_e} \\frac{d\\zeta'}{\\zeta'-\\zeta} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')} \\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\frac{d\\mathcal{X}_e(\\zeta')}{\\mathcal{X}_e(\\zeta')}\\right.\\\\\n& + \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta'-\\zeta)}\\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\frac{d\\mathcal{X}_e(\\zeta')}{\\mathcal{X}_e(\\zeta')}\\\\\n& + \\int_0^{-\\zeta_e} \\frac{d\\zeta'}{\\zeta'-\\zeta} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')} \\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\frac{d\\mathcal{X}_e(\\zeta')}{\\mathcal{X}_e(\\zeta')}\\\\\n& + \\left. \\int_{-\\zeta_e}^{-\\zeta_e \\infty} \\frac{\\zeta d\\zeta'}{\\zeta'(\\zeta'-\\zeta)}\\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\frac{d\\mathcal{X}_e(\\zeta')}{\\mathcal{X}_e(\\zeta')} \\right\\}\n\\end{align*}\n\nIn the ``inside'' part of the wall of marginal stability. A similar equation holds in the other side. We can simplify the wedge products above by taking instead\n\\begin{equation}\n\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} \\wedge \\left(\\frac{d\\mathcal{X}_e(\\zeta)}{\\mathcal{X}_e(\\zeta)} - \\frac{d\\mathcal{X}_e(\\zeta')}{\\mathcal{X}_e(\\zeta')} \\right) = \\pi R \\left[ \\left( \\frac{1}{\\zeta} - \\frac{1}{\\zeta'}\\right)da + (\\zeta - \\zeta')d\\overline{a}\\right] +i \\left( d\\Phi_e(\\zeta) - d\\Phi_e(\\zeta') \\right)\n\\end{equation}\n\nRecall that $\\Phi_e$ represents the corrections to $\\theta_e$, so $\\Upsilon_e = \\theta_e + \\Phi_e$. By \\S \\ref{solut}, $\\Phi_e$ and $d\\Phi_e$ are defined for $\\zeta = 0$ $\\zeta = \\infty$ even if $a = 0$, since $\\lim_{a \\to 0} Z_m(a) \\neq 0$ and the exponential decay in $\\mathcal{X}_m^\\text{sf}$ still present guarantees convergence of the integrals in \\ref{inteq1}. Hence, the terms involving $d\\Phi_e(\\zeta) - d\\Phi_e(\\zeta')$ are holomorphic for any $\\zeta \\in \\mathbb{P}^1$. It thus suffices to consider the other terms. After simplifying the integration kernels, we obtain\n\n\\begin{align*}\n\\frac{\\pi R da}{\\zeta} \\int_0^{\\zeta_e} \\frac{d\\zeta'}{\\zeta'} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')} & +\\pi R da \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{d\\zeta'}{(\\zeta')^2} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\\\\n\\frac{\\pi R da}{\\zeta} \\int_0^{-\\zeta_e} \\frac{d\\zeta'}{\\zeta'} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')} & +\\pi R da \\int_{-\\zeta_e}^{-\\zeta_e \\infty} \\frac{d\\zeta'}{(\\zeta')^2} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\\\\n-\\pi R d\\overline{a} \\int_0^{\\zeta_e} d\\zeta' \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')} & -\\pi R \\zeta d\\overline{a} \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{d\\zeta'}{\\zeta'} \\frac{\\mathcal{X}_e(\\zeta')}{1-\\mathcal{X}_e(\\zeta')}\\\\\n-\\pi R d\\overline{a} \\int_0^{\\zeta_e} d\\zeta' \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')} & -\\pi R \\zeta d\\overline{a} \\int_{\\zeta_e}^{\\zeta_e \\infty} \\frac{d\\zeta'}{\\zeta'} \\frac{\\mathcal{X}^{-1}_e(\\zeta')}{1-\\mathcal{X}^{-1}_e(\\zeta')}\\\\\n\\end{align*}\n\nThe only dependence on $\\zeta$ is in the factors $\\zeta, 1\/\\zeta$. Thus $\\varpi(\\zeta)$ has only simple poles at $\\zeta = 0$ and $\\zeta = \\infty$.\n\nFinally, the estimates in Lemma \\ref{estbadfib} show that if we recover the hyperk\\\"{a}hler metric $g$ from the holomorphic symplectic form $\\varpi(\\zeta)$ as in \\S \\ref{extmetric} and \\S \\ref{genextmtr}, we obtain that the hyperk\\\"{a}hler metric for the Pentagon case is the metric obtained in \\ref{extmetric} for the Ooguri-Vafa case plus smooth corrections near $a = 0, \\theta_e = 0$, so it extends to this locus.\n\n This gives Theorem \\ref{smfrm}.\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nA triple $(E,+,\\circ)$, where $(E,+)$ and $(E, \\circ)$ are groups is said to be a left skew brace if \n$$a \\circ (b+c)=a\\circ b-a+a \\circ c$$\nholds for all $a,b,c \\in E$, where $-a$ denotes the inverse of $a$ in $(E, +)$. In 2007, Rump \\cite{WR07} introduced classical braces to study involutive and non-degenerate solutions of the Yang-Baxter equation. Later, Guarnieri and Vendramin \\cite{GV17} generalized this concept to skew brace to study the non-degenerate solution of the Yang-Baxter equation, which is further generalized to semi-braces by Catino, Colazzo, and Stefanelli in \\cite{FMP21} to study non-bijective solutions of the Yang-Baxter equation. In \\cite{DG16}, Ben David and Ginosar investigated extensions of bijective $1$-cocycles. Carter, Elhambadi and Satio in \\cite{CES} developed homology and cohomology theories for solution sets of the Yang-Baxter equations. Different homology theories for various structures related to solutions of the Yang-Baxter equations were investigated extensively by Lebed and Vendramin \\cite{LV17}. Cohomology and extensions of linear cycle sets with trivial actions is studied by Lebed and Vendramin \\cite{LV16}. Recently generalized by Jorge A. Guccione, Juan J. Guccione and Christian Valqui \\cite{GG21} to non trivial actions. Various type of products like matched product, semi-direct product, asymmetric product has been defined for the solutions of Yang-Baxter equation [see \\cite{DB18}, \\cite{BCJO19}, \\cite{CCS}, \\cite{CCS1}, ,\\cite{CCS2}, \\cite{WR08}]. In \\cite{NMY}, M. K. Yadav and author developed the theory of skew brace extensions for skew brace extensions by an abelian group and developed the Well's type exact sequence for skew braces. This work can be thought as a generalization of \\cite{DB18}, \\cite{LV17} at the level of extensions. The fundamental exact sequence of Wells for groups was introduced by C Wells in \\cite{W71}. The fundamental exact sequence of Wells with various applications is carried out in all fine details in \\cite[Chapter 2]{PSY18}. A similar exact sequence for cohomology, extensions and automorphisms of quandles was constructed in \\cite{BS20}.\n In this paper, we define a new product for the skew braces and construct few examples. We give constructions for skew braces similar to that of group theory and generalize the Well's type exact sequence for the trivial skew brace.\n\n\n\n\n\\section{Preliminaries}\n\nAn algebraic structure $(E, + , \\circ)$ is said to be a \\emph{left skew brace} if $(E, +)$ and $(E, \\circ)$ are a group and the following compatibility condition holds:\n\\begin{equation}\\label{bcomp}\na \\circ (b + c ) = a \\circ b -a + a\\circ c\n\\end{equation}\nfor all $ a, b , c \\in E$, where $-a$ denotes the inverse of $a$ with respect to `$+ $'. \nNotice that the identity element $0$ of $(E, +)$ coincides with the identity element of $(E, \\circ)$.\n\nFor a left skew brace $E$ and $a \\in E$, define a map $\\lambda_a : E \\to E$ by\n$$\\lambda_a(b) = -a + (a \\circ b)$$\nfor all $b \\in E$. The automorphism group of a group $G$ is denoted by $\\operatorname{Aut} (G)$. \nWe have the following result for skew braces .\n\\begin{lemma}\nLet $(E,+, \\circ)$ be a left skew brace, then for each $a \\in E$, the map $\\lambda_a$ is an automorphism of $(E, +)$ and the map $\\lambda : (E, \\circ) \\to \\operatorname{Aut} (E, +)$ given by $\\lambda(a) = \\lambda_a$ is a group homomorphism.\n\\end{lemma}\n\nA sub skew brace $I$ of a left skew brace $E$ is said to be a \\emph{left ideal} of $E$ if $\\lambda_a(y) \\in I$ for all $a \\in E$ and $y \\in I$. A left ideal of $E$ is said to be an \\emph{ideal} if $(I, \\circ)$ is a normal subgroup of $(E, \\circ)$. The Socle of a skew brace $E$ is defined as $\\operatorname{Soc} (E) = \\operatorname{Ker} \\lambda$ $\\cap $ $\\operatorname{Z} (E, +)$, where $\\operatorname{Z} (E, +)$ represents the centre of the group $(E, +)$ and the annihilator of $E$ is defined as $\\operatorname{Ann} (E)=\\operatorname{Soc} (E) \\cap \\operatorname{Z} (E, \\circ)$.\n\n\nThe following is an easy but important observation, which will be used several times in what follows.\n\\begin{lemma}\nLet $E$ be a left skew brace. Then for all $a, b \\in E$, the following hold:\n\n(i) $a + b = a \\circ \\lambda^{-1}_a(b)$.\n\n(ii) $a \\circ b = a + \\lambda_a(b)$.\n\\end{lemma}\n\n\nLet $E_1$ and $E_2$ be two left skew braces. A map $f : E_1 \\to E_2$ is said to be a \\emph{skew brace homomorphism} if $f(a + b) = f(a) + f(b)$ and $f(a\\circ b) = f(a) \\circ f(b)$ for all $a, b \\in E_1$. A one-to-one and onto skew brace homomorphism from $E_1$ to itself is called an \\emph{automorphism} of $E_1$. The \\emph{kernel} of a homomorphism $f : E_1 \\to E_2$ is defined to be the subset $\\{a \\in E_1 \\mid f(a) = 0\\}$ of $E_1$. It turns out that $\\operatorname{Ker} (f)$, the kernel of $f$, is an ideal of $E_1$. The set of all skew brace automorphisms of a left skew brace $E$, denoted by $\\operatorname{Autb} (E)$, is a group.\n\n\n\nLet $H$ and $I$ be two left skew braces. By an \\emph{extension} of $H$ by $I$, we mean a left skew brace $E$ with an exact sequence \n$$\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to} E \\stackrel{\\pi}{\\to} H \\to 0,$$ \nwhere $i$ and $\\pi$ are injective and surjective brace homomorphisms, respectively. Thereafter, we denote the image of $y$ under $i$ by $y$ itself for all $y \\in I$. A set map $s : H \\to E$ is called a \\emph{set-theoretic section} of $\\mathcal{E}$ if $\\pi(s(h)) = h$ for all $h \\in H$ and $s(0) = 0$. The abbreviation `st-section' will be used for `set-theoretic section' throughout. We call $\\mathcal{E}$ to be split exact sequence of skew braces if there exist a st-section of $\\mathcal{E}$ which is a skew brace homomorphism. \n\n\\section{split extensions of skew brace}\n\nLet $H$ and $I$ be two left skew braces. Let $ \\mu: (H, +) \\rightarrow Aut(I, +)$, $\\sigma : (H, \\circ) \\rightarrow Aut(I, \\circ)$ be anti-homomorphisms, and $\\nu: (H, \\circ) \\rightarrow Aut (I, +)$ be a homomorphism. Let $\\mu, \\sigma $ and $\\nu$ satisfy the following compatibility condition\n\\noindent \\begin{align}\\label{SE}\n\\nu_{h_1\\circ(h_2 + h_3)}(\\sigma_{h_2 + h_3}(\\nu^{-1}_{h_1}(y_1)) \\circ \\nu^{-1}_{h_2 + h_3}(\\mu_{h_3}(y_2)+ y_3))& = \\mu_{-h_1+(h_2 \\circ h_3)}(\\nu_{h_1 \\circ h_2}(\\sigma_{h_2}(\\nu^{-1}_{h_1}(y_1)) \\circ \\nu^{-1}_{h_2}(y_2))-y_1) \\notag \\\\\n& + \\nu_{h_1 \\circ h_3}(\\sigma_{h_3}(\\nu^{-1}_{h_1}(y_1)) \\circ \\nu^{-1}_{h_3}(y_3)) \n \\end{align}\n \nfor all $y_1, y_2 , y_3 \\in I $ and $ h_1, h_2, h_3 \\in H$.\n\n\\begin{thm}\nLet $H$ and $I$ be two skew braces with $(\\nu ,\\mu,\\sigma),$ as defined above and satisfying \\eqref{SE}, then the operations \n\\begin{align}\n (h_1, y_1) + (h_2, y_2)&=(h_1 + h_2, \\mu_{h_1}(y_1) + y_2), \\label{sb+} \\\\\n (h_1, y_1) \\circ (h_2, y_2)&=(h_1 \\circ h_2, \\nu_{h_1 \\circ h_2}(\\sigma_{h_2}(\\nu^{-1}_{h_1}(y_1)) \\circ \\nu^{-1}_{h_2}(y_2)) \\label{sbcirc}\n\\end{align}\ndefine a left skew brace structure on $ H \\times I $.\n\\end{thm}\n\\begin{proof}\nIt is easy to check that the given operations define group structure on $H \\times I$ and the condition \\eqref{bcomp} follows from the compatibility condition of $(\\nu, \\mu, \\sigma)$. \n\\hfill $\\Box$\n\n\\end{proof}\n\nWe call this structure a \\emph{split semi-direct product} of $H$ by $I$ with respect to the triplet $( \\nu,\\mu, \\sigma)$ and denote it by $(H, I, \\nu,\\mu, \\sigma)$.\n\n\\begin{lemma}\nLet $(H, I, \\nu,\\mu, \\sigma)$ be split semi-direct product of $H$ by $I$ with respect to some triplet $(\\nu,\\mu,\\sigma)$. Then the following short exact sequence of skew braces\n$$ \\mathcal{E} := 0 \\to I \\stackrel{i}{\\to} (H, I, \\nu, \\mu,\\sigma) \\stackrel{\\pi}{\\to} H \\to 0 $$\n splits, where $i$ and $\\pi$ are natural injection and projection respectively.\n\\end{lemma}\n\n\\begin{proof}\nIt is easy to check that the map $ s: H \\rightarrow (H, I, \\mu, \\sigma,\\nu)$ given by $s(h)=(h,0)$ is both a homomorphism of skew braces and a st-section of $\\mathcal{E}$ simultaneously. \n\\hfill $\\Box$\n\n\\end{proof}\n\n\\begin{thm}\nLet $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to} E \\stackrel{\\pi}{\\to} H \\to 0$ be a split short exact sequence of skew braces. Then $E$ is a split semi-direct product of $H$ by $I$.\n\\end{thm}\n\n\\begin{proof}\nLet the short exact sequence $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to} E \\stackrel{\\pi}{\\to} H \\to 0$ split. Then there exists a st-section $s :H \\rightarrow E$, which is also a skew brace homomorphism. Define $\\mu: H \\longrightarrow Aut(I, +)$, $\\sigma : H \\rightarrow Aut(I, \\circ)$, and $\\nu : H \\rightarrow Aut(I, +) $ by\n\\begin{align}\\label{actions}\n\\nu_h(y) & =-s(h) + (s(h) \\circ y),\\notag\\\\\n\\mu_h(y) & =-s(h)+ y+ s(h), \\\\\n\\sigma_h(y) & =s(h)^{-1} \\circ y \\circ s(h).\\notag\n\\end{align}\nSince $E$ is a skew brace, we have \n\\begin{equation}\\label{sbc1}\n(s(h_1) + y_1) \\circ \\big{(}s(h_2) + y_2 + s(h_3) +y_3 \\big{)} = (s(h_1)+y_1) \\circ (s(h_2) + y_2) - (s(h_1) + y_1) + (s(h_1) + y_1) \\circ (s(h_3) + y_3). \n\\end{equation}\nUsing \\eqref{sbc1} and linearity of $s$ in `$+$' and `$\\circ$', we can easily establish that $(\\nu, \\mu, \\sigma)$ satisfies \\eqref{SE}. Hence we have semi-direct product $(H,I,\\mu,\\sigma,\\nu)$. We know that every element $ x \\in E$ can be uniquely written as $x=s(h)+ y$. Define $\\phi : E \\rightarrow (H,I,\\mu,\\sigma,\\nu)$ by $\\phi(s(h)+ y)=(h, y)$. Then $\\phi$ is an isomorphism of skew braces and the diagram\n$$\\begin{CD}\n 0 @>i>> I @>>> E @>{{\\pi} }>> H @>>> 0\\\\\n && @V{\\text{Id}}VV @V{\\phi}VV @ VV{ \\text{Id}}V \\\\\n 0 @>i'>> I @>>> (H,I,\\nu,\\mu,\\sigma) @>{{\\pi^\\prime} }>> H @>>> 0\n\\end{CD}$$\ncommutes, where $i^\\prime$ and $\\pi^\\prime$ are natural injection and projection, respectively. This completes the proof.\n\n\\hfill $\\Box$\n\n\\end{proof}\n\n\\section{Examples}\n\n\nIn this section we provide some examples of split semi direct product of skew braces. We have used GAP to compute $\\nu, \\mu$ and $\\sigma$.\n\n\\noindent \\textbf{Example 1} Let $\\mathbb{Z}$ and $\\mathbb{C}$ be trivial skew braces, Define $\\nu, \\mu, \\sigma : \\mathbb{Z} \\rightarrow Aut(\\mathbb{C})$ by $\\nu_1(x)=\\mu_1(x)=\\sigma_1(x)=-x$. Using \\eqref{sb+} and \\eqref{sbcirc}, we can define a skew brace structure on $\\mathbb{Z} \\times \\mathbb{C}$ by \n\\begin{align*}\n(l, \\ y_1)+(m, \\ y_2) & = \\big(l+m, \\ (-1)^{m+n}y_{1}+y_2\\big),\\\\\n(l, \\ y_1)\\circ(m, \\ y_2) & = \\big(l+m, \\ y_1+(-1)^{m+n}y_2\\big).\n\\end{align*}\n\n\n\\noindent \\textbf{Example 2} Let $H=D_{2n}= \\langle a,b\\hspace{.1cm}|\\hspace{.1cm} a^{2n}=b^2=e, bab=a^{-1}\\rangle$ and $I=\\mathbb{Z}_p$ be trivial skew braces, where $D_{2n}$ and $\\mathbb{Z}_p$ denotes dihederal group of order $4n$ and cyclic group of order $p$ respectively. Define $\\nu, \\mu, \\sigma : D_{2n} \\rightarrow Aut(\\mathbb{Z}_p)$ by $\\nu_a(x)=\\mu_a(x)=\\sigma_a(x)=-x$ and $\\nu_b(x)=\\mu_b(x)=\\sigma_b(x)=-x$. Using \\eqref{sb+} and \\eqref{sbcirc}, we can define a skew brace structure on $D_{2n} \\times \\mathbb{Z}_p$ by \n\\begin{align*}\n(a^{i}b^{j}, y_1)+(a^{m} b^{n}, y_2) & = \\big(a^{i}b^{j}a^{m} b^{n}, \\ y_2+(-1)^{m+n}y_{1}\\big),\\\\\n(a^{i}b^{j}, y_1)\\circ(a^{m} b^{n}, y_2) & = \\big(a^{i}b^{j}a^{m} b^{n}, \\ y_1+(-1)^{i+j}y_2\\big).\n\\end{align*}\nIf we have trivial skew brace $H=D_n$, where $n$ is odd and $I$ be the same as above, then we can define $\\nu, \\mu, \\sigma : D_{n} \\rightarrow Aut(\\mathbb{Z}_p)$ by $\\nu_a(x)=\\mu_a(x)=\\sigma_a(x)=x$ and $\\nu_b(x)=\\mu_b(x)=\\sigma_b(x)=-x$.\n\n\\\n\n\\noindent \\textbf{Example 3} Let $H=\\mathbb{Z}_8$ be trivial skew brace and $I=\\mathbb{Z}_3 \\times \\mathbb{Z}_2$ be skew brace of order $6$ defined in \\cite{EM20} by the following operations\n\\begin{align*}\n (n,m)+(s,t) & =(n+2^{m}s, m+t),\\\\\n(n,m) \\circ (s,t) & =(2^{t}n+2^{m}s,m+t).\n\\end{align*}\nWe have $(I,+)=\\langle (1,0), (0,1) \\rangle \\cong S_3$ and $(I, \\circ)=\\langle (1,1) \\rangle \\cong \\mathbb{Z}_6$. We take $\\mu_{a}(n,m) = (n,m)$, $ \\sigma_{a}(n,m)=(n,m)^{-1}=(2n,m), $ $\\nu_{a}(n,m)=(2n,m) $, for all $(n,m) \\in I$, where $a$ is a generator of $H$. Hence the additive group of skew brace structure on $(H, I, \\nu, \\mu, \\sigma)$ is just direct product of their respective additive groups and multiplicative group is given as follows\n\\begin{eqnarray*}\n(a^k, (n,m)) \\circ (a^l,(s,t))=(a^{k+l}, ((2n,m) \\circ (2s,t)^{k}).\n\\end{eqnarray*} \n\\\n\n\n\\noindent \\textbf{Example 4} Let $H$ be brace of order $4$ defined in \\cite{DB15} by $(H,+)=\\mathbb{Z}_2 \\times \\mathbb{Z}_2$, $(H, \\circ)=\\langle(0,1) \\rangle \\cong \\mathbb{Z}_4$ and $I$ be a brace such that $(I,+)=\\mathbb{Z}_4$, $(I,\\circ)=\\langle 1,2 \\rangle \\cong \\mathbb{Z}_2 \\times \\mathbb{Z}_2 $. Consider $\\mu_x=\\operatorname{Id}$ for all $x \\in H$, where $\\operatorname{Id}$ denotes the identity mapping on $I$ and $\\nu_{(0,1)}(x)=\\sigma_{(0,1)}(x)=-x$. Then split semi- direct product of $I$ by $H$ is given by the skew brace with additive group as direct product of $H$ and $I$ and multiplicative group structure is given as follows\n\\begin{eqnarray*}\n\\big((0,1)^k, l) \\circ ((0,1)^n, m\\big)=\\big((0,1)^{k+n}, l + (-1)^{k}m+(-1)^{n}lm\\big).\n\\end{eqnarray*}\n\\\n\n\\noindent \\textbf{Example 5} Let $H$ be the brace of order $8$ with additive group $\\mathbb{Z}_8$ having Socle of order $2$ and $(H,\\circ)=\\langle 1,2 \\rangle \\cong \\mathbb{Z}_4 \\times \\mathbb{Z}_2$ defined in \\cite{DB15} and $I$ be brace of order $4$ as defined in Example 3. Then we have total $8$ different split semi-direct products of $I$ by $H$, interestingly with $\\mu_x=Id$ for all $x \\in H$ in all cases. We list few cases\n\\begin{enumerate}\n\\item[(i)]\n$\\nu_1(x)=x^{-1}, \\hspace{.1cm} \\nu_2(x)=x \\hspace{.1cm} and \\hspace{.1cm} \\sigma_h(x)=x \\hspace{.1cm}for \\hspace{.1cm}all \\hspace{.1cm} x \\in I, \\hspace{.1cm} h \\in H. $\\\\\n\n\n\\item[(ii)]\n$\\nu_1(x)=x^{-1}, \\hspace{.1cm} \\nu_2(x)=x^{-1} \\hspace{.1cm} and \\hspace{.1cm} \\sigma_2(2)=3 ,\\hspace{.1cm} \\sigma_1(x)=x \\hspace{.1cm} for \\hspace{.1cm}all \\hspace{.1cm} x \\in I.$ \n\n\\end{enumerate}\n\n\n\n\n\n\n\\section{general extensions of skew braces}\n\nLet $(H, +, \\circ)$ and $(I, + , \\circ) $ be two skew braces, $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to} E \\stackrel{\\pi}{\\to} H \\to 0$ be an extension of $H$ by $I$. and let $s : H \\rightarrow E$ be an st-section of $\\mathcal{E}$. Corresonding to $s$, consider the pair $(\\beta, \\tau)$, where $\\beta$ and $\\tau$ are defined as \n\\begin{align}\n\\beta(h_1, h_2) &:= - s(h_1 + h_2) + s(h_1) + s(h_2),\\label{cocycle1 sb}\\\\\n\\tau(h_1, h_2) &:= s(h_1 \\circ h_2)^{-1} \\circ s(h_1) \\circ s(h_2).\\label{cocycle2 sb}\n\\end{align} \nIt is easy to see that $ \\nu, \\mu $ and $ \\sigma$, defined in \\eqref{actions}, need not be homomorphisms in general, but they satisfy the following identities\n\n\\begin{align}\n\\nu_{h_1 \\circ h_2}&=\\nu_{h_1} \\nu_{h_2} \\lambda^{-1}_{\\tau(h_1, h_2)},\\label{action1 }\\\\\n \\mu_{h_1 + h_2} &= i^{+}_{-\\beta(h_1, h_2)} \\mu_{h_2} \\mu_{h_1}, \\label{action2 }\\\\\n\\sigma_{h_1 \\circ h_2}&=i^{\\circ}_{\\tau(h_1, h_2)^{-1}} \\sigma_{h_2} \\sigma_{h_1}, \\label{action3}\n \\end{align}\nwhere \n \n\\begin{align}\ni^{+}_y(z)&:=y+z-y, \\\\\ni^{\\circ}_y(z)&:=y \\circ z \\circ y^{-1},\n \\end{align}\nare inner automorphisms of $(H,+)$ and $(H, \\circ)$, respectively, and $\\beta$ and $\\tau$ are as defined above in \\eqref{cocycle1 sb} and \\eqref{cocycle2 sb}.\n \nLet $N$ be the smallest normal subgroup of $\\operatorname{Aut} (I, +)$ generated by the set $\\{ \\lambda_y \\hspace{.1cm}| \\hspace{.1cm} y \\in I\\}$. Let $ \\operatorname{Inn} (I , +)$ and $ \\operatorname{Inn} (I, \\circ)$ be the inner automorphism subgroups of $\\operatorname{Aut} (I, +)$ and $\\operatorname{Aut} (I, \\circ)$ respectively. Then we have the maps $\\bar{\\nu} : (H, \\circ ) \\rightarrow \\operatorname{Aut} (I, +)\/N$, $\\bar{\\mu} : (H, +) \\rightarrow \\operatorname{Aut} (I, )\/ \\operatorname{Inn} (I, +)$ and $\\bar{\\sigma} : (H, \\circ) \\rightarrow \\operatorname{Aut} (I, \\circ)\/ \\operatorname{Inn} (I, \\circ)$ defined by $\\nu , \\mu$ and $\\sigma$ composing with natural projections respectively. We call the triplet $\\chi:=(\\nu, \\mu, \\sigma)$ satisfying (\\ref{action1 }), (\\ref{action2 }), and (\\ref{action3}) an action of $H$ on $I$ and corresponding triplet $\\bar{\\chi}:=(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})$ will be called a coupling from $H$ to $I$ corresponding to $(\\nu,\\mu, \\sigma)$. Let $\\chi =(\\nu, \\mu, \\sigma)$ and $\\chi^\\prime=(\\nu^\\prime, \\mu^\\prime, \\sigma^\\prime)$ be two actions. Then we say that $\\bar{\\chi}^\\prime\\approx\\bar{\\chi}$ if there exists a map $\\theta: H \\rightarrow I$ such that $\\theta(0)=0$ and $\\nu^{\\prime}_{h}=\\nu_{h} \\lambda_{\\theta(h)}$, $\\mu^\\prime_{h}=i^+_{\\nu_h(-\\theta(h))} \\mu_h$ and $\\sigma^\\prime=i^{\\circ}_{\\theta(h)^{-1}} \\sigma_{h}.$\n\n{\\bf{Remark:}} If $\\bar{\\chi}^\\prime\\approx\\bar{\\chi}$ then $\\bar{\\chi}^\\prime =\\bar{\\chi}$ but converse need not be true. Note that the map $\\theta : H \\rightarrow I$ mentioned above need not be unique. \n\nWith this setting, we have\n\\begin{prop}\\label{well-def-act-coc}\nLet $0 \\to I \\stackrel{}{\\to} E \\stackrel{\\pi}{\\rightarrow} H \\to 1$ be an extension of a left skew brace $I$ by $H$. Then the following hold:\n \n(1) The coupling $\\bar{\\chi}$ is independent of the choice of an st-section. \n \n(2) Equivalent extensions have the same coupling.\n\n\\end{prop}\n\\begin{proof}\n(1) Let $s_1$ and $s_2$ be two st-sections of $\\pi$. We know that two sections differ by an element of $I$, hence for an element $h \\in H$, there exist $y_h \\in I$ such that $s_2(h)=s_1(h)\\circ y_h$. Let $\\chi=(\\nu, \\mu , \\sigma)$ and $\\chi^\\prime=(\\nu^\\prime, \\mu^\\prime, \\sigma^\\prime)$ be actions corresponding to $s_1$ and $s_2$ respectively. Define $\\theta: H \\rightarrow I$ be $\\theta(h)=y_h$. It can be easily seen that $\\bar{\\chi}^\\prime \\approx \\bar{\\chi}$ using $\\theta$ as a required map.\n \n(2) Let $E$ and $E^\\prime$ be two equivalent extensions. Then there exist a skew brace homomorphism $\\phi: E^\\prime \\rightarrow E$ such that the following diagram commutes\n$$\\begin{CD}\n 0 @>>> I @>>> E^\\prime @>{{\\pi^\\prime} }>> H @>>> 0\\\\\n && @V{\\text{Id}}VV @V{\\phi}VV @ VV{ \\text{Id}}V \\\\\n 0 @>>> I @>>> E @>{{\\pi} }>> H @>>> 0.\n\\end{CD}$$\nLet $s : H \\rightarrow E^\\prime$ be any st-section of the extension $E^\\prime$. Then $\\phi s : H \\rightarrow E$ is a st-section of extension the $E$. Let $\\chi=(\\nu, \\mu, \\sigma)$ be actions of $E$ corresponding to $\\phi s$ and $\\chi^\\prime=(\\nu^\\prime, \\mu^\\prime, \\sigma^\\prime)$ be actions of $E^\\prime$ corresponding to $ s$, respectively. Then we have $\\nu=\\nu^\\prime$, $\\mu = \\mu^\\prime$ and $\\sigma=\\sigma^\\prime$. Hence $\\bar{\\chi} \\approx \\bar{\\chi^\\prime}$ by taking $\\theta: H \\rightarrow I$ to be $\\theta(h)=0$ for all $h \\in H$. As we have already proved that coupling is independent of an st-section, so this holds for all st-sections of $E$ and $E^\\prime$.\n\\hfill $\\Box$\n\n\\end{proof}\nLet $\\operatorname{Ext} (H,I)$ denote the set of equivalence classes of all extensions of $H$ by $I$. Equivalence class of an extension $\\mathcal{E} : 0 \\to I \\to E \\to H \\to 0$ is denoted by $[\\mathcal{E}]$. As a consequence of the preceding proposition, it follows that each equivalence class of extension of $H$ by $I$ admits a unique coupling $\\bar{\\chi}=(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})$ corresponding to actions $\\chi=(\\nu, \\mu, \\sigma)$ of $H$ on $I$. Let $\\operatorname{Ext} _{(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})}(H, I)$ denote the equivalence class of those extensions of $H$ by $I$ whose corresponding coupling is $(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})$. We can easily establish\n \n\n\\begin{cor}\\label{cor 1}\n $\\operatorname{Ext} (H, I) = \\bigsqcup_{(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})} \\operatorname{Ext} _{(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})}(H, I)$.\n \\end{cor}\n\n\n\\begin{prop}\\label{prop 2}\nLet $\\mathcal{E}$ be a extension of $H$ by $I$. Then the following hold \\\\\n\n\\begin{itemize}\n\\item[1)]\nLet $s$ be an st-section of $\\mathcal{E}$. Then the pair $(\\beta, \\tau)$ corresponding to $s$ together with action defined in \\eqref{actions} satisfies\n\n\\begin{equation}\\label{cocycle 1}\n\\beta(h_1, h_2+h_3)+\\beta(h_2, h_3)-\\beta(h_1+h_2, h_3)-\\mu_{h_3}(\\beta(h_1, h_2))=0,\n\\end{equation}\nand \n\\begin{equation}\\label{cocycle 2}\n\\tau(h_1, h_2 \\circ h_3) \\circ \\tau(h_2, h_3) \\circ \\tau(h_1 \\circ h_2, h_3)^{-1} \\circ (\\sigma_{h_3}(\\tau(h_1, h_2)))^{-1}=0.\n\\end{equation}\n\n\\item[2)]\nLet $s_1$ and $s_2$ be two st-sections of $\\mathcal{E}$, and let $(\\beta_1, \\tau_1)$ and $(\\beta_2, \\tau_2)$ be the pairs corresponding to $s_1$ and $s_2$, respectively. Let $(\\prescript{}{1}{\\nu}, \\prescript{}{1}{\\mu}, \\prescript{}{1}{\\sigma})$ and $(\\prescript{}{2}{\\nu}, \\prescript{}{2}{\\mu}, \\prescript{}{2}{\\sigma})$ be actions corresponding to $s_1$ and $s_2$, respectively. Then there exists a map $\\theta : H \\rightarrow I$ such that \n$$\ns_2(h)=s_1(h) \\circ \\theta(h)=s_1(h) + \\prescript{}{1}{\\nu}_{h}(\\theta(h)),\n$$ \n\\begin{equation}\\label{equi 1}\n\\prescript{}{1}{\\nu}_{h_1+h_2}(-\\theta(h_1+h_2))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta(h_1)))+\\prescript{}{1}{\\nu}_{h_2}(\\theta(h_2))=\\beta_2(h_1, h_2),\n\\end{equation}\nand\n\\begin{equation}\\label{equi 2}\n\\theta(h_1 \\circ h_2)^{-1} \\circ \\tau_1(h_1, h_2) \\circ \\prescript{}{1}{\\sigma}_{h_2}(\\theta(h_1)) \\circ \\theta(h_2)=\\tau_2(h_1, h_2),\n\\end{equation}\nfor all $h, h_1, h_2 \\text{ and } h_3 \\in H$.\\\\\n\n\\item[3)]\nLet $\\mathcal{E}_1$ and $\\mathcal{E}_2$ be two equivalent extensions of $H$ by $I$, and let $s_1$ and $s_2$ be st-sections of $\\mathcal{E}_1$ and $\\mathcal{E}_2$, respectively. Let $(\\beta_1, \\tau_1)$ and $(\\beta_2, \\tau_2)$ be the pairs corresponding to $s_1$ and $s_2$, respectively. Then there exists a map $\\theta : H \\rightarrow I$ satisfying \\eqref{equi 1} and \\eqref{equi 2}.\n\n\\end{itemize}\n\n\n\n\\end{prop}\n\\begin{proof}\nIt is easy to see that (1) and (2) follows directly from definitions. Now we will prove (3). \nSince $\\mathcal{E}_1 := 0 \\to I \\stackrel{}{\\to} E_1 \\stackrel{\\pi_1}{\\to} H \\to 0$ and $\\mathcal{E}_2:= 0 \\to I \\stackrel{i}{\\to} E_2 \\stackrel{\\pi_2}{\\to} H \\to 0$ are two equivalent extensions, there exists an isomorphism $\\phi : E_1 \\rightarrow E_2$ such that the following diagram commutes\n$$\n\\begin{CD}\n 0 @>>> I @>>> E_1 @>{{\\pi_1} }>> H @>>> 0\\\\\n && @V{\\text{Id}}VV @V{\\phi}VV @ VV{ \\text{Id}}V \\\\\n 0 @>>> I @>>> E_2 @>{{\\pi_2} }>> H @>>> 0.\n\\end{CD}\n$$\n\nLet $s_1$ be an st-section of $\\mathcal{E}_1$. Then $\\phi s_1$ is an st-section of $\\mathcal{E}_2$. Let $(\\beta_1, \\tau_1)$ and $(\\beta^\\prime, \\tau^\\prime)$ be the pairs corresponding to $s_1$ and $\\phi s_1$, respectively. By the commutativity of the above diagram, we have $\\phi(y)=y$, for all $y \\in I$, hence $\\beta_1=\\beta^\\prime$ and $\\tau_1=\\tau^\\prime$. Let $s_2$ be an st-section of $\\mathcal{E}_2$ and $(\\beta_2, \\tau_2)$ be the pair corresponding to $s_2$. Using \\eqref{equi 1}, \\eqref{equi 2} for $s_2$ and $\\phi s_1$, we get the desired result. \n\\hfill $\\Box$\n\n\\end{proof}\n\\begin{defn}\nLet $\\chi=(\\nu, \\mu, \\sigma)$ be an action of $H$ on $I$, and $\\beta, \\tau : H \\times H \\rightarrow I$ such that $\\beta$ and $\\tau$ together with $\\chi$ satisfies \\eqref{cocycle 1} and \\eqref{cocycle 2}, respectively. Then the ordered pair $(\\beta, \\tau)$ is a $2$-cocycle with action $\\chi$. \n\n{\\bf{Remark:}} Note that the pair $(\\beta, \\tau)$ defined by the \\eqref{cocycle1 sb} and \\eqref{cocycle2 sb} is a $2$-cocycle corresponding to st-section $s$ with action defined by \\eqref{actions}. \n\\end{defn}\n\nLet $\\mathcal{E}:= 0 \\to I \\stackrel{}{\\to} E\\stackrel{\\pi}{\\to} H \\to 0$ be an skew brace extension of $H$ by $I$. Let $s : H \\rightarrow E$ be an st-section of $\\mathcal{E}$. Due to the compatiblity condition of a left skew brace $E$, we have\n$$\n(s(h_1) \\circ y_1) \\circ (s(h_2) \\circ y_2+s(h_3) \\circ y_3)=s(h_1) \\circ y_1) \\circ (s(h_2) \\circ y_2)-(s(h_1) \\circ y_1)+(s(h_1) \\circ y_1) \\circ (s(h_3) \\circ y_3).\n$$ \nFrom the above equality we see that the triple $(\\nu, \\mu, \\sigma)$ (defined in \\eqref{actions}) together with $(\\beta, \\tau)$ satisfy\n\\begin{eqnarray}\\label{parent relation}\n \\nu_{h_1 \\circ (h_2+h_3)}(\\tau(h_1, h_2+h_3) \\circ \\sigma_{h_2+h_3}(y_1) \\circ \\nu^{-1}_{h_2+h_3}(\\beta(h_2, h_3) +\\mu_{h_3}(\\nu_{h_2}(y_2))+\\nu_{h_3}(y_3))) = A,\n\\end{eqnarray}\nwhere\n\\begin{align*}\nA = &\\beta (h_1 \\circ h_3-h_1, h_1 \\circ h_3)+\\mu_{h_1 \\circ h_3}(\\beta(h_1 \\circ h_2, -h_1)+ \\mu_{-h_1}(\\nu_{h_1 \\circ h_2}(\\tau(h_1, h_2) \\circ \\sigma_{h_2}(y_1) \\circ y_2))\\\\\n& -\\nu_{h_1}(y_1)- \\beta(h_1, -h_1)) +\\nu_{h_1 \\circ h_3}(\\tau(h_1, h_3) \\circ \\sigma_{h_3}(y_1) \\circ y_3),\n\\end{align*}\nfor all $h_1, h_2, h_3 \\in H$ and $y_1, y_2, y_3 \\in I.$\n\n\n{\\bf{Remark:}} Note that if $I$ is an abelian group equipped with trivial skew brace structure, then the above condition will simplify to the condition defined for good triplet of actions in \\cite[Pg.5]{NMY}.\n\n For $\\alpha=(\\bar{\\nu},\\bar{\\mu}, \\bar{\\sigma})$ a coupling from $H$ to $I$. Define\n\\begin{align*}\\label{stZ^2}\n\\mathcal{Z}^2_{\\alpha}(H, I):=\\Bigg\\{(\\chi,\\beta, \\tau)\\hspace{.1cm} \\Big| \\hspace{.1cm} \\hspace{.1cm}\\substack{\\chi \\mbox{ is an action of } H \\mbox{ on } I, \\hspace{.1cm} \\bar{\\chi} \\approx \\alpha, \\hspace{.1cm} \\mbox{and} \\hspace{.1cm}(\\beta,\\tau)\\hspace{.1cm} \\mbox{ia a 2-cocycle} \\hspace{.1cm}\\mbox{with action} \\\\ \\hspace{.1cm} \\chi \\hspace{.1cm} \\mbox{and satisfy} \\hspace{.1cm} \\eqref{parent relation}\n }\\Bigg\\}.\n\\end{align*}\nLet $(\\chi_1, \\beta_1, \\tau_1)$ and $(\\chi_2 \\beta_2, \\tau_2)$ be two elements of $\\mathcal{Z}^2_{\\alpha}(H, I)$, where $\\chi_1=(\\prescript{}{1}{\\nu}, \\prescript{}{1}{\\mu}, \\prescript{}{1}{\\sigma})$ and $\\chi_2=(\\prescript{}{2}{\\nu}, \\prescript{}{2}{\\mu}, \\prescript{}{2}{\\sigma})$. We say that $(\\chi_1, \\beta_1, \\tau_1)$ $\\sim$ $(\\chi_2, \\beta_2, \\tau_2)$ if there exits a map $\\theta: H \\rightarrow I$ such that $\\bar{\\chi_2} \\approx \\bar{\\chi_1} $ by $\\theta$ and $\\beta_1$, $\\beta_2$ satisfy (\\ref{equi 1}), $\\tau_1, \\tau_2$ satisfy (\\ref{equi 2}) with respect to $\\theta$. \\\\\n\n\n\\begin{prop}\nThe relation `$\\sim$' defined in above para is an equivalence relation.\n\\end{prop}\n\n\\begin{proof}\nReflexivity is easy to see by taking $\\theta:H\\rightarrow I$ given by $\\theta(h)=0$ for all $h \\in H$. Now we will show that the above relation is symmetric. Let $ (\\chi_1, \\beta_1, \\tau_1)$ $\\sim$ $(\\chi_2, \\beta_2, \\tau_2)$, where $\\chi_i=(\\prescript{}{i}{\\nu}, \\prescript{}{i}{\\mu}, \\prescript{}{i}{\\sigma})$ for $i=1,2$. We know that there exist a map $\\theta : H \\rightarrow I$ such that $\\bar{\\chi_1} \\approx\\bar{\\chi_2}$ by $\\theta$ and \\eqref{equi 1}, \\eqref{equi 2}. Define $\\psi:H \\rightarrow I$ by $\\psi(h)=\\theta(h)^{-1}$. Then we have $\\prescript{}{1}{\\nu}_h=\\prescript{}{2}{\\nu}_h \\lambda_{\\psi(h)}$, and hence $\\prescript{}{2}{\\nu}_h(-\\psi(h))=\\prescript{}{1}{\\nu}_h(\\theta(h))$. This proves that $\\bar{\\chi_1}=\\bar{\\chi_2} $ by $\\psi$. Similarly we can prove that $\\beta_2, \\beta_1$ satisfy (\\ref{equi 1}) and $\\tau_1, \\tau_2$ satisfy (\\ref{equi 2}) with respect to $\\psi$. Next we prove transitivity. Let $(\\chi_1, \\beta_1, \\tau_1)$ $\\sim$ $(\\chi_2, \\beta_2, \\tau_2)$ (by $\\theta_1$) and $ (\\chi_2, \\beta_2, \\tau_2)$ $\\sim$ $(\\chi_3, \\beta_3, \\tau_3)$ (by $\\theta_2$), where\n$\\chi_i=(\\prescript{}{i}{\\nu}, \\prescript{}{i}{\\mu}, \\prescript{}{i}{\\sigma})$ for $i=1,2,3$. We claim that $\\bar{\\chi_3} \\approx \\bar{\\chi_1}$ by $\\phi:H \\rightarrow I $ defined by $\\psi(h)=\\theta_1(h) \\circ\\theta_2(h)$. We have \n\\begin{eqnarray}\\label{first relation}\n\\prescript{}{2}{\\nu_h} =\n\\prescript{}{1}{\\nu_h} \\lambda_{\\theta_1(h)},\n\\end{eqnarray}\nand \n\\begin{eqnarray*}\n\\prescript{}{3}{\\nu}_h =\n\\prescript{}{2}{\\nu_h} \\lambda_{\\theta_2(h)}.\n\\end{eqnarray*}\n\nCombining these two equations we have \n\\begin{eqnarray*}\n\\prescript{}{3}{\\nu}_h &=&\n\\prescript{}{1}{\\nu_h} \\lambda_{\\theta_1(h)}\\lambda_{\\theta_2(h)}\\\\\n&=& \\prescript{}{1}{\\nu_h} \\lambda_{\\phi(h)}.\n\\end{eqnarray*} \nFor additive action we have\n\n\\begin{eqnarray*}\n\\prescript{}{2}{\\mu}_{h}=i^+_{\\prescript{}{1}{\\nu_h}(-\\theta_1(h))} \\prescript{}{1}{\\mu}_h\n\\end{eqnarray*}\n\n\\begin{eqnarray*}\n\\prescript{}{3}{\\mu}_{h}=i^+_{\\prescript{}{2}{\\nu_h}(-\\theta_2(h))} \\prescript{}{2}{\\mu}_h.\n\\end{eqnarray*}\n\nCombining the above two equations we have \n\\begin{eqnarray*}\n\\prescript{}{3}{\\mu}_{h} &=& i^+_{\\prescript{}{2}{\\nu_h}(-\\theta_2(h))}i^+_{\\prescript{}{1}{\\nu_h}(-\\theta_1(h))} \\prescript{}{1}{\\mu}_h.\\\\\n\\end{eqnarray*}\nUsing (\\ref{first relation}) we have \n\\begin{eqnarray*}\n\\prescript{}{3}{\\mu}_{h} &=& i^+_{\\prescript{}{1}{\\nu_h} \\lambda_{\\theta_1(h)(-\\theta_2(h))}}\ni^+_{\\prescript{}{1}{\\nu_h}(-\\theta_2(h))} \\prescript{}{1}{\\mu}_h.\\\\\n\\end{eqnarray*}\n\nFinally, using the relation $a+\\lambda_a(b)=a \\circ b$, we get \n\\begin{eqnarray*}\n\\prescript{}{3}{\\mu}_{h} &=& i^+_{\\prescript{}{1}{\\nu_h}(-\\phi(h))}\\prescript{}{1}{\\mu}_h.\n\\end{eqnarray*}\n\n\nSimilarly we can prove that $\\prescript{}{3}{\\sigma}_h=i^{\\circ}_{\\phi(h)^{-1}} \\prescript{}{1}{\\sigma_h}$, which shows that $\\bar{\\chi_3} \\approx \\bar{\\chi_1}$ by $\\phi$. Next we prove that $\\beta_1$ and $\\beta_3$ also satisfy (\\ref{cocycle 1}) with respect to $\\phi$. We have\n\n\\begin{align*}\n\\prescript{}{1}{\\nu}_{h_1+h_2}(-\\theta_1(h_1+h_2))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta_1(h_1)))+\\prescript{}{1}{\\nu}_{h_2}(\\theta_1(h_2))=\\beta_2(h_1, h_2),\\\\\n\\prescript{}{2}{\\nu}_{h_1+h_2}(-\\theta_2(h_1+h_2))+\\beta_2(h_1, h_2)+ \\prescript{}{2}{\\mu}_{h_2}(\\prescript{}{2}{\\nu}_{h_1}(\\theta_2(h_1)))+\\prescript{}{2}{\\nu}_{h_2}(\\theta_2(h_2))=\\beta_3(h_1, h_2).\n\\end{align*}\n\nCombining these two equations and using the fact that $\\prescript{}{1}{\\nu}_{h}(\\theta_1(h))+\\prescript{}{2}{\\nu}_{h}(\\theta_2(h))=\\prescript{}{1}{\\nu}_{h}(\\theta_1(h)\\circ \\theta_2(h)) $, we have \n\n\\begin{align*}\n\\beta_3(h_1, h_2)& = \\prescript{}{2}{\\nu}_{h_1+h_2}(-\\theta_2(h_1+h_2))+\\prescript{}{1}{\\nu}_{h_1+h_2}(-\\theta_1(h_1+h_2))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta_1(h_1)))\n \\\\ \n &\\hspace*{4mm} +\\prescript{}{1}{\\nu}_{h_2}(\\theta_1(h_2))+ \\prescript{}{2}{\\mu}_{h_2}(\\prescript{}{2}{\\nu}_{h_1}(\\theta_2(h_1)))+\\prescript{}{2}{\\nu}_{h_2}(\\theta_2(h_2))\\\\\n&= \\prescript{}{1}{\\nu}_{h_1+h_2}(-(\\theta_2(h_1+h_2)\\circ\\theta_1(h_1+h_2)))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta_1(h_1)))\\\\\n& \\hspace*{4mm} + \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{2}{\\nu}_{h_1}(\\theta_2(h_1)))+ \\prescript{}{1}{\\nu}_{h_2}(\\theta_1(h_2))+\\prescript{}{2}{\\nu}_{h_2}(\\theta_2(h_2))\\\\\n&= \\prescript{}{1}{\\nu}_{h_1+h_2}(-(\\theta_2(h_1+h_2)\\circ\\theta_1(h_1+h_2)))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta_1(h_1) \\circ \\theta_2(h_1) ))\\\\\n& \\hspace*{4mm} + \\prescript{}{1}{\\nu}_{h_2}(\\theta_1(h_2)\\circ \\theta_2(h_2))\\\\\n&= \\prescript{}{1}{\\nu}_{h_1+h_2}(-\\phi(h_1+h_2))+\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\phi(h_1)))+\\prescript{}{1}{\\nu}_{h_2}(\\phi(h_2)).\n\\end{align*}\n\n\n\n\nSimilar calculation shows that $\\tau_1, \\tau_3$ satisfy (\\ref{cocycle 2}) with respect to $\\phi$. Hence the relation `$\\sim$' is an equivalence relation.\n\\hfill $\\Box$\n\n\\end{proof}\n\nDefine $$\\mathcal{H}^2_{\\alpha}(H, I):=\\mathcal{Z}^2_{\\alpha}(H, I)\/ \\sim$$ and denote $[(\\chi,\\beta, \\tau)] \\in \\mathcal{H}^2_{\\alpha}(H, I)$, the equivalence class of $(\\chi,\\beta, \\tau)$. This concept will be used in next section.\n\n\\section{action of cohomology group on extensions}\nIn this section, we define a faithful group action of $\\operatorname{H} _{N}^2(H, \\operatorname{Z} (I))$ \\cite[Pg.6]{NMY} on $\\operatorname{Ext} _{\\alpha}(H, I)$ and we will show that this action is transitive whenever $I$ is trivial skew brace.\n\\begin{thm} \\label{main}\nLet $\\alpha$ be a coupling from $H$ to $I$. Then there exists a bijection between $\\operatorname{Ext} _{\\alpha}(H, I)$ and $\\mathcal{H}^2_{\\alpha}(H, I)$.\n\\end{thm}\n\n\\begin{proof}\nDefine $\\phi : \\operatorname{Ext} _{\\alpha}(H, I) \\rightarrow \\mathcal{H}^2_{\\alpha}(H, I)$ as follows. Let $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to} E \\stackrel{\\pi}{\\to} H \\to 0$ be an extension with coupling $\\alpha$. Fix an st-section $s$, then there exists an action $\\chi=(\\nu, \\mu, \\sigma)$ as we defined in \\eqref{actions} such that $\\bar{\\chi}=\\alpha$; also we have $(\\beta, \\tau)$ as we defined in (\\ref{cocycle1 sb}) and (\\ref{cocycle2 sb}), together they satisfy equation (\\ref{parent relation}). Set \n$$ \n\\phi([\\mathcal{E}])=[(\\chi, \\beta, \\tau)].\n$$\nThen by Proposition \\ref{well-def-act-coc} and Proposition \\ref{prop 2} the map $\\phi$ is well defined.\nNext we define a map $\\psi : \\mathcal{H}^2_{\\alpha}(H, I) \\rightarrow Ext_{\\alpha}(H, I)$ as follows. Given an element $(\\chi, \\beta, \\tau)$ of $\\mathcal{Z}^2_{\\alpha}(H, I),$ we define binary operations on the set $H \\times I$ by setting\n\n(1) $(h_1, y_1)+(h_2, y_2)=(h_1+h_2, \\nu^{-1}_{h_1+h_2}(\\beta(h_1, h_2)+\\mu_{h_2}(\\nu_{h_1}(y_1))+\\nu_{h_2}(y_2)))$,\n\n(2) $(h_1, y_1) \\circ (h_2, y_2)=(h_1 \\circ h_2,\\tau(h_1, h_2) \\circ \\sigma_{h_2}(y_1) \\circ y_2).$\n\nIt is easy to check that (\\ref{cocycle 1}) and (\\ref{cocycle 2}) gives the associativity of `$+$' and `$\\circ$', respectively, which is enough to see that $(H \\times I, +)$ and $(H \\times I , \\circ)$ are groups and (\\ref{parent relation}) proves that `$+$' and `$\\circ$' defined here satisfy \\eqref{bcomp}. We denote this left skew brace structure by $(H, I , \\chi, \\beta, \\tau)$. Now Consider the extension \n$$\\mathcal{E}(\\chi, \\beta, \\tau) := 0 \\to I \\stackrel{i}{\\to} (H, I , \\chi, \\beta, \\tau) \\stackrel{\\pi}{\\to} H \\to 0,$$ where $i(y)=(0, y)$ and $\\pi(h, y)=h$ for all $h \\in H$ and $y \\in I$. Define $\\psi$ by setting $$\\psi([(\\chi, \\beta, \\tau)])= [\\mathcal{E}(\\chi, \\beta, \\tau)].$$ \nWe show that the map $\\psi$ is well defined. Let $(\\chi_1, \\beta_1, \\tau_1)$ $\\sim$ $(\\chi_2, \\beta_2, \\tau_2)$, then there exist a map $\\theta : H \\rightarrow I $ such that $\\bar{\\chi_1} \\approx \\bar{\\chi_2}$ by $\\theta$ and $\\beta_1, \\beta_2$ satisfy \\eqref{equi 1} and $\\tau_1, \\tau_2$ satisfy \\eqref{equi 2}, respectively. Define $\\zeta: \\mathcal{E}(\\chi_2, \\beta_2, \\tau_2) \\rightarrow \\mathcal{E}(\\chi_1, \\beta_1, \\tau_1)$ given by \n$$\n\\zeta(h, y)=(h, \\theta(h) \\circ y).\n$$\n\nWe have\n\\begin{align*}\n\\zeta((h_1, y_1)+(h_2, y_2))=& (h_1+h_2, \\ \\theta(h_1+h_2) \\circ \\prescript{}{2}{\\nu}^{-1}_{h_1+h_2}(\\beta_2(h_1, h_2)+ \\prescript{}{2}{\\mu}_{h_2}( \\prescript{}{2}{\\nu}_{h_1}(y_1))+ \\prescript{}{2}{\\nu}_{h_2}(y_2)))\\\\\n=&(h_1+h_2, \\ \\theta(h_1+h_2)+ \\prescript{}{1}{\\nu}^{-1}_{h_1+h_2}(\\beta_2(h_1, h_2)+ \\prescript{}{2}{\\mu}_{h_2}( \\prescript{}{2}{\\nu}_{h_1}(y_1))+ \\prescript{}{2}{\\nu}_{h_2}(y_2)))\\\\\n=& (h_1+h_2, \\ \\prescript{}{1}{\\nu}^{-1}_{h_1+h_2}(\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta(h_1)))+\\prescript{}{1}{\\nu}_{h_2}(\\theta(h_2))\\\\\n&+ \\prescript{}{2}{\\mu}_{h_2}( \\prescript{}{2}{\\nu}_{h_1}(y_1))+ \\prescript{}{2}{\\nu}_{h_2}(y_2)))\\\\\n=& (h_1+h_2, \\ \\prescript{}{1}{\\nu}^{-1}_{h_1+h_2}(\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta(h_1)))+ \\prescript{}{1}{\\mu}_{h_2}( \\prescript{}{2}{\\nu}_{h_1}(y_1))\\\\\n&+ \\prescript{}{1}{\\nu}_{h_2}(\\theta(h_2))+ \\prescript{}{2}{\\nu}_{h_2}(y_2)))\\\\\n=& (h_1+h_2, \\ \\prescript{}{1}{\\nu}^{-1}_{h_1+h_2}(\\beta_1(h_1, h_2)+ \\prescript{}{1}{\\mu}_{h_2}(\\prescript{}{1}{\\nu}_{h_1}(\\theta(h_1) \\circ y_1))\\\\\n&+ \\prescript{}{1}{\\nu}_{h_2}(\\theta(h_2) \\circ y_2)))\\\\\n=&(h_1, \\ \\theta(h_1) \\circ y_1)+(h_2, \\ \\theta(h_2) \\circ y_2).\n\\end{align*}\nWhich shows that $\\zeta$ is linear in `$+$'. Similarly we can show that $\\zeta$ is linear in `$\\circ$' as well. It is easy to see that $\\zeta$ is an isomorphism and the following diagram commutes\n$$\n\\begin{CD}\n 0 @>>> I @>>> \\mathcal{E}(\\chi_2, \\beta_2, \\tau_2) @>{{\\pi_1} }>> H @>>> 0\\\\\n && @V{\\text{Id}}VV @V{\\zeta}VV @ VV{ \\text{Id}}V \\\\\n 0 @>>> I @>>> \\mathcal{E}(\\chi_2, \\beta_2, \\tau_2) @>{{\\pi_2} }>> H @>>> 0,\n\\end{CD}\n$$\nwhere $\\pi_1$ and $\\pi_2$ are natural projections. Hence, $\\mathcal{E}(\\chi_1, \\beta_1, \\tau_1)$ and $\\mathcal{E}(\\chi_2, \\beta_2, \\tau_2)$ are equivalent extensions. That shows that the map $\\psi$ is well defined. It is easy to check that $\\psi$ is well-defined and $\\psi$ and $\\phi$ are inverses of each other. The proof is now complete.\n\\hfill $\\Box$\n\n\\end{proof}\n\nThe elements of $\\mathcal{Z}^2_{\\alpha}(H, I)$ are called associated triplets as every element of $\\mathcal{Z}^2_{\\alpha}(H, I)$ is associated to some extension in view of Theorem \\ref{Main thm}.\n\n\\begin{thm}\\label{action change}\nLet $H$ and $I$ be two skew braces and let $(\\chi, \\beta, \\tau) \\in \\mathcal{Z}^2_{\\alpha}(H, I)$ be an associated triplet. If $\\chi^\\prime$ is an action of $H$ on $I$ for which $\\bar{\\chi} \\approx\\bar{\\chi}^\\prime$, then there exist maps $\\beta^\\prime, \\tau^\\prime : H \\rightarrow I$ such that $(\\chi^\\prime, \\beta^\\prime, \\tau^\\prime)$ is an associated triplet and $[(\\chi, \\beta, \\tau)]= [ (\\chi^\\prime, \\beta^\\prime, \\tau^\\prime)]$.\n\\end{thm}\n \\begin{proof}\n In the view of Theorem \\ref{main} there exists an extension $$\\mathcal{E}(\\chi, \\beta, \\tau) := 0 \\to I \\stackrel{i}{\\to} E \\stackrel{\\pi}{\\to} H \\to 0,$$ corresponding to the associated triplet $(\\chi, \\beta, \\tau)$. Let $s : H \\rightarrow E$ be a st-section inducing $(\\chi, \\beta, \\tau)$. Since $\\bar{\\chi} \\approx\\bar{\\chi^\\prime}$, there exist a map $\\theta : H \\rightarrow I$ such that $\\theta(0)=0$, $\\nu^{\\prime}_{h}=\\nu_{h} \\lambda_{\\theta(h)}$, $\\mu^\\prime_{h}=i^+_{\\nu_h(-\\theta(h))} \\mu_h$ and $\\sigma^\\prime=i^{\\circ}_{\\theta(h)^{-1}} \\sigma_{h}$. Define an st-section $s^\\prime(h)=s(h) \\circ \\theta(h)$ for all $ x \\in H$. Consequently we have an associated triplet $(\\chi_1, \\beta^\\prime, \\tau^\\prime)$ corresponding to the st-section $s^\\prime$ and it is easy to see that $\\chi^\\prime=\\chi_1$. Hence $[(\\chi, \\beta, \\tau)]=[(\\chi_1, \\beta^\\prime, \\tau^\\prime)])=[(\\chi^\\prime, \\beta^\\prime, \\tau^\\prime)]$ as $(\\chi, \\beta^\\prime, \\tau^\\prime)$ and $(\\chi_1, \\beta^\\prime, \\tau^\\prime)$ are associated triplet of the same extension by different st-sections. This completes the proof.\n \\hfill $\\Box$\n\n\\end{proof}\n We now state a theorem analogous to \\cite[Theorem 3.6]{NMY}. Let $H$ be a skew brace and $I$ be an abelian group. \n \n Define\n\\begin{align*}\n \\operatorname{Z} _N^2(H, I)=\\Bigg\\{(g ,f) \\hspace{.1cm} \\Big \\vert \\hspace{.1cm}g,f:H \\times H \\rightarrow I, \\hspace{.1cm} \\substack{ g,f \\hspace{.1cm} \\mbox{sastisy}\\hspace{.1cm} (\\ref{cocycle 1}) \\hspace{.1cm}\\mbox{and}\\hspace{.1cm} (\\ref{cocycle 2}),\\hspace{.1cm} \\mbox{respectively, and} \\\\ \\hspace{.1cm}\\mbox{vanish on degenerate tupples}} \\Bigg\\},\n\\end{align*}\nand $\\operatorname{B} _N^2(H, I)$ is the collection of the pairs $(g, f) \\in \\operatorname{Z} _N^2(H, I)$ such that there exists a map $\\theta$ from $H$ to $I$ with $g= \\nu_{h_1+h_2}(-\\theta(h_1+h_2))+\\mu_{h_2}((\\nu_{h_1}(\\theta(h_1)))+\\nu_{h_2}(\\theta(h_2))$ and $f=-\\theta(h_1 \\circ h_2) + \\sigma_{h_2}\\theta(h_1)) + \\theta(h_2)$.\n\nPut\n\\begin{align*}\n\\operatorname{Z} _N^1(H, I)=\\Bigg\\{\\substack{ \\theta \\hspace{.1cm}\\mbox{is a map from} \\hspace{.1cm} H \\hspace{.1cm} \\mbox{to} \\hspace{.1cm} I \\hspace{.1cm} \\mbox{such that }\\hspace{.1cm} \\theta(h_1 \\circ h_2)= \\sigma_{h_2}(\\theta(h_1)+\\theta(h_2) \\hspace{.1cm} \\\\ \\mbox{and} \\hspace{.1cm} \\nu_{h_1+h_2}(\\theta(h_1 + h_2))= \\mu_{h_2}(\\nu_{h_1}(\\theta(h_1))+\\nu_{h_2}(\\theta(h_2))} \\Bigg\\},\n\\end{align*}\nthe set $\\operatorname{Z} _N^1(H, I)$ is called as the set of derivations, and\n$$\n\\operatorname{H} ^2_N(H, I):=\\operatorname{Z} _N^2(H, I)\/\\operatorname{B} _N^2(H, I)\n$$\n is the second cohomology group of $H$ by $I$.\n\n\\begin{thm} \nLet $H$ be a skew brace and let $I$ be an abelian group equiped with trivial brace structure. Let $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to} E \\stackrel{\\pi}{\\to} H \\to 0$ be an extension. Then the coupling and action are same, and there is a bijection between $\\operatorname{Ext} _{(\\nu, \\mu, \\sigma)}(H, I)$ and $\\operatorname{H} ^2_N(H, I)$. \n\\end{thm}\n\n\n\n\\begin{thm}\\label{Main thm}\nLet $[\\mathcal{E}] \\in \\operatorname{Ext} _{\\alpha}(H, I)$ and $(\\chi, \\beta, \\tau)$ be an associated triplet of $\\mathcal{E}$. Then for $[(\\beta_1, \\tau_1)] \\in \\operatorname{H} ^2_{N}(H, \\operatorname{Ann} (I))$, the operation $$[(\\beta_1, \\tau_1)] [\\mathcal{E}(\\chi, \\beta, \\tau)]= [\\mathcal{E}(\\chi, \\beta_1+\\beta, \\tau_1+\\tau)] $$ defines a free action of the group $\\operatorname{H} ^2_{N}(H, \\operatorname{Ann} (I))$ on the set $\\operatorname{Ext} _{\\alpha}(H, I)$. If $I$ is trivial skew brace, then this action becomes transtivite. \n\\end{thm}\n\n\\begin{proof}\nIt is easy to check that the action under consideration is well defined. Let $[\\mathcal{E}(\\chi, \\beta, \\tau)] \\in Ext_{\\alpha}(H, I)$ and $[(\\beta_1, \\tau_1)] \\in \\operatorname{H} _N^2(H, \\operatorname{Ann} (I))$ be such that $[(\\beta_1, \\tau_1)] [\\mathcal{E}(\\chi, \\beta, \\tau)]=[\\mathcal{E}(\\chi, \\beta, \\tau)]$. Then $$[(\\chi, \\beta_1+\\beta, \\tau_1+\\tau)]=[(\\chi, \\beta, \\tau)],$$ and therefore there exist a map $\\theta: H \\rightarrow I$ such that $\\theta(0)=0$ and $\\bar{\\chi} \\approx\\bar{\\chi}$ by $\\theta$, which implies that $\\theta(h) \\in \\operatorname{Ann} (I)$ for all $h \\in H$, and\n\\begin{equation*}\n-\\nu_{h_1+h_2}(\\theta(h_1+h_2))+\\beta(h_1, h_2)+ \\mu_{h_2}(\\nu_{h_1}(\\theta(h_1)))+\\nu_{h_2}(\\theta(h_2))=\\beta(h_1, h_2)+\\beta_1(h_1, h_2)\n\\end{equation*}\nand\n\\begin{equation*}\n\\theta(h_1 \\circ h_2)^{-1} \\circ \\tau(h_1, h_2) \\circ \\sigma_{h_2}(\\theta(h_1)) \\circ \\theta(h_2)=\\tau(h_1, h_2)+\\tau_1(h_1, h_2)\n\\end{equation*}\nfor all $h_1, h_2 , h_3 \\in H$(using the fact that $\\theta(h) \\in \\operatorname{Ann} (I) $ for all $h \\in H$).\n\n We have \n \\begin{align*}\n \\beta_1(h_1, h_2)&= \\nu_{h_1+h_2}(-\\theta(h_1+h_2))+\\mu_{h_2}((\\nu_{h_1}(\\theta(h_1)))+\\nu_{h_2}(\\theta(h_2)),\\\\ \n\\tau_1(h_1, h_2)&=-\\theta(h_1 \\circ h_2) + \\sigma_{h_2}\\theta(h_1)) + \\theta(h_2).\n \\end{align*}\nThus $[(\\tau_1, \\beta_1)]=1$, and hence the action is free.\n\nLet $\\mathcal{E}_1$ and $\\mathcal{E}_2$ be two elements in $Ext_{\\alpha}(H,I)$. Then for $i=1,2$, $[\\mathcal{E}_i]=[\\mathcal{E}_i(\\chi_i, \\beta_i,\\tau_i)]$ for some associated trilpet $(\\chi_i, \\beta_i,\\tau_i)$, where $\\chi_i= (\\prescript{}{i}{\\nu}, \\prescript{}{i}{\\mu}, \\prescript{}{i}{\\sigma})$. By Theorem \\ref{action change}, we can construct $(\\beta^\\prime,\\tau^\\prime)$ such that $(\\chi_2, \\beta^\\prime, \\tau^\\prime)$ is an associated triplet with\n$$ [\\mathcal{E}_1(\\chi_1, \\beta_1, \\tau_1)]=[\\mathcal{E}_1(\\chi_2, \\beta^\\prime, \\tau^\\prime)].$$\nWe set \n$$\\beta_3(h_1, h_2)=\\beta^\\prime(h_1,h_2)-\\beta_2(h_1, h_2)$$ \nand \n$$\\tau_3(h_1, h_2)=\\tau^\\prime(h_1, h_2)\\circ \\tau_2(h_1, h_2)^{-1}.$$\nNow $(\\beta^\\prime, \\tau^\\prime)$ and $(\\beta_2, \\tau_2)$ are $2$-cocycles with the same action $\\chi_2$. Then from (\\ref{action1 }), (\\ref{action2 }) and (\\ref{action3}) it follows that $\\beta_3(h_1, h_2) \\in Z(I,+)$ and $\\tau_3(h_1, h_2) \\in Soc(I)$ for all $h_1 , h_2 \\in H$. If we take $I$ to be trivial skew brace, then $\\operatorname{Z} (I,+)= \\operatorname{Soc} (I)=\\operatorname{Ann} (I)$. Finally we get $\\beta_3, \\tau_3 : H \\rightarrow Ann(I)$. It is easy to see that $(\\beta_3, \\tau_3)$ is $2-cocycle$ with respect to the action $\\chi_2$ and $(\\beta_3, \\tau_3)[\\mathcal{E}(\\chi_2, \\beta_2, \\tau_2)]=[\\mathcal{E}(\\chi_2, \\beta^\\prime, \\tau^\\prime)]=[\\mathcal{E}(\\chi_1, \\beta_1, \\tau_1)]\n$. Hence the action is transtive, which completes the proof.\n\\hfill $\\Box$\n\n\\end{proof}\n\nAs a consequence, we get\n\\begin{thm} \\label{main 1}\nLet $H$ be a skew brace and let $I$ be a trivial skew brace with a fixed coupling $\\alpha$. Then there exists a bijection between $Ext_{\\alpha}(H,I)$ and $Ext_{\\alpha}(H, \\operatorname{Z} (I))$.\n\\end{thm}\n\n\\section{action of automorphism group on extensions}\n\nThroughout this section we consider $I$ to be a trivial skew brace and $\\mathcal{E} := 0 \\to I \\stackrel{i}{\\to} E \\stackrel{\\pi}{\\to} H \\to 0$ be an extension of skew braces. Then $ \\nu :H \\rightarrow Aut(I,+)$ as defined in (\\ref{actions}), is independent of the choice of an st-section. Also $\\bar{\\nu}= \\nu$ and $\\bar{\\mu}:(H,+) \\rightarrow \\operatorname{Out} (I)$, $\\bar{\\sigma}: (H, \\circ) \\rightarrow \\operatorname{Out} (I)$, where $Out(I)$ represents the group of outer-automorphisms of $I$. For a pair $(\\phi, \\theta) \\in \\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$ of skew brace automorphisms and an extension \n$$\\mathcal{E} : 0 \\rightarrow I \\stackrel{i}{\\rightarrow} E \\stackrel{\\pi}{\\rightarrow} H \\rightarrow 0$$\nof $H$ by $I$, we can define a new extension\n$$\\mathcal{E}^{(\\phi, \\theta)} : 0 \\rightarrow I \\stackrel{i\\theta}{\\longrightarrow} E \\stackrel{\\phi^{-1} \\pi}{\\longrightarrow} H \\rightarrow 0$$\nof $H$ by $I$. Thus, for a given $(\\phi, \\theta) \\in \\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$, we can define a map from $\\operatorname{Ext} (H, I)$ to itself given by \n\\begin{equation}\\label{act1 sb}\n[\\mathcal{E}] \\mapsto [ \\mathcal{E}^{(\\phi, \\theta)}].\n\\end{equation}\n If $\\phi$ and $\\theta$ are identity automorphisms, than obviously $\\mathcal{E}^{(\\phi, \\theta)} = \\mathcal{E}$. It is also easy to see that \n$$[\\mathcal{E}] ^{(\\phi_1, \\theta_1) (\\phi_2, \\theta_2)}= \\big([\\mathcal{E}]^{(\\phi_1, \\theta_1)}\\big)^{(\\phi_2, \\theta_2)}.$$\nWe conclude that the association \\eqref{act1 sb} gives an action of the group $\\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$ on the set $\\operatorname{Ext} (H, I)$. From Corollary \\ref{cor 1} , we know that $\\operatorname{Ext} (H, I) = \\bigsqcup_{(\\nu, \\bar{\\mu}, \\bar{\\sigma})} \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$. \\emph{Let $(\\nu, \\bar{\\mu}, \\bar{\\sigma})$ be an arbitrary but fixed coupling from $H$ to $I$.} Let $\\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}$ denote the stabiliser of $\\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$ in $\\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$; more explicitly\n$$\\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})} = \\{ (\\phi, \\theta) \\in \\operatorname{Autb} (H) \\times \\operatorname{Autb} (I) \\mid \\nu_h=\\theta^{-1}\\nu_{\\phi(h)}\\theta, \\bar{\\mu}_h = \\theta^{-1}\\bar{\\mu}_{\\phi(h)}\\theta \\mbox{ and } \\bar{\\sigma}_h = \\theta^{-1}\\bar{\\sigma}_{\\phi(h)}\\theta \\}.$$ \nIt is easy to see that $\\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}$ is a subgroup of $\\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$. For details see \\cite[Pg.15]{NMY}.\n\nNext we consider an action of $ \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})}$ on $\\operatorname{H} ^2_N(H, Z(I))$ (same as in \\cite{NMY}) by \n\\begin{equation} \\label{act3 sb}\n[(g, f)] \\mapsto [\\big(g^{(\\phi, \\theta)}, f^{(\\phi, \\theta)}\\big)],\n\\end{equation}\nwhere $g^{(\\phi, \\theta)}(h_1, h_2)=\\theta^{-1}(g(\\phi(h_1), \\phi(h_2))$. This action of $ \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})}$ on $\\operatorname{H} ^2_N(H, Z(I))$ is same as the action of $ \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})}$ on $\\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$ transferred on $\\operatorname{H} ^2_N(H, Z(I))$ through bijection of Theorem \\ref{main 1} . Using this action we can define the semi-direct product $\\Gamma = \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})} \\ltimes \\operatorname{H} ^2_N(H,Z(I))$. We wish to define an action of $\\Gamma$ on $\\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$. For $(c, h) \\in \\Gamma$ and $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$, define \n\\begin{equation}\\label{act4 sb}\n[\\mathcal{E}]^{(c, h)} = ([\\mathcal{E}]^c)^h.\n\\end{equation}\n\n\\begin{lemma}\\label{wells2 sb}\nThe rule in \\eqref{act4 sb} gives an action of $\\Gamma$ on $\\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows on the lines of \\cite[Lemma 5.2]{NMY}.\n\n\\end{proof}\n\n Let $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$ be a fixed extension. Since the action of $\\operatorname{H} ^2_N(H,Z(I))$ on $\\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$ is transitive and faithful, for each $c \\in \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})}$, there exists a unique element (say) $h_c$ in $\\operatorname{H} ^2_N(H,Z(I))$ such that \n $$[\\mathcal{E}]^{c} = [\\mathcal{E}]^{h_c}.$$\n We thus have a well defined map $ \\omega(\\mathcal{E}): \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})} \\rightarrow \\operatorname{H} ^2_N(H,Z(I))$ given by\n \\begin{equation}\\label{wells-map sb}\n \\omega(\\mathcal{E})(c)=h_c\n \\end{equation}\n for $c \\in \\operatorname{C} _{(\\nu, \\bar{\\mu,} \\bar{\\sigma})}$. \n \n\\begin{lemma}\\label{wells3 sb}\nThe map $ \\omega(\\mathcal{E}): \\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})} \\rightarrow \\operatorname{H} ^2_N(H,Z(I))$ defined in \\eqref{wells-map sb} is a derivation with respect to the action of $\\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}$ on $\\operatorname{H} ^2_N(H,Z(I))$ given in \\eqref{act3 sb}.\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows on the lines of \\cite[Lemma 5.3]{NMY}.\n\\end{proof}\n\nLet \n$$\\mathcal{E}: 0 \\rightarrow I \\rightarrow E \\overset{\\pi}\\rightarrow H $$\nbe an extension of a left skew brace $H$ by a trivial skew brace $I$ such that $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\mu, \\sigma)}(H,I)$.\nLet $\\operatorname{Autb} _I(E)$ denote the subgroup of $\\operatorname{Autb} (E)$ consisting of all automorphisms of $E$ which normalize $I$, that is,\n$$\\operatorname{Autb} _I(E) := \\{ \\gamma \\in \\operatorname{Autb} (E) \\mid \\gamma(y) \\in I \\mbox{ for all } y \\in I\\}.$$ \nFor $\\gamma \\in \\operatorname{Autb} _I(E)$, set $\\gamma_I := \\gamma |_I$, the restriction of $\\gamma$ to $I$, and $\\gamma_H$ to be the automorphism of $H$ induced by $\\gamma$. More precisely, $\\gamma_H(h) = \\pi(\\gamma(s(h)))$ for all $h \\in H$, where $s$ is an st-section of $\\pi$. Notice that the definition of $\\gamma_H$ is independent of the choice of an st-section. Define a map $\\rho(\\mathcal{E}) : \\operatorname{Autb} _I(E) \\rightarrow \\operatorname{Autb} (H) \\times \\operatorname{Autb} (I)$ by\n$$\\rho(\\mathcal{E})(\\gamma)=(\\gamma_H, \\gamma_I).$$ \nAlthough $\\omega(\\mathcal{E})$ is not a homomorphism, but we can still talk about its set theoretic kernel, that is,\n$$\\operatorname{Ker} (\\omega(\\mathcal{E})) = \\{c \\in C_{(\\nu, \\bar{\\mu},\\bar{\\sigma})} \\mid [\\mathcal{E}]^c=[\\mathcal{E}]\\}.$$\n\n\\begin{prop}\\label{wells4 sb}\nFor the extension $\\mathcal{E}$, $\\operatorname{Im} (\\rho(\\mathcal{E})) \\subseteq \\operatorname{C} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}$ and \n$\\operatorname{Im} (\\rho(\\mathcal{E})) = \\operatorname{Ker} (\\omega(\\mathcal{E}))$.\n\\end{prop}\n\n\\begin{proof}\nThe proof follows on the lines of \\cite[Proposition 5.4]{NMY}.\n\\end{proof}\n\nContinuing with the above setting, set $\\operatorname{Autb} ^{H, I}(E) := \\{\\gamma \\in \\operatorname{Autb} _I(E) \\mid \\gamma_I = \\operatorname{Id}, \\gamma_H = \\operatorname{Id}\\}$. Notice that $\\operatorname{Autb} ^{H,I}(E)$ is precisely the kernel of $\\rho(\\mathcal{E})$. Hence, using Proposition \\ref{wells4 sb}, we get\n\n\n\\begin{thm}\\label{wells5 sb}\nLet $\\mathcal{E}: 0 \\rightarrow I \\rightarrow E \\overset{\\pi}\\rightarrow H$ be a extension of a left skew brace $H$ by a trivial skew brace $I$ such that $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H,I)$. Then we have the following exact sequence of groups \n$$0 \\rightarrow \\operatorname{Autb} ^{H,I}(E) \\rightarrow \\operatorname{Autb} _I(E) \\stackrel{\\rho(\\mathcal{E})}{\\longrightarrow} \\operatorname{C} _{(\\nu,\\bar{\\mu}, \\bar{\\sigma})} \\stackrel{\\omega(\\mathcal{E})}{\\longrightarrow} \\operatorname{H} ^2_N(H,Z(I)),$$\nwhere $\\omega(\\mathcal{E})$ is, in general, only a derivation.\n\\end{thm}\n\nFurther we have\n\\begin{prop}\\label{wells6 sb}\nLet $\\mathcal{E} : 0 \\rightarrow I \\rightarrow E \\overset{\\pi}\\rightarrow H$ be an extension of $H$ by $I$ such that $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H, I)$. Then $\\operatorname{Autb} ^{H,I}(E) \\cong \\operatorname{Z} ^1_N(H,Z(I))$.\n\\end{prop}\n\n\\begin{proof}\nThe map $\\psi : \\operatorname{Z} ^1_N(H,Z(I)) \\rightarrow \\operatorname{Autb} ^{H,I}(E)$ defined by $\\psi(\\lambda)(s(h) \\circ y)= s(h) \\circ \\lambda(h) \\circ y$ is the required isomorphism. Rest proof follows on the lines of \\cite[Proposition 5.6]{NMY}\n\\end{proof}\n\nWe finally get the following Wells' like exact sequence for skew braces.\n\\begin{thm}\\label{wells7 sb}\nLet $\\mathcal{E}: 0 \\rightarrow I \\rightarrow E \\overset{\\pi}\\rightarrow H$ be an extension of a left skew brace $H$ by a trivial skew brace $I$ such that $[\\mathcal{E}] \\in \\operatorname{Ext} _{(\\nu, \\bar{\\mu}, \\bar{\\sigma})}(H,I)$. Then we have the following exact sequence of groups \n$$0 \\rightarrow \\operatorname{Z} ^1_N(H,\\operatorname{Z} (I)) \\rightarrow \\operatorname{Autb} _I(E) \\stackrel{\\rho(\\mathcal{E})}{\\longrightarrow} \\operatorname{C} _{(\\nu,\\bar{\\mu}, \\bar{\\sigma})} \\stackrel{\\omega(\\mathcal{E})}{\\longrightarrow} \\operatorname{H} ^2_N(H,\\operatorname{Z} (I)),$$\nwhere $\\omega(\\mathcal{E})$ is, in general, only a derivation.\n\\end{thm}\n\n\\section{Acknowledgements}\n\nI am grateful to my supervisor Prof. M. K. Yadav for his constant support, comments and suggestions while doing this project. I would like to thank Prof. L. Vendramin for his kind help in writing GAP code. The author acknowledge Harish-Chandra Research institute for fantastic facilities and for the serene ambience that it facilitates.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nUnderstanding the universal behavior of classical many-body systems near their critical points is a central goal %\nof classical statistical mechanics. Although this is a difficult problem in general, in one and two spatial dimensions, significant insights have been provided by exactly solved models~\\cite{baxter2016exactly}. One important open problem is to generalize these solutions to three-dimensional~(3D) systems with realistic short-range interactions. Despite a long effort with some preliminary results~\\cite{suzuki1972solution,zamolodchikov1980tetrahedra,bazhanov1992new,huang1997exact,dhar2008exact,mangazeev2013integrable}, no physical 3D model has been exactly solved that displays a genuinely 3D phase transition~\\footnote{Among the models constructed in Refs.~\\cite{suzuki1972solution,zamolodchikov1980tetrahedra,bazhanov1992new,huang1997exact,dhar2008exact,mangazeev2013integrable}, only the models in Refs.~\\cite{suzuki1972solution,huang1997exact} have phase transitions, and in these the 3D partition function factorizes into a product of partition functions of 2D systems, giving the phase transitions an essentially 2D character.}. \n\n\n\nIn this paper, we make progress in this direction by exactly solving a classical Ising model on a special 3D lattice, as depicted in Fig.~\\ref{fig:hc0}, \nalthough with the caveat that the model has imaginary coupling constants. \nThe transfer matrix of this system has a structure similar to a non-Hermitian version of the 2D Kitaev honeycomb model~\\cite{Kitaev2006}, and the partition function can be obtained using the representation theory of the so($2N$) Lie algebra and the corresponding Lie group.\nThe solution displays a third order phase transition between two distinct phases, and near the critical point we can exactly obtain a critical exponent of the model. \n\nThe phases are interesting in their own right, as they are distinguished by topological properties. Specifically, there is a family of loop observables whose \n expectation values distinguish the two phases and are equal to some rational numbers~($0$, $1$, or $1\/3$) depending on the topology of the loop. \n\nDespite its complex coupling constants~(also a complication of some previous approaches~\\cite{zamolodchikov1980tetrahedra,bazhanov1992new}), our findings have physical relevance. First, we show in Sec.~\\ref{sec:physical_model} that the topological features discovered in one of the phases of the model with complex couplings also exist in a similar exactly solvable model with real-valued couplings. More speculatively, it is possible more generally that the long-distance property of our model belongs to the same universality class of certain physical 3D classical systems. \nIt remains an open question whether the other phase of our model can also be reproduced in a physical system, but if there indeed exists a physical classical system that has the two phases mentioned above and a phase transition between them, then the concept of universality suggests that the long-distance behaviors and the critical exponent we obtain here will apply to\nsuch physical systems.\n\nAs another point of physical relevance for the model with complex couplings, in Sec.~\\ref{sec:quantum_amplitude} we show two constructions that realize the partition function $Z$ of our model in certain dynamical processes of a 3D quantum system: one is to map $Z$ to the transition amplitude between a family of product states, the other is to realize $Z$ as the coherence of a probe spin coupled to the whole 3D system. Both constructions in principle allow the free energy to be experimentally measured, albeit with an exponentially small signal. Under these mappings, the phase transition of our model corresponds to a dynamical quantum phase transition~(DQPT)~\\cite{heyl2013dynamical,Heyl_2018}, a phenomenon that has gained much attention recently. Statistical mechanics with complex configuration energies also appears in the study of Lee-Yang zeros~\\cite{yang1952statistical,lee1952statistical,wei2012lee,peng2015experimental}, non-Hermitian quantum systems~\\cite{moiseyev2011non,Gong2018Topological,Ashida2020Non}, and complex conformal field theories~\\cite{faedo2020holographic}. \n\n\n\n\n\nOur paper is organized as follows. In Sec.~\\ref{sec:model} we define our model and a family of loop observables of interest. In Sec.~\\ref{sec:solution} we present the exact solution of the model: in Sec.~\\ref{sec:TM} we derive the transfer matrix of the classical model, in Sec.~\\ref{sec:map_fermion} we use a spin-fermion mapping to reduce the problem to a free fermion problem, in Sec.~\\ref{sec:solve_fermion_TM} we solve the eigenvalues of the free fermion transfer matrix and calculate the thermodynamic free energy, in Sec.~\\ref{sec:phase_boundary} we obtain the phase diagram, in Sec.~\\ref{sec:critical_exp} we calculate a critical exponent, and in Sec.~\\ref{sec:TPloop} and Sec.~\\ref{sec:loop_observables} we calculate the expectation values of loop observables and demonstrate their topological properties. In Sec.~\\ref{sec:justification} we give two physical implications of our model: the existence of a physical classical phase with similar topological behaviors~(Sec.~\\ref{sec:physical_model}), and realizations of the partition function in quantum dynamical processes~(Sec.~\\ref{sec:quantum_amplitude}). %\nIn Sec.~\\ref{sec:summary} we summarize our results. \nThe Appendices contain technical results used throughout our arguments. \n\n\\section{The Model}\\label{sec:model}\n\\begin{figure}\n\t\\center{\\includegraphics[width=0.9\\linewidth]{3dbrickwall-yellowthin-loops-large-RGB.png}}%\n\t\\caption{\\label{fig:hc0} Definition of the model and loop observables. The classical system sits on a 3D stacking of the brick wall lattice, of arbitrarily large extent in each direction. Classical spins lie on vertices, and they only interact via the thicker links. The horizontal links~(red, blue, black) have real coupling constants $J_x,J_y,J_z$, for $x$-planes, $y$-planes, and $z$-planes, respectively. The coupling constant $J_\\perp$ for the vertical links~(pink) and the external field $h$ are imaginary when the solvability condition Eq.~\\eqref{cond:exact} is met. The yellow shaded cuboid shows an example of the loop observable $\\sigma[\\mathfrak{L}_{(xy)}]$ for a contractible loop $\\mathfrak{L}$~(here being an elementary plaquette), which is equal to the product of Ising spins on the larger yellow vertices~[see Eq.~\\eqref{eq:def_loop_observable}]. Similarly, the green shaded rectangle shows an example of $\\sigma[\\mathfrak{L}_{(yz)}]$ for a noncontractible loop, extended infinitely to the right and to the left.\n\t}\n\\end{figure}\nIn this section we define our model and the class of physical observables we are interested in.\nThe model is defined on a 3D stacking of the 2D brick wall lattice, with classical Ising spins, $\\sigma_j \\in \\{-1, +1\\}$, lying on vertices $j$, as shown in Fig.~\\ref{fig:hc0}, and we use periodic boundary conditions~(PBC) for all the three directions for simplicity. Nearest neighbor Ising-type interactions exist only on a subset of links in this lattice, which are shown in Fig.~\\ref{fig:hc0} as thick red, blue, black, and pink links. The energy of the system for a specific classical spin configuration is\n\\begin{eqnarray}\\label{eq:HIK}\n H[\\{\\sigma\\}]&=&-J_x\\sum_{\\langle ij\\rangle\\in \\mathbf{X}}\\sigma_{i}\\sigma_{j}-J_y\\sum_{\\langle ij\\rangle\\in \\mathbf{Y}}\\sigma_{i}\\sigma_{j}\\\\\n &&-J_z\\sum_{\\langle ij\\rangle\\in \\mathbf{Z}}\\sigma_{i}\\sigma_{j}-J_\\perp\\sum_{\\langle ij\\rangle\\in\\boldsymbol{\\perp}}\\sigma_{i}\\sigma_{j}+h\\sum_i\\sigma_i,\\nonumber\n \\end{eqnarray}\nwhere $\\mathbf{X}$ denotes the set of all thick links on $x$-planes, and similarly for $\\mathbf{Y},\\mathbf{Z}$, while $\\boldsymbol{\\perp}$ is the set of all the vertical links in Fig.~\\ref{fig:hc0}, and the external field $h$ acts on all spins. \nThe goal is to find the partition function\n\\begin{eqnarray}\\label{eq:Z}\nZ(K_x,K_y,K_z,K_\\perp,\\beta h)=\\sum_{\\{\\sigma\\}}e^{-\\beta H[\\{\\sigma\\}]},\n\\end{eqnarray}\nwhere $K_i=\\beta J_i,i=x,y,z,\\perp$. The free energy is related to the partition function by\n\\begin{equation}\\label{def:free_energy0}\n\tF=-k_B T \\ln Z.\n\\end{equation}\nThe model is exactly solvable when the following conditions hold:\n\\begin{equation}\\label{cond:exact}\n4J_\\perp\\beta\\equiv \\pi i~(\\mathrm{mod}~2\\pi i),~~~2h\\beta\\equiv \\frac{\\pi i}{2}~(\\mathrm{mod}~ 2\\pi i).\n\\end{equation}\nAfter imposing these solvability conditions, there remains a three-dimensionless-parameter space~$(K_x,K_y,K_z)$ of solutions.\n\n\nBeyond the free energy~(and its derivatives), we also consider the thermal expectation values of a family of loop observables that are products of $\\sigma_j$s on closed loops, defined by the following procedure:\\\\\n(1) Choose a loop $\\mathfrak{L}$ on the 2D brick wall lattice~($\\mathfrak{L}$ must consist of edges of the brick wall lattice);\\\\\n(2) Choose two nearest neighbor planes of type $\\alpha$ and $\\beta$ of the 3D lattice, denoted $(\\alpha \\beta)$, which can be $(xy),(yz)$ or $(zx)$;\\\\\n(3) Denote by $\\mathfrak{L}_{(\\alpha\\beta)}$ the graph consisting of all sites in the loop $\\mathfrak{L}$ of both $\\alpha$ and $\\beta$ planes and the edges of the lattice joining pairs of these sites;\\\\ \n(4) For a lattice site $i\\in \\mathfrak{L}_{(\\alpha\\beta)}$, denote by $\\bar{i}$ the same site of the other plane~(if $i\\in \\alpha$, then $\\bar{i}\\in \\beta$ and vice versa);\\\\\n(5) For $i\\in \\mathfrak{L}_{(\\alpha\\beta)}$, define $n(i)$ to be the number of thick horizontal edges in $\\mathfrak{L}_{(\\alpha\\beta)}$ linked to $i$~[notice that $n(i)\\in\\{0,1\\}$];\\\\\n(6) The loop product is defined as \n\\begin{equation}\\label{eq:def_loop_observable}\n\t\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]=\\prod_{i\\in \\mathfrak{L}_{(\\alpha\\beta)}} \\sigma_i^{n(\\bar{i})}.\n\\end{equation}\nIn Fig.~\\ref{fig:hc0} we illustrate the definition of $\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]$ for a contractible and a noncontractible loop $\\mathfrak{L}$.\nIn Sec.~\\ref{sec:TPloop} we will compute their thermal expectation values\n\\begin{eqnarray}\\label{eq:thermal_loop}\n\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle=\\frac{1}{Z}\\sum_{\\{\\sigma\\}}\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}] e^{-\\beta H[\\{\\sigma\\}]}.\n\\end{eqnarray}\nWe will see that the expectation values of these observables are sensitive to the topology of the loop $\\mathfrak{L}_{(\\alpha\\beta)}$. Namely, for a contractible loop $\\mathfrak{L}_{(\\alpha\\beta)}$ we have $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle=\\pm 1$~(and the same loop $\\mathfrak{L}_{(\\alpha\\beta)}$ takes the same value for different phases), while for a non-contractible loop $\\mathfrak{L}_{(\\alpha\\beta)}$, $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ is equal to $0$ in one phase~(the $A$-phase) and $-1\/3$ in another phase~(the $B$-phase). Therefore, noncontractible loop observables can be used as order parameters of this model.\n\n\\section{The solution}\\label{sec:solution}\n\\begin{figure}\n\t\\center{\\includegraphics[width=0.75\\linewidth]{brickwall-unitcell-RGB.png}}\n\t\\caption{\\label{fig:brickwall-unitcell} The 2D brick wall lattice on which the transfer matrix Eq.~\\eqref{eq:def_TM} is defined. A unit cell is shown in the shaded square. The conserved loop operator $\\hat{W}_p$ acts on the six spins of the elementary plaquette $p$, and the conserved noncontractible loop $\\hat{\\Phi}_x$~($\\hat{\\Phi}_y$) acts on a row~(column) of spins. The $\\pm \\kappa$ shown next to each link is the real part of the small perturbation to the link coupling constant needed to gap the fermionic spectrum of the $B$-phase. }\n\\end{figure} \n\\subsection{The Transfer Matrix}\\label{sec:TM}\nThe first step to solve this model is to find the transfer matrix $\\hat{T}$ for each period of $x,y,z$ planes, as shown in Fig.~\\ref{fig:hc0}, defined so that $Z=\\mathrm{Tr}[\\hat{T}^M]$, where $M$ is the total number of periods. We will show that when the conditions~\\eqref{cond:exact} are satisfied, the transfer matrix is\n\\begin{eqnarray}\\label{eq:def_TM}\n \\hat{T}&=&\\exp\\left(K_x\\sum_{\\langle ij\\rangle\\in X_{2D}}\\hat{\\sigma}^x_{i}\\hat{\\sigma}^x_{j}\\right)\\exp\\left(K_y\\sum_{\\langle ij\\rangle\\in Y_{2D}}\\hat{\\sigma}^y_{i}\\hat{\\sigma}^y_{j}\\right)\\nonumber\\\\\n &&\\times\\exp\\left(K_z\\sum_{\\langle ij\\rangle\\in Z_{2D}}\\hat{\\sigma}^z_{i}\\hat{\\sigma}^z_{j}\\right),\n\\end{eqnarray}\nwhere $\\hat{\\sigma}^{x,y,z}_i$ are Pauli operators acting on the spin located at site $j$ of the 2D brick wall lattice shown in Fig.~\\ref{fig:brickwall-unitcell}, $X_{2D}$ denotes the set of all the $x$-links shown in Fig.~\\ref{fig:brickwall-unitcell}, and similarly for $Y_{2D},Z_{2D}$. Henceforth, we use $\\sum_{x}$, $\\sum_{y}$, and $ \\sum_{z}$ as abbreviations for $\\sum_{(i,j) \\in X_{2D}}$, $\\sum_{(i,j) \\in Y_{2D}}$, and $\\sum_{(i,j) \\in Z_{2D}}$, respectively.\nWe prove Eq.~\\eqref{eq:def_TM} by inserting resolutions of identity on each plane in $Z=\\mathrm{Tr}[\\hat{T}^M]$ in Eq.~\\eqref{eq:def_TM} and showing that it reproduces\nEq.~\\eqref{eq:Z}. The trick here is that when inserting resolution of identity, we use the $\\hat{\\sigma}^x$ basis $|X\\rangle\\equiv\\otimes_j|\\sigma_j\\rangle_x$ on $x$-planes, $\\hat{\\sigma}^y$ basis $|Y\\rangle\\equiv\\otimes_j|\\sigma_j\\rangle_y$ on $y$-planes, and $\\hat{\\sigma}^z$ basis $|Z\\rangle\\equiv\\otimes_j|\\sigma_j\\rangle_z$ on $z$-planes, where $\\hat{\\sigma}^x_j|\\sigma_j\\rangle_x=\\sigma_j|\\sigma_j\\rangle_x$ and similarly for $|\\sigma_j\\rangle_y,|\\sigma_j\\rangle_z$. Therefore, we have\n\\begin{widetext}\n\\begin{eqnarray}\\label{eq:quantum_classical_map}\n \\mathrm{Tr}[\\hat{T}^M]&=&\\sum_{\\substack{X_1,Y_1,Z_1,\\ldots,\\\\ X_M,Y_M,Z_M}}\\langle X_1|e^{K_x\\sum_{x}\\hat{\\sigma}^x_i\\hat{\\sigma}^x_j}|Y_1\\rangle \\langle Y_1|e^{K_y\\sum_{y}\\hat{\\sigma}^y_i\\hat{\\sigma}^y_j}|Z_1\\rangle \\langle Z_1|e^{K_z\\sum_{z}\\hat{\\sigma}^z_i\\hat{\\sigma}^z_j}|X_2\\rangle\\langle X_2|\\cdots\\nonumber\\\\\n &&~~~~~~~~{}\\times|X_M\\rangle\\langle X_M|e^{K_x\\sum_{x}\\hat{\\sigma}^x_i\\hat{\\sigma}^x_j}|Y_M\\rangle \\langle Y_M|e^{K_y\\sum_{y}\\hat{\\sigma}^y_i\\hat{\\sigma}^y_j}|Z_M\\rangle \\langle Z_M|e^{K_z\\sum_{z}\\hat{\\sigma}^z_i\\hat{\\sigma}^z_j}|X_1\\rangle\\nonumber\\\\\n &=&\\sum_{\\{\\sigma\\}}\\exp\\left(K_x\\sum_{\\langle ij\\rangle\\in \\mathbf{X}} \\sigma_i \\sigma_j+K_y\\sum_{\\langle ij\\rangle\\in \\mathbf{Y}} \\sigma_i \\sigma_j+K_z\\sum_{\\langle ij\\rangle\\in \\mathbf{Z}} \\sigma_i \\sigma_j\\right)\\nonumber\\\\\n &&~~~~{}\\times\\langle X_1|Y_1\\rangle \\langle Y_1|Z_1\\rangle \\langle Z_1|X_2\\rangle\\cdots\n \\langle X_M|Y_M\\rangle \\langle Y_M|Z_M\\rangle \\langle Z_M|X_1\\rangle.\n\\end{eqnarray}\n\\end{widetext}\nThe first factor corresponds to the classical Boltzmann weight contributed by all the horizontal links. For the overlap matrices in the last line of Eq.~\\eqref{eq:quantum_classical_map}, using a suitable phase convention for basis states \n\\begin{eqnarray}\n\t|\\pm 1\\rangle_z&=&\\{|\\uparrow\\rangle,e^{\\pi i\/4}|\\downarrow\\rangle \\},\\nonumber\\\\\n\t|\\pm 1\\rangle_x&=&\\{\\frac{|\\uparrow\\rangle+|\\downarrow\\rangle}{\\sqrt{2}}e^{3\\pi i\/4},\\frac{|\\uparrow\\rangle-|\\downarrow\\rangle}{\\sqrt{2}}e^{\\pi i\/2} \\}\\nonumber\\\\\n\t|\\pm 1\\rangle_y&=&\\{\\frac{|\\uparrow\\rangle+i|\\downarrow\\rangle}{\\sqrt{2}}e^{-3\\pi i\/4},\\frac{|\\uparrow\\rangle-i|\\downarrow\\rangle}{\\sqrt{2}}e^{-\\pi i\/2} \\},\\nonumber\n\\end{eqnarray}\nwe have $${}_x\\langle \\sigma|\\sigma'\\rangle_y={}_y\\langle \\sigma|\\sigma'\\rangle_z ={}_z\\langle \\sigma|\\sigma'\\rangle_x =\\frac{1}{\\sqrt{2}}e^{\\frac{\\pi i}{4}(\\sigma\\sigma'-\\frac{\\sigma+\\sigma'}{2}+3)}.$$ The overlaps give the Boltzmann weights contributed by the vertical links and external field terms with $\\beta J_\\perp=\\pi i\/4,\\beta h=\\pi i\/4$, up to an irrelevant constant shift of the energy. Also, one can show that adding $2\\pi i $ to $4J_\\perp\\beta$ or $2h\\beta$ will only multiply the partition function by an irrelevant overall constant phase factor, since whenever we flip a spin $\\sigma_j$, the imaginary part of $\\beta H[\\{\\sigma\\}]$ changes by $\\pm 2( 2s J_\\perp- h )\\beta $, where $s=(\\sigma'_j+\\sigma''_j)\/2\\in\\{-1,0,+1\\}$, and $\\sigma'_j$~($\\sigma''_j$) is the neighbor of $\\sigma_j$ lying above~(below) it. Therefore the model Eq.~\\eqref{eq:HIK} has transfer matrix Eq.~\\eqref{eq:def_TM} when Eq.~\\eqref{cond:exact} is satisfied. \n\n\n \nNow that we have obtained the transfer matrix $\\hat{T}$ of our model, the next step is to calculate the largest~(in magnitude) eigenvalue $\\Lambda_{\\max}$ of $\\hat{T}$, which governs the free energy in the thermodynamic limit \n\\begin{equation}\\label{def:free_energy}\n\tF =-M k_B T \\ln\\Lambda_{\\max}+O(\\Lambda_1^M\/\\Lambda_{\\max}^M),\n\\end{equation}\nwhere $\\Lambda_1$ is the next-to-largest~(in magnitude) eigenvalue of $\\hat{T}$. We will calculate the eigenvalues of $\\hat{T}$ in two steps: in Sec.~\\ref{sec:map_fermion} we map the transfer matrix $\\hat{T}$ to a free fermion transfer matrix $\\hat{T}'$ in Eq.~\\eqref{eq:Tbondfermion}, and then in Sec.~\\ref{sec:solve_fermion_TM} we solve the eigenvalues of this free fermion transfer matrix.\n\n\n\\subsection{Mapping to a free fermion problem}\\label{sec:map_fermion}\nOur goal in this section is to map the transfer matrix $\\hat{T}$ to a free fermion transfer matrix $\\hat{T}'$, written in terms of Majorana fermion bilinear operators. While this can be accomplished by Kitaev's original technique~\\cite{Kitaev2006}, or by using a Jordan-Wigner transformation~\\cite{Feng2007JordanWigner},\nhere we use the algebraic method developed in Refs.~\\cite{Nussinov2009bond,Cobanera2011bond,Chapman2020characterizationof,Ogura2020geometric}, which is far simpler. The key idea of this technique is that, instead of considering the mapping of each individual spin operators, we view the interaction term on each link $\\langle ij\\rangle$ as a whole, and consider the algebra generated by all these terms. We write the transfer matrix as \n\\begin{equation}\\label{eq:Tbond}\n\t\\hat{T}=e^{K_x\\sum_{x}\\hat{\\gamma}_{ij}}e^{K_y\\sum_{y}\\hat{\\gamma}_{ij}}e^{K_z\\sum_{z}\\hat{\\gamma}_{ij}},\n\\end{equation}\nwhere the bond operators are defined as $\\hat{\\gamma}_{ij}=\\hat{\\sigma}_i^\\alpha\\hat{\\sigma}_j^\\alpha$ if $\\langle ij\\rangle$ is an $\\alpha$-link in the 2D brick wall lattice. We now construct another transfer matrix\n\\begin{eqnarray}\\label{eq:Tbondfermion}\n\t\\hat{T}'&=&e^{K_x\\sum_{x}\\hat{\\gamma}'_{ij}}e^{K_y\\sum_{y}\\hat{\\gamma}'_{ij}}e^{K_z\\sum_{z}\\hat{\\gamma}'_{ij}}\\nonumber\\\\\n\t&\\equiv& e^{K_x\\sum_{x}u_{ij}i\\hat{c}_i\\hat{c}_j}e^{K_y\\sum_{y}u_{ij}i\\hat{c}_i\\hat{c}_j}e^{K_z\\sum_{z}u_{ij}i\\hat{c}_i\\hat{c}_j},\n\\end{eqnarray}\nwhich has exactly the same exponential structure and the same set of parameters as $\\hat{T}$, but has the bond operators replaced by Majorana fermion bilinears $\\hat{\\gamma}'_{ij}\\equiv u_{ij}i\\hat{c}_i\\hat{c}_j$ on each link, where $\\hat{c}^\\dagger_i=\\hat{c}_i$, and $\\{\\hat{c}_i,\\hat{c}_j\\}=2\\delta_{ij}$.\nHere $u_{ij}$ is a real number defined independently on each link, whose value is to be determined later. Notice that the ordering of Majorana operators $\\hat{c}_i\\hat{c}_j$ matters in the sum since they anti-commute; throughout this paper, we use the convention that whenever we sum~(or product) over links, each link $\\langle ij\\rangle$ appears only once in the sum, with $i$ representing an even site~(black dots in Fig.~\\ref{fig:brickwall-unitcell}) and $j$ representing an odd site ~(white open circles in Fig.~\\ref{fig:brickwall-unitcell}), and we always order $\\hat{c}_i$ to the left unless otherwise stated. \n\nThe goal now is to choose these real coefficients $\\{u_{ij}\\}$ \\textit{such that the algebra generated by $\\{\\hat{\\gamma}_{ij}\\}$ is isomorphic to the algebra generated by $\\{\\hat{\\gamma}'_{ij}\\}$.} Once this is done, Refs.~\\cite{Nussinov2009bond,Cobanera2011bond,Chapman2020characterizationof,Ogura2020geometric} claim that there exists a unitary mapping $\\hat{U}$ between the two systems such that $\\hat{\\gamma}'_{ij}=\\hat{U} \\hat{\\gamma}_{ij}\\hat{U}^\\dagger$ for all links $\\langle ij\\rangle$~(we will also need to check that the Hilbert space dimensions of the two systems are the same), leading to $\\hat{T}'=\\hat{U}\\hat{T}\\hat{U}^\\dagger$, i.e. $\\hat{T}$ and $\\hat{T}'$ have the same eigenvalues. Requiring the two algebras to be isomorphic means that any algebraic relation satisfied by the generators $\\{\\hat{\\gamma}_{ij}\\}$, say $f(\\{\\hat{\\gamma}_{ij}\\})=0$, must be satisfied by $\\{\\hat{\\gamma}'_{ij}\\}$ as well, $f(\\{\\hat{\\gamma}'_{ij}\\})=0$, and vice versa. In our case, this leads to the following four families of relations:\\\\\n\\textit{Relation 1.} We have $\\hat{\\gamma}^2_{ij}=1$ for each link $\\langle ij\\rangle$, and therefore we must require $\\hat{\\gamma}'^2_{ij}=u_{ij}^2=1$, which constrains $u_{ij}$ to be $\\pm 1$.\\\\\n\\textit{Relation 2.} Two bond operators anti-commute if and only if they share exactly one vertex, otherwise, they commute. It is straightforward to check that this is satisfied by both $\\{\\hat{\\gamma}_{ij}\\}$ and $\\{\\hat{\\gamma}'_{ij}\\}$, so this condition puts no constraints on $\\{u_{ij}\\}$.\\\\\n\\textit{Relation 3.} The product of $\\hat{\\gamma}'_{ij}$ on any closed loop $\\mathfrak{L}$ is equal to a constant, so the product of $\\hat{\\gamma}_{ij}$ on $\\mathfrak{L}$ must be equal to the same constant. It is enough to require this constraint only on all the elementary plaquettes $\\mathfrak{L}_p$ along with two large loops $\\mathfrak{L}_x$ and $\\mathfrak{L}_y$ winding around the torus~(as shown in Fig.~\\ref{fig:brickwall-unitcell}),\nsince the product on other loops decompose into products on these elementary loops. The product of $\\hat{\\gamma}'_{ij}$ on these loops are equal to \n\\begin{eqnarray}\\label{eq:Wpphixphiy}\n\tW_p&\\equiv& \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_p} \\hat{\\gamma}'_{ij}=\\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_p} u_{ij},\\nonumber\\\\%\\hat{\\gamma}'_{01}\\hat{\\gamma}'_{21}\\hat{\\gamma}'_{23}\\hat{\\gamma}'_{43}\\hat{\\gamma}'_{45}\\hat{\\gamma}'_{05}\n\t\\Phi_x&\\equiv & \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_x} \\hat{\\gamma}'_{ij}=\\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_x} u_{ij},\\nonumber\\\\\n\t\\Phi_y&\\equiv& \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_y} \\hat{\\gamma}'_{ij}=\\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_y} u_{ij},\n\\end{eqnarray}\nfor every plaquette $p$, and we order the product of operators according to their linear order in the loop~(the orientation of the loop and the initial point do not affect the result of the product).\n\nThe product of $\\hat{\\gamma}_{ij}$ on these loops are equal to \n\\begin{eqnarray}\\label{eq:phixphiy}\n\t\\hat{W}_p&\\equiv& \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_p} \\hat{\\gamma}_{ij}=-\\hat{\\sigma}^z_0\\hat{\\sigma}^y_1\\hat{\\sigma}^y_2\\hat{\\sigma}^z_3\\hat{\\sigma}^x_4\\hat{\\sigma}^x_5,\\nonumber\\\\\n\t\\hat{\\Phi}_x&\\equiv & \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_x} \\hat{\\gamma}_{ij}=-\\prod_{i\\in \\mathfrak{L}_x} \\hat{\\sigma}^y_i,\\nonumber\\\\\n\t\\hat{\\Phi}_y&\\equiv& \\prod_{\\langle ij\\rangle\\in \\mathfrak{L}_y} \\hat{\\gamma}_{ij}=-\\prod_{i\\in \\mathfrak{L}_y} \\hat{\\sigma}^z_i,\n\\end{eqnarray}\nwhere $0,1,2,3,4,5$ label the sites of the plaquette $p$, as shown in Fig.~\\ref{fig:brickwall-unitcell}~(and similarly for all other plaquettes).\nAlthough the RHS of Eq.~\\eqref{eq:phixphiy} are not constants, one can check that these operators mutually commute, and they commute with all the bond operators $\\hat{\\gamma}_{ij}$, and therefore they commute with the transfer matrix $\\hat{T}$. They play the role of conserved observables, and their common eigenspaces are invariant under the action of $\\hat{T}$. Further, since $\\hat{W}_p^2=\\hat{\\Phi}_x^2=\\hat{\\Phi}_y^2=1$, their eigenvalues can only be $\\pm 1$. \nTo guarantee the algebraic isomorphism between the algebras $\\{\\hat{\\gamma}_{ij}\\}$ and $\\{\\hat{\\gamma}'_{ij}\\}$, we need to \nmap the spin model transfer matrix $\\hat{T}$ in each common eigenspace of $\\{\\hat{W}_p,\\hat{\\Phi}_x,\\hat{\\Phi}_y\\}$ to a different fermionic transfer matrix $\\hat{T}'$, with the $u_{ij}$ chosen in such a way that their loop products $\\{W_p,\\Phi_x,\\Phi_y\\}$ equal the eigenvalues of $\\{\\hat{W}_p,\\hat{\\Phi}_x,\\hat{\\Phi}_y\\}$.\\\\\n\\textit{Relation 4.}\nOn a closed manifold, the product of all $\\{\\hat{\\gamma}_{ij}\\}$ on the lattice equals a constant:\n\\begin{eqnarray}\\label{eq:prod_all_bond}\n\\prod_{\\text{all }\\langle ij\\rangle} \\hat{\\gamma}_{ij}=i^{4L_xL_y}=1,\n\\end{eqnarray}\nwhere $L_x$~($L_y$) is the system size in the $x$-~($y$-) direction.\nSimilarly, the product of all $\\{\\hat{\\gamma}'_{ij}\\}$ is\n\\begin{eqnarray}\\label{eq:prod_all_bond_f}\n\\prod_{\\text{all }\\langle ij\\rangle} \\hat{\\gamma}'_{ij}=\n\\hat{P}_f\\prod_{\\text{all }\\langle ij\\rangle} u_{ij},\n\\end{eqnarray}\nwhere\n$\\hat{P}_f\\equiv \\prod_{z}\\left(-i\\hat{c}_i\\hat{c}_j\\right)$\nis the conserved fermion parity operator. Therefore the algebraic isomorphism restricts the fermion model to the eigen-subspace of $\\hat{P}_f$ with eigenvalue\n\\begin{equation}\\label{eq:parity_restriction}\n\tP_f= \\prod_{\\text{all }\\langle ij\\rangle} u_{ij}.\n\\end{equation}\n\\textit{Summary and consistency check.} In summary, the mutually commuting conserved operators $\\{\\hat{W}_p,\\hat{\\Phi}_x,\\hat{\\Phi}_y\\}$ split the full Hilbert space into a direct sum of their common eigen-subspaces, and the transfer matrix $\\hat{T}$ leaves each subspace invariant. In the subspace labeled by the conserved eigenvalues $\\{W_p,\\Phi_x,\\Phi_y\\}$, the transfer matrix $\\hat{T}$ is mapped to a fermionic transfer matrix $\\hat{T}'$ defined in Eq.~\\eqref{eq:Tbondfermion} where the parameters $u_{ij}=\\pm 1$ are chosen to satisfy Eq.~\\eqref{eq:Wpphixphiy}~\\footnote{While there are exponentially many solutions $\\{u_{ij}\\}$ to Eq.~\\eqref{eq:Wpphixphiy} for a fixed configuration $\\{W_p,\\Phi_x,\\Phi_y\\}$, all of them are equivalent up to a gauge transformation, and the spectrum of $\\hat{T}'$ only depends on the values of $\\{W_p,\\Phi_x,\\Phi_y\\}$.}, and $\\hat{T}'$ is restricted to a fixed fermion parity sector satisfying Eq.~\\eqref{eq:parity_restriction}. \n\nAs a consistency check, let us verify that the subspace dimension of the spin and fermionic systems, mapped to each other by the above algebraic isomorphism, are the same. For the spin system, we have $4L_xL_y$ qubit degrees of freedom~(d.o.f.) in total; in each subspace, the constraint Eq.~\\eqref{eq:phixphiy} removes $2L_x L_y-1+2$ qubit d.o.f~($-1$ because the product of all $\\hat{W}_p$ is a constant, so only $2L_x L_y-1$ of them are independent), leaving us with $2L_x L_y-1$ qubit d.o.f. For the fermionic system, we have $4L_xL_y$ Majorana fermions in total, which amounts to $2L_xL_y$ Dirac fermion d.o.f.; the fermion parity restriction Eq.~\\eqref{eq:parity_restriction} further removes one of them, leaving us $2L_x L_y-1$ Dirac fermion d.o.f.. Therefore the Hilbert space dimension of the two systems are the same, both equal to $2^{2L_x L_y-1}$.\n\n\\subsection{Solving the free fermion transfer matrix}\\label{sec:solve_fermion_TM}\nIn the last section we mapped the transfer matrix $\\hat{T}$ in each sector labeled by $\\{W_p,\\Phi_x,\\Phi_y\\}$ to a free fermion transfer matrix $\\hat{T}'$ in Eq.~\\eqref{eq:Tbondfermion}, where $u_{ij}=\\pm 1$ are chosen to satisfy Eq.~\\eqref{eq:Wpphixphiy}, and the fermion parity satisfies Eq.~\\eqref{eq:parity_restriction}. Now we solve these free fermion problems in each sector to get the full spectrum of $\\hat{T}$. The difficulty here is that there are exponentially many such sectors~($2^{2L_x L_y+1}$ in total), most of which are not translationally invariant and can only be solved numerically. Fortunately we are most interested in the sector that contains the principal eigenvalue $\\Lambda_\\mathrm{max}$ of $\\hat{T}$, i.e. the sector $\\{W_p,\\Phi_x,\\Phi_y\\}$ where the principal eigenvalue of $\\hat{T}'$ is largest, since $\\Lambda_\\mathrm{max}$~(and the corresponding principal eigenstate $|\\Lambda_\\mathrm{max}\\rangle$) determines the thermodynamic properties of the original classical system. \nIn App.~\\ref{appen:Lieb} we prove a generalization of Lieb's optimal flux theorem~\\cite{lieb1994flux} for the transfer matrix $\\hat{T}'$, which shows that for real $K_x,K_y,K_z$, the principal eigenvalue of $\\hat{T}'$ is maximized by a configuration $\\{W_p,\\Phi_x,\\Phi_y\\}$ where all $W_p$ are equal to $+1$. From now on we will call such a configuration vortex-free, and for a configuration with some $W_p=-1$ we say it has a vortex excitation at $p$. This leaves four sectors to consider, corresponding to $(\\Phi_x,\\Phi_y)=(++),(+-),(-+),(--)$~[we use $(++)$ as a shorthand for $(+1,+1)$, and similarly for the other three]. These four sectors can be treated in an identical way, which we do in the following.\n\nWe first need to find a solution $\\{u_{ij}\\}$ to Eq.~\\eqref{eq:Wpphixphiy}. For the $(++)$ sector, we can simply take $u_{ij}=+1$ for all links $\\langle ij\\rangle$. To obtain solutions for the other three vortex-free sectors, %\nnotice that we can flip the sign of $\\Phi_x$ or $\\Phi_y$ by flipping the signs of $u_{ij}$ on a large~(i.e. non-contractible) loop of links, without changing the value of any $W_p$. For example, if we flip all the $z$-links between $x=L_x-1\/2$ and $x=0$~(denote this set of links by $Z_{L_x-1\/2,0}$), then we can flip the sign of $\\Phi_x$ without flipping any of the $W_p$. Similarly we can flip the sign of $\\Phi_y$ by flipping the signs of all the $y$-links between $y=L_y-1\/2$ and $y=0$~(denote this set of links by $Y_{L_y-1\/2,0}$). In this way, the solution for the sector $(\\Phi_x, \\Phi_y)$ can be taken as $u_{ij}=1$ for $\\langle ij\\rangle\\notin Z_{L_x-1\/2,0} \\cup Y_{L_y-1\/2,0} $, $u_{ij}=\\Phi_x$ for $\\langle ij\\rangle\\in Z_{L_x-1\/2,0} $, and $u_{ij}=\\Phi_y$ for $\\langle ij\\rangle\\in Y_{L_y-1\/2,0}$. %\n\nThe transfer matrix defined in Eq.~\\eqref{eq:Tbondfermion} for all these four sectors can be written in a translationally invariant way provided that we use suitable boundary conditions for the Majorana operators. To this end, we use $i=(\\vec{r},\\lambda)$ to label lattice sites, where $\\vec{r}$ labels the unit cells, and $\\lambda=0,1,2,3$ label the sites in a unit cell, as shown in Fig.~\\ref{fig:brickwall-unitcell}. We define $\\hat{c}_{(L_x,y),\\lambda}=\\Phi_x\\hat{c}_{(0,y),\\lambda}$ and $\\hat{c}_{(x,L_y),\\lambda}=\\Phi_y\\hat{c}_{(x,0),\\lambda}$, corresponding to periodic or antiperiodic boundary conditions.\nThen the transfer matrices for all the four vortex-free sectors have the same expression\n\\begin{equation}\\label{eq:Ttilde}\n\t\\hat{T}'=e^{K_x\\sum_{x}i\\hat{c}_i\\hat{c}_j}e^{K_y\\sum_{y}i\\hat{c}_i\\hat{c}_j}e^{K_z\\sum_{z}i\\hat{c}_i\\hat{c}_j}.\n\\end{equation}\nwhere the above boundary condition on $\\hat{c}_i,\\hat{c}_j$ is used, and it is understood that the lattice coordinates of $i,j$ for each link $\\langle ij\\rangle$ should be consecutive numbers, e.g. the term on a flipped $z$-link is understood as $ \\hat{c}_{(L_x-1,y),\\lambda} \\hat{c}_{(L_x,y),\\lambda'}$ instead of $\\hat{c}_{(L_x-1,y),\\lambda} \\hat{c}_{(0,y),\\lambda'}$.\n\nThe rest of the task is to find the eigenvalues of the translationally invariant vortex-free transfer matrix $\\hat{T}'$ in Eq.~\\eqref{eq:Ttilde} under the four possible boundary conditions $(++),(+-),(-+),(--)$. To this end, we introduce the Fourier transform of the Majorana operators\n\\begin{eqnarray}\\label{eq:Ftrans}\n \\hat{a}_{\\vec{q},\\lambda}&=&\\frac{1}{\\sqrt{2N}}\\sum_{\\vec{r}} e^{-i\\vec{q}\\cdot\\vec{r}}\\hat{c}_{\\vec{r},\\lambda},\\nonumber\\\\\n \\hat{c}_{\\vec{r},\\lambda}&=&\\sqrt{\\frac{2}{N}}\\sum_{\\vec{q}} e^{i\\vec{q}\\cdot\\vec{r}}\\hat{a}_{\\vec{q},\\lambda},\n\\end{eqnarray}\nwhere $N=L_x L_y$ is the total number of unit cells. \nThe quasi-momentum in the $\\alpha$-direction $q_\\alpha$ is quantized as $2n\\pi\/L_\\alpha$ where $n\\in\\mathbb{Z}$ if $\\Phi_\\alpha=+1$ and $n\\in\\mathbb{Z}+1\/2$ if $\\Phi_\\alpha=-1$ . The operators $\\hat{a}_{\\vec{q},\\lambda}$ satisfy $\\hat{a}_{\\vec{q},\\lambda}^\\dagger=\\hat{a}^{\\phantom{\\dagger}}_{-\\vec{q},\\lambda}$ and $\\{\\hat{a}^{\\phantom{\\dagger}}_{\\vec{p},\\lambda},\\hat{a}^\\dagger_{\\vec{q},\\mu}\\}=\\delta_{\\vec{p},\\vec{q}}\\delta_{\\lambda,\\mu}$. We can now rewrite $\\hat{T}'$ as\n\\begin{eqnarray}\\label{eq:TtildeFF}\n \\hat{T}'&=&\\exp\\left[2K_x\\sum_{\\vec{q}}(i\\hat{a}_{\\vec{q},0}\\hat{a}_{-\\vec{q},1}+i\\hat{a}_{\\vec{q},2}\\hat{a}_{-\\vec{q},3})\\right]\\nonumber\\\\\n &&\\times\\exp\\left[2K_y\\sum_{\\vec{q}}(i\\hat{a}_{\\vec{q},0}\\hat{a}_{-\\vec{q},1}e^{iq_y}+i\\hat{a}_{\\vec{q},2}\\hat{a}_{-\\vec{q},3}e^{-iq_y})\\right]\\nonumber\\\\\n &&\\times\\exp\\left[2K_z\\sum_{\\vec{q}}(i\\hat{a}_{\\vec{q},2}\\hat{a}_{-\\vec{q},1}+i\\hat{a}_{\\vec{q},0}\\hat{a}_{-\\vec{q},3}e^{iq_x})\\right],\\nonumber\\\\\n &\\equiv &\\tilde{T}_{0}\\prod_{\\vec{q}+}\\tilde{T}_{\\vec{q}},\n\\end{eqnarray}\nwhere $\\tilde{T}_{0}$ contains all the terms with $\\vec{q}\\equiv -\\vec{q}~(\\mathrm{mod}~ 2\\pi)$, and $\\prod_{\\vec{q}+}$ is the product over $\\vec{q}$ with $\\vec{q}\\not\\equiv -\\vec{q}~(\\mathrm{mod}~ 2\\pi)$ such that each pair $\\pm\\vec{q}$ appears exactly once, and in the last line we have rearranged terms of different $\\vec{q}$ modes using $[\\tilde{T}_{\\vec{q}},\\tilde{T}_{0}]=0$, and $[\\tilde{T}_{\\vec{q}},\\tilde{T}_{\\vec{p}}]=0$ for $\\vec{q}\\neq\\pm \\vec{p}$. Because of this commutativity, all the $\\tilde{T}_{\\vec{q}}$ and $\\tilde{T}_{0}$ can be simultaneously diagonalized. We treat $\\tilde{T}_{\\vec{q}}$ first, which can be written as \n\\begin{eqnarray}\\label{def:Tq}\n\\tilde{T}_{\\vec{q}}&=&e^{2K_x \\sum_{\\lambda,\\mu}P^{(\\vec{q})}_{\\lambda\\mu}\\hat{a}^\\dagger_{\\vec{q}\\lambda}\\hat{a}^{\\phantom{\\dagger}}_{\\vec{q}\\mu}}e^{2K_y \\sum_{\\lambda,\\mu} Q^{(\\vec{q})}_{\\lambda\\mu}\\hat{a}^\\dagger_{\\vec{q}\\lambda}\\hat{a}^{\\phantom{\\dagger}}_{\\vec{q}\\mu}}\\nonumber\\\\\n&&\\times e^{2K_z \\sum_{\\lambda,\\mu} R^{(\\vec{q})}_{\\lambda\\mu}\\hat{a}^\\dagger_{\\vec{q}\\lambda}\\hat{a}^{\\phantom{\\dagger}}_{\\vec{q}\\mu}},\n\\end{eqnarray}\nwhere the $4\\times 4$ matrices $P^{(\\vec{q})},Q^{(\\vec{q})},R^{(\\vec{q})}$ are~(we drop the superscript $\\vec{q}$ when there is no confusion)\n\\begin{eqnarray}\\label{def:PQR}\nP&=&\\begin{pmatrix}\n\t0 & i & 0 & 0 \\\\\n\t-i& 0 & 0 & 0 \\\\\n\t0 & 0 & 0 & i \\\\\n\t0 & 0 & -i& 0\n\\end{pmatrix},\nR=\\begin{pmatrix}\n\t0 & 0 & 0 & ie^{-iq_x} \\\\\n\t0 & 0 & -i & 0 \\\\\n\t0 & i & 0 & 0 \\\\\n\t-ie^{iq_x} & 0 & 0 & 0\n\\end{pmatrix},\\nonumber\\\\\nQ&=&\\begin{pmatrix}\n\t0 & ie^{-iq_y} & 0 & 0 \\\\\n\t-ie^{iq_y}& 0 & 0 & 0 \\\\\n\t0 & 0 & 0 & ie^{iq_y} \\\\\n\t0 & 0 & -ie^{-iq_y}& 0\n\\end{pmatrix}.\n\\end{eqnarray}\nNotice that the fermion bilinears $\\hat{a}^\\dagger_{\\vec{q}\\lambda}\\hat{a}^{\\phantom{\\dagger}}_{\\vec{q}\\mu}$ in Eq.~\\eqref{def:Tq} form the basis of an $\\mathfrak{sl}(4)$ Lie algebra, so $\\tilde{T}_{\\vec{q}}$ is an element of the corresponding $\\mathrm{SL}(4)$ Lie group. Using the relation between the fundamental representation and the free fermion representation of this Lie algebra and group, (similar to the method in App.~\\ref{appen:numerical_method}), one can show that \n\\begin{equation}\\label{eq:Tq_diagonal}\n\t\\tilde{T}_{\\vec{q}}=e^{\\epsilon_{\\vec{q},1}(\\hat{n}_{\\vec{q},1}-\\hat{n}_{\\vec{q},\\bar{1}})+\\epsilon_{\\vec{q},2}(\\hat{n}_{\\vec{q},2}-\\hat{n}_{\\vec{q},\\bar{2}})},\n\\end{equation}\nwhere $ e^{\\pm\\epsilon_{\\vec{q},1}}, e^{\\pm\\epsilon_{\\vec{q},2}}$ are the eigenvalues of the matrix $T_{\\vec{q}}=e^{2K_x P}e^{2K_yQ}e^{2K_zR}$, which is the representation of $\\tilde{T}_{\\vec{q}}$ in the fundamental representation of the $\\mathrm{SL}(4)$ Lie group, and $\\hat{n}_{\\vec{q},j},\\hat{n}_{\\vec{q},\\bar{j}}$ ~(with $j\\in \\{1,2\\}$) are mutually commuting fermion number operators. The single mode energies $\\epsilon_{\\vec{q},j}$ can be analytically calculated by solving the quartic equation $P_{T_{\\vec{q}}}(x)=0$, where $P_{T_{\\vec{q}}}(x)$ is the degree four characteristic polynomial of the $4\\times 4$ matrix $T_{\\vec{q}}$. This quartic equation can be simplified to a quadratic one $z^2+Az+B=0$, where $z=(x+1\/x)\/2=\\cosh\\epsilon_{\\vec{q},j}$~(for $j=1,2$), and \n\\begin{eqnarray}\\label{eq:chepsilon}\nA&=&-2c_3(c_1c_2+s_1s_2 \\cos q_y),\\nonumber\\\\\nB&=&\\frac{1}{8}S_1S_2(3+C_3-2s_3^2\\cos q_x)\\cos q_y+\\frac{1}{2} s_1^2 s_2^2\\cos(2q_y)\\nonumber\\\\\n&&{}+\\frac{1}{4}s_3^2(1-C_1C_2)\\cos(q_x)+\\frac{C_1+C_2+3C_3}{8}\\nonumber\\\\\n&&{}+\\frac{C_1C_2}{4}+\\frac{C_1C_2C_3}{8},\n\\end{eqnarray}\nwhere $c_j=\\cosh 2K_j,s_j=\\sinh 2K_j,C_j=\\cosh4K_j$, and $S_j=\\sinh4K_j$.\nSince the eigenvalues of $T_{\\vec{q}}$ come in pairs $\\pm\\epsilon_{\\vec{q},1}, \\pm\\epsilon_{\\vec{q},2}$, we can assume without loss of generality that $0\\leq \\mathrm{Re}[\\epsilon_{\\vec{q},1}]\\leq \\mathrm{Re}[\\epsilon_{\\vec{q},2}]$. Then the maximal eigenvalue of $\\tilde{T}_{\\vec{q}}$ is $e^{\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2}}$.\n\nThe term $\\tilde{T}_{0}$ in the last line of Eq.~\\eqref{eq:TtildeFF} is defined by %\n$\\tilde{T}_{0}=\\prod_{\\vec{q}\\equiv \\vec{0}~(\\mathrm{mod}~\\pi)}\\tilde{T}_{0,\\vec{q}}$ with\n\\begin{equation}\\label{eq:T_0qxqy}\n\t\\tilde{T}_{0,\\vec{q}}=e^{2(K_x+K_ye^{iq_y})i\\hat{a}_{\\vec{q}0}\\hat{a}_{\\vec{q}1}(1-\\hat{P}_{\\vec{q}})} e^{2K_zi\\hat{a}_{\\vec{q}1}\\hat{a}_{\\vec{q}2}(1+\\hat{P}_{\\vec{q}}e^{iq_x})},\n\\end{equation}\nwhere $\\hat{P}_{\\vec{q}}=4\\hat{a}_{\\vec{q},0}\\hat{a}_{\\vec{q},1}\\hat{a}_{\\vec{q},2}\\hat{a}_{\\vec{q},3}$. \nUsing $\\hat{a}^\\dagger_{\\vec{q},\\lambda}=\\hat{a}_{\\vec{q},\\lambda}, \\hat{a}_{\\vec{q},\\lambda}^2=1\/2$,\nthe eigenvalues of $\\tilde{T}_{0,\\vec{q}}$ can be straightforwardly obtained by diagonalizing Eq.~\\eqref{eq:T_0qxqy}, and one can show that the largest one happens to be equal to $e^{(\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2})\/2}$. %\n\nWe have not yet taken into account the fermion parity restriction in Eq.~\\eqref{eq:parity_restriction}. However, as we will see in Sec.~\\ref{sec:TPloop},\nthis constraint changes $\\ln\\Lambda_{\\mathrm{max}}$ by at most $O(\\epsilon_{\\vec{q},j})$, and therefore does not affect the free energy density in the thermodynamic limit. The largest eigenvalue $\\Lambda_{\\mathrm{max}}^{(\\Phi_x,\\Phi_y)}$ of $\\hat{T}'$ is\n\\begin{eqnarray}\\label{eq:lamdba_max}\n\t\\ln\\Lambda_{\\mathrm{max}}^{(\\Phi_x,\\Phi_y)}&=&\\frac{1}{2}\\sum_{\\vec{q}}(\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2}),%\n\\end{eqnarray}\nwhere $(\\Phi_x,\\Phi_y)\\in\\{(++),(+-),(-+),(--)\\}$, and the RHS implicitly depends on $(\\Phi_x,\\Phi_y)$ through the quantization of $\\vec{q}$. %\nThe largest eigenvalue $\\Lambda_{\\mathrm{max}}$ of $\\hat{T}$ is the largest of these four. Regardless of which one is the largest, the free energy density~(per site) in the thermodynamic limit is\n\\begin{eqnarray}\\label{eq:free_energy}\nf\\equiv\\frac{F}{12MN}&=&-\\frac{k_BT}{24N}\\sum_{\\vec{q}}(\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2})\\\\\n&=&-\\frac{k_BT}{96\\pi^2}\\iint_{[-\\pi,\\pi]^2}(\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2})~d^2q,\\nonumber\n\\end{eqnarray}\nwhere the free energy $F$ is defined in Eq.~\\eqref{def:free_energy}.\n\n\\subsection{Excitations and phase boundaries}\\label{sec:phase_boundary}\n\\begin{figure}\n\t\\center{\\includegraphics[width=0.7\\linewidth]{critical.png}}\n\t\\caption{\\label{fig:critical} A 2D section of the 3D phase diagram of our model, with the intersecting plane $K_x+K_y+K_z=\\beta(J_x+J_y+J_z)=\\mathrm{const.}$ The parameters $(K_x,K_y,K_z)$ of an arbitrary point in the diagram is given by the distance from that point to the three sides of the triangle. \n\t%\n\tThe $B$-phase~(shaded) has a gapless transfer matrix, which acquires a gap after a small perturbation is introduced~(Sec.~\\ref{sec:phase_boundary}).\n\tThe $A$-region has a gapped transfer matrix and consists of three disjoint phases $A_x, A_y,A_z$. In Sec.~\\ref{sec:critical_exp} we study the critical behavior of the free energy as we approach the phase boundary from the $A_z$-phase $\\eta\\equiv K_z - K_x - K_y \\to 0^+$. }\n\\end{figure}\nIn this section we study other eigenvalues of the transfer matrix $\\hat{T}$ beyond the principal eigenvalue, and, using this, determine the phase diagram of our model. It is useful to define an effective non-Hermitian Hamiltonian \n$\t\\hat{H}=-\\ln \\hat{T}.$\nIn this way the principal eigenstates of $\\hat{T}$ are mapped to the ground states of $\\hat{H}$ and the eigenvalues $\\Lambda_j$ of $\\hat{T}$ are related to excitation energies $E_j-E_0$ of $\\hat{H}$ by $E_j-E_0=\\ln \\Lambda_{\\text{max}}-\\ln\\Lambda_j$. For the rest of this paper, we use the term ``excitation spectrum of $\\hat{T}$'' to mean the excitation spectrum of $\\hat{H}$, and call the transfer matrix ``gapped''~(``gapless'') if $\\mathrm{Re}[E_j-E_0]$ is gapped~(gapless) in the thermodynamic limit. The spectral gap $\\Delta=\\min_{j\\neq 0}\\mathrm{Re}[E_j-E_0]$ plays an important role in the physical properties of the original classical Ising model. First, as we will see in a moment, the phase boundary of our model is determined by regions where $\\Delta$ vanishes. Secondly, although we do not calculate in this paper, we claim that two point connected correlations $\\langle \\sigma_i\\sigma_j\\rangle_c$~(or more generally, $\\langle O_i O_j\\rangle_c$ where $O_i$ is a product of classical spins in a local region) decay exponentially in distance when $\\Delta>0$, while there are algebraically decaying correlations when $\\Delta=0$. \n\n\n\nThere are two types of excitations: fermionic excitations, corresponding to the positive energy eigenmodes of the fermionic transfer matrix $\\hat{T}'$, %\nand vortex excitations, corresponding to eigenstates of $\\hat{T}'$ in a different sector $\\{W_p,\\Phi_x,\\Phi_y\\}$ where some of $W_p$s are equal to $-1$. Vortices can only be created in pairs. \nA pair of vortices can be created by first drawing a segment connecting the two vortices~(the segment should avoid passing through lattice sites) and then flipping $u_{ij}$ on all the lattice edges intersecting with this segment~(similar to Kitaev's honeycomb~\\cite{Kitaev2006} and toric code~\\cite{Kitaev2003Fault} models). Our analysis in App.~\\ref{appen:Lieb} and the numerical results in App.~\\ref{appen:numerical_vgap} suggest that the vortices have gapped and positive excitation energies. %\nOn the other hand, the fermionic excitations can become gapless for certain values of $(K_x,K_y,K_z)$, and this determines the phase boundary of our model. \n\nWe emphasize that it is the gap closing of the real part of $\\epsilon_{\\vec{q},1}$ that determines the phase boundary~\\footnote{In fact, for real $K_x,K_y,K_z$, the single fermion energies $\\epsilon_{\\vec{q},1},\\epsilon_{\\vec{q},2}$ are real; so the distinction between $\\epsilon_{\\vec{q},1}$ and $\\mathrm{Re}[\\epsilon_{\\vec{q},1}]$ is unimportant here. In particular, one obtains the same phase diagram even if $\\epsilon_{\\vec{q},1}=0$ is used as a criterion for phase transition. }. This claim is based on the analysis in App.~\\ref{appen:proof_analyticity}, where we rigorously prove that the free energy $f$ defined in Eq.~\\eqref{eq:free_energy} is complex analytic in all its parameters when $\\mathrm{Re}[\\epsilon_{\\vec{q},1}]>0~\\forall \\vec{q}\\in [-\\pi,\\pi]^2$. The proof also suggests that when the gap closes $\\mathrm{Re}[\\epsilon_{\\vec{q},1}]=0$, there are branch points in $\\epsilon_{\\vec{q},1}+\\epsilon_{\\vec{q},2}$ that leads to non-analytic behavior of $f$, which we calculate directly in Sec.~\\ref{sec:critical_exp}.%\n\nWe find two distinct phases corresponding to whether $\\mathrm{Re}[\\epsilon_{\\vec{q},1}]$ is gapped or gapless. The phase boundary is determined as follows. One can show that for fixed $q_y$ the minimum of \n$\\mathrm{Re}[\\epsilon_{\\vec{q},1}]$ occurs at $q_x=0$~[since $\\partial_{q_x}\\epsilon_{\\vec{q}}=f(\\epsilon_{\\vec{q}},q_y) \\sin q_x $ for some positive function $f(\\epsilon_{\\vec{q}},q_y) $]. Furthermore, in the gapped phase the minimum of $\\mathrm{Re}[\\epsilon_{(0,q_y),1}]$ occurs either at $q_y=0$ or $q_y=\\pi$. %\nTherefore, the phase transition occurs when $\\mathrm{Re}[\\epsilon_{\\vec{q},1}]$ vanishes at either $\\vec{q}=(0,0)$ or $\\vec{q}=(0,\\pi)$, \nwhich happens when one of $K_x,K_y,K_z$ equals the sum of the other two~[this can be seen by diagonalizing $\\tilde{T}_{\\vec{q}}$ in Eq.~\\eqref{def:Tq} at $\\vec{q}=(0,0)$ or $(0,\\pi)$]. When $K_x,K_y,K_z$ form three sides of a triangle~(we call this the $B$-region, shown as the shaded triangle in Fig.~\\ref{fig:critical}), the spectrum is gapless, and when one of $K_x,K_y,K_z$ is bigger than the sum of the other two, the spectrum is gapped~(we call this the $A$-region, consisting of three disjoint white triangles in Fig.~\\ref{fig:critical}). The phase diagram in terms of $K_x,K_y, K_z$ is shown in Fig.~\\ref{fig:critical}, which is identical to the phase diagram of Kitaev's honeycomb model~\\cite{Kitaev2006}. \n\nThe fermionic spectrum of the $B$-phase can be gapped by adding suitable perturbations. For example, we can add small imaginary parts to $J_x, J_y$, so that $K_x\\to K_x+i\\kappa,K_y\\to K_y-i\\kappa$, and then add a small real part to the coupling constants of the $x,y$ links that break the lattice reflection symmetry, in the pattern shown in Fig.~\\ref{fig:brickwall-unitcell}. Here $\\kappa$ is a small real number $|\\kappa|\\ll |K_{i}|,i=x,y,z$. (Notice that this corresponds to modifying the link coupling constants of the original classical statistical model on all the $x$ and $y$ planes, which breaks the reflection symmetry of the 3D lattice.) App.~\\ref{appen:gap_phases_B} proves that a subregion of the $B$-phase is gapped by this perturbation. More specifically, when $|K_z|\/2<|K_x|=|K_y|$, we have $\\Delta=\\min_{\\vec{q}}\\mathrm{Re}[\\epsilon_{\\vec{q},1}]\\propto\\kappa^2$. This fact will be useful for Sec.~\\ref{sec:TPloop} where we calculate the topological degeneracy of $\\Lambda_{\\max}$ and Sec.~\\ref{sec:loop_observables} where we find loop observables whose expectation values distinguish the two phases. Notice that while our proof of Lieb's theorem in App.~\\ref{appen:Lieb} assumes real $K_x,K_y,K_z$, as long as the vortices are gapped, the principal eigenstate is still in the vortex-free sector if $\\kappa$ is sufficiently small, which we assume throughout this paper. \n\n\\subsection{Critical exponents}\\label{sec:critical_exp}\nIn this section we study the critical behavior of our model near the phase boundary between the $A$ and $B$ phases, and show that this is a third order phase transition. Specifically, we parameterize the distance to the phase boundary by $\\eta = K_z - K_x - K_y$ and show that as the phase boundary is approached from the $A$-phase side, $\\eta \\to 0^+$, the leading singular part of the free energy is $f \\sim \\eta^{5\/2}$~\\footnote{We are approaching the phase boundary strictly inside the big triangle, i.e. the parameters $K_x,K_y,K_z$ are all nonzero. If one instead approaches the point where two phase boundaries meet from along a side of the big triangle, then one can show that the transition is in 2D Ising universality class, where $f \\sim \\eta^{2}\\ln\\eta$.}.\n\nWe start from the expression in Eq.~\\eqref{eq:free_energy}. Near the phase boundary, the leading singular part of $f$ is contributed by the integration near $\\vec{q}=(0,\\pi)$ where $\\epsilon_{\\vec{q},1}$ approaches zero. Letting $\\vec{q}=(p_x,\\pi+p_y)$ where $p_x,p_y\\ll 1$, we expand $\\epsilon^2_{\\vec{q},1}$ in powers of the small parameters $p_x,p_y$, and $\\eta$. Using Eq.~\\eqref{eq:chepsilon} and $\\cosh \\epsilon\\approx 1+\\epsilon^2\/2$ for $\\epsilon\\ll 1$, we have\n\\begin{eqnarray}\n\t\\epsilon^2_{\\vec{q},1}%\n\t&=& \\frac{s^2_3}{4} p_x^2+2 \\frac{s_1s_2}{s_3} \\eta p_y^2+4\\eta^2+\\frac{s_1^2s_2^2}{4s_3^2}p_y^4\\\\\n\t&&{}+O(p_x^4)+O(p_x^2\\eta)+O(p_x^2p_y^2)+O(p_y^4\\eta), \\nonumber\n\\end{eqnarray}\nwhere the neglected terms will not affect the leading-order singularity. The leading singular part of $f$ is \n\\begin{eqnarray}\\label{eq:f_singular}\n\tf&\\sim&-\\frac{1}{48\\pi^2\\beta s_1s_2}\\iint \\sqrt{ p_x^2+2 \\eta p_y^2+4\\eta^2+\\frac{p_y^4}{4}} ~dp_x dp_y\\nonumber\\\\\n\t&\\sim &\\frac{1}{48\\pi^2\\beta s_1s_2}\\int 2(p_y^2+4\\eta)^2\\ln(p_y^2+4\\eta) ~ dp_y\\nonumber\\\\\n\t&\\sim &\\frac{64}{45\\beta\\pi s_1s_2}\\eta^{\\frac{5}{2}},\n\\end{eqnarray}\nwhere in the first line we rescale the integration variables $p_x,p_y$, the integration range is a fixed-length interval passing through the origin, say $[-\\epsilon,\\epsilon]^2$ with $0<\\epsilon\\ll 1$, and we use $\\sim$ to indicate that an unimportant analytic part has been ignored. %\nTherefore, the third derivative $\\partial^3_\\eta f$ diverges as $\\eta\\to 0^+$, i.e., the phase transition is third order. \n\\subsection{Topological degeneracy}\\label{sec:TPloop}\nIn this section we show that the largest eigenvalues of the transfer matrix $\\hat{T}$ of our original spin model are topologically degenerate, and the degeneracy depends on the phase. This topological degeneracy gives rise to the topological behaviors of the loop observables presented in the next section.\n \n %\nTo this end, we need to compare the values $\\Lambda_{\\mathrm{max}}^{(\\Phi_x,\\Phi_y)}$ of the four sectors $(\\Phi_x,\\Phi_y)\\in\\{(++),(+-),(-+),(--)\\}$, given in Eq.~\\eqref{eq:lamdba_max}. Let us focus on regions where $\\mathrm{Re}[\\epsilon_{\\vec{q}}]$ is gapped, i.e. the $A$-region and the $B$-region with the perturbation discussed in Sec.~\\ref{sec:phase_boundary}. \nIn App.~\\ref{appen:FSE} we show that the largest eigenvalues\n $\\ln\\Lambda_{\\mathrm{max}}^{(\\Phi_x,\\Phi_y)}$ of each of the four sectors are equal up to an exponentially small correction $O(e^{-L\/\\xi})$, where $\\xi$ is a fixed correlation length. %\nThis suggests a 4-fold topological degeneracy since all the fermion and vortex excitations are gapped. However, we have not taken into account the fermion parity constraint yet. \nAs we discussed in Sec.~\\ref{sec:map_fermion}, only those eigenstates of $\\hat{T}'$ that satisfy the fermion parity constraint Eq.~\\eqref{eq:parity_restriction} correspond to eigenstates of $\\hat{T}$. So the actual degeneracy of $\\hat{T}$ is the number of ``parity-compatible'' sectors, i.e. sectors whose principal eigenstate $|\\Lambda_{\\mathrm{max}}^{(\\Phi_x,\\Phi_y)}\\rangle$ satisfies the fermion parity constraint.\nThe fermion parity constraint Eq.~\\eqref{eq:parity_restriction}, written in terms of $\\hat{a}_{\\vec{q}\\lambda},\\Phi_x,\\Phi_y$, becomes\n\\begin{equation}\\label{eq:parity_restriction2}\n(-1)^{(L_x-1)L_y}\\prod_{\\substack{\\vec{q}+}}P_{\\vec{q},0} P_{\\vec{q},1}P_{\\vec{q},2}P_{\\vec{q},3}\\prod_{\\vec{q}\\equiv -\\vec{q}} P_{\\vec{q}} \n= \\Phi_x^{L_y},\n\\end{equation}\nwhere $P_{\\vec{q},\\lambda}=(1-2n_{\\vec{q},\\lambda})$, $P_{\\vec{q}}=4\\hat{a}_{\\vec{q},0}\\hat{a}_{\\vec{q},1}\\hat{a}_{\\vec{q},2}\\hat{a}_{\\vec{q},3}$ and $\\equiv$ is equality $\\mathrm{mod}~2\\pi$. As we mentioned above Eq.~\\eqref{eq:free_energy}, the principal eigenstate of $\\tilde{T}_{\\vec{q}}$ in Eq.~\\eqref{def:Tq} always has $n_{\\vec{q},1}=n_{\\vec{q},2}=1$ and $n_{\\vec{q},{\\bar 1}}=n_{\\vec{q},{\\bar 2}}=0$, so we have $P_{\\vec{q},0} P_{\\vec{q},1}P_{\\vec{q},2}P_{\\vec{q},3}=+1$ for $\\vec{q}\\not\\equiv -\\vec{q}$. Therefore, whether a sector $(\\Phi_x, \\Phi_y)$ is parity-compatible or not is determined by the values of $P_{\\vec{q}}$ where $\\vec{q}\\equiv -\\vec{q}$. \n\nThere are only four possible $\\vec{q}$ that can satisfy $\\vec{q}\\equiv -\\vec{q}$~: $(0,0),(0,\\pi),(\\pi,0),(\\pi,\\pi)$. For the rest of this section, we assume that $L_x,L_y$ are both even numbers~[we treat the other cases in App.~\\ref{appen:TPD}; the conclusions are the same], in which case these four modes appear in the $(++)$ sector only. This means that Eq.~\\eqref{eq:parity_restriction2} is trivially satisfied for the sectors $(+-),(-+),(--)$, i.e. $\\hat{T}$ has at least a 3-fold degeneracy. For the $(++)$ sector, Eq.~\\eqref{eq:parity_restriction2} becomes $P_{00}P_{0\\pi}P_{\\pi 0}P_{\\pi\\pi}=+1$.\nThe value of $P_{\\vec{q}}$ for these four Majorana modes in the principal eigenstate $|\\Lambda_{\\mathrm{max}}^{(++)}\\rangle$ is determined by maximizing the $\\tilde{T}_{0,\\vec{q}}$ term in Eq.~\\eqref{eq:T_0qxqy}.\nIt is straightforward to see that $P_{\\pi0}=P_{\\pi\\pi}=-1$, $P_{00}=[K_z>K_x+K_y]$, and $P_{0\\pi}=[K_z>|K_x-K_y|]$, where $[S]=+1$ if the statement $S$ is true and $[S]=-1$ otherwise. In the $A$-phases, $P_{00}, P_{0\\pi}$ are both $-1$~(for $A_x, A_y$) or both $+1$~(for $A_z$), so $P_{00}P_{0\\pi}P_{\\pi 0}P_{\\pi\\pi}=+1$ and $\\hat{T}$ has a 4-fold degeneracy. In the $B$-phase we have $P_{00}=-1, P_{0\\pi}=+1$, so $P_{00}P_{0\\pi}P_{\\pi 0}P_{\\pi\\pi}=-1$, i.e. the sector $(++)$ is parity-incompatible, and $\\hat{T}$ has a 3-fold degeneracy.\n\n\n\n\n\\subsection{Loop Observables}\\label{sec:loop_observables}\nIn this section we compute the thermal expectation value of the family of loop observables $\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]$ defined in Eq.~\\eqref{eq:def_loop_observable}, and verify our earlier claim that it is equal to $\\pm 1$ for contractible loops, $0$ for large loops in the $A$-phase, and $1\/3$ for large loops in the gapped $B$-phase. \n\nWe begin with a contractible loop $\\mathfrak{L}_p$ being an elementary plaquette of the brickwall lattice. Using the transfer matrix method, we find\n\\begin{eqnarray}\\label{eq:loop_transfer_matrix}\n\t\\langle\\sigma[\\mathfrak{L}_{p,(\\alpha\\beta)}]\\rangle&=&\\mathrm{Tr}[\\hat{W}_p \\hat{T}^M]\/\\mathrm{Tr}[\\hat{T}^M]\\nonumber\\\\%\\equiv \\langle \\hat{W}_p \\rangle,\n\t&\\underset{M\\to\\infty}{=}& \\frac{1}{D}\\sum^D_{j=1}\\langle \\Lambda^{(L)}_{\\max,j}|\\hat{W}_p|\\Lambda^{(R)}_{\\max,j}\\rangle\\nonumber\\\\\n\t&=&+1. \n\\end{eqnarray}\nwhere $(\\alpha\\beta)\\in\\{(xy),(yz),(zx)\\}$, the sum is over all the $D$-fold degenerate principal eigenstates, $\\langle \\Lambda^{(L)}_{\\max,j}|$ and $|\\Lambda^{(R)}_{\\max,j}\\rangle$ are the left and right principal eigenstates of $\\hat{T}$, respectively. The last line of Eq.~\\eqref{eq:loop_transfer_matrix} follows from the fact that the principal eigenstates of $\\hat{T}$ are eigenstates of the conserved operator $\\hat{W}_p$ with eigenvalue $+1$. The value of $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ on larger contractible loops can be calculated in a similar way, and the result is~(up to a possible minus sign) the expectation value of the product of $\\hat{W}_p$ for all the plaquette $p$ enclosed by $\\mathfrak{L}$. Since the $\\hat{W}_p$ mutually commute and have eigenvalue $+1$ on the principal eigenstates, $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ is $\\pm 1$ for contractible loops. %\n\nThe behavior of $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ is more interesting on non-contractible loops. For a large loop $\\mathfrak{L}_y$ parallel to the $y$-direction, as shown in Fig.~\\ref{fig:hc0}, we have\n\\begin{eqnarray}\\label{eq:loop_Ly_transfer_matrix}\n\t\\langle\\sigma[\\mathfrak{L}_{y,(\\alpha\\beta)}]\\rangle&=&-\\mathrm{Tr}[\\hat{\\Phi}_y \\hat{T}^M]\/\\mathrm{Tr}[\\hat{T}^M]\\nonumber\\\\%\\equiv \\langle \\hat{W}_p \\rangle,\n\t&\\underset{M\\to\\infty}{=}& -\\frac{1}{D}\\sum^D_{j=1}\\langle \\Lambda^{(L)}_{\\max,j}|\\hat{\\Phi}_y|\\Lambda^{(R)}_{\\max,j}\\rangle.\t\n\\end{eqnarray} \nFor $A$-phases, this is\n\\begin{eqnarray}\\label{eq:loop_Ly_A}\n\t\\langle\\sigma[\\mathfrak{L}_{y,(\\alpha\\beta)}]\\rangle&=&-\\frac{\\langle \\hat{\\Phi}_y\\rangle_{++}+\\langle \\hat{\\Phi}_y\\rangle_{+-}+\\langle \\hat{\\Phi}_y\\rangle_{-+}+\\langle \\hat{\\Phi}_y\\rangle_{--}}{4}\\nonumber\\\\\n\t&=&0\n\\end{eqnarray} \nwhile for the gapped $B$-phase,\n\\begin{eqnarray}\\label{eq:loop_Ly_B}\n\t\\langle\\sigma[\\mathfrak{L}_{y,(\\alpha\\beta)}]\\rangle&=&-\\frac{\\langle \\hat{\\Phi}_y\\rangle_{+-}+\\langle \\hat{\\Phi}_y\\rangle_{-+}+\\langle \\hat{\\Phi}_y\\rangle_{--}}{3}\\nonumber\\\\\n\t&=&\\frac{1}{3}.\n\\end{eqnarray} \nThe value of $\t\\langle\\sigma[\\mathfrak{L}_{x,(\\alpha\\beta)}]\\rangle$ for a large loop $\\mathfrak{L}_x$ parallel to the $x$-direction is mapped to $-\\langle \\hat{\\Phi}_x\\rangle$ [Eq.~\\eqref{eq:phixphiy}] and can be calculated in an identical way, leading to the same result. \nWe see that the value of $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ indeed distinguish between contractible and non-contractible loops, are always quantized at rational values, and can be used as a (nonlocal) order parameter that distinguishes the phases. \n\nIn order for the topological features to be a universal characteristic of the phase, rather than an accidental property (arising, for example, due to the model's solvability), they must be in some way robust against small, local perturbations. We argue that this is likely the case.\nNotice that a local perturbation, e.g. a small real magnetic field term $B\\sum_j \\sigma_j$, in the original classical Ising model can be mapped to a local perturbation in the transfer matrix in Eq.~\\eqref{eq:def_TM}. The classical loop observables defined in Eq.~\\eqref{eq:def_loop_observable} stills maps to the loop operators $\\hat{W}_p,\\hat{\\Phi}_x,\\hat{\\Phi}_y$, but they no longer commute with the perturbed $\\hat{T}$, and when they act on $|\\Lambda^{(R)}_{\\max,j}\\rangle$ they create excitations along the loop. Consequently we expect the expectation value $\\langle\\sigma[\\mathfrak{L}_{(\\alpha\\beta)}]\\rangle$ to decay exponentially in the length of $\\mathfrak{L}$. \n\nHowever, based on the robustness of the topological phases of the 2D quantum systems~(defined by the transfer matrix $\\hat{T}$), we expect that there exists a family of perturbed loop observables~(whose definition depends on the perturbation) that have exactly the same properties shown above. The argument is based on the idea of quasi-adiabatic continuation~\\cite{Hastings2005Quasiadiabatic}. For simplicity, let us assume $K_x,K_y,K_z\\ll 1$ so that $\\hat{T}$ can be approximated as a Hermitian operator. Then Ref.~\\cite{Hastings2005Quasiadiabatic} shows that there exists a quasi-local unitary transformation $\\hat{U}_\\lambda$ that evolves the unperturbed principal eigenstates to the perturbed ones $|\\Lambda^{(R)}_{\\max,j}\\rangle_\\lambda=\\hat{U}_\\lambda|\\Lambda^{(R)}_{\\max,j}\\rangle_{\\lambda=0}$, where $\\lambda$ is the strength of the perturbation. [Roughly speaking, $\\hat{U}_\\lambda$ is a finite-time evolution by a locally-interacting Hamiltonian $\\sum_{i}\\hat{h}_i$ such that $t\\|\\hat{h}_i\\|=O(\\lambda)$, where $t$ is the total time duration.] Then the perturbed loop operators $\\hat{U}_\\lambda\\{\\hat{W}_p,\\hat{\\Phi}_x,\\hat{\\Phi}_y\\}\\hat{U}_\\lambda^\\dagger$ have exactly the same expectation values in the perturbed principal eigenstates as in the unperturbed solvable model shown above. And due to the quasi-locality of $\\hat{U}_\\lambda$, Lieb-Robinson bounds~\\cite{Lieb1972,hastings2010locality} show that these perturbed operators are finite-width~(of order $v_{\\text{LR}}t$, where $v$ is the Lieb-Robinson speed) extensions of the unperturbed ones. So we do expect robustness in this sense, essentially the same robustness of loop observables in quantum topological phases.\n\n\n \n\n\n\\iffalse\n\\subsection{Correlation functions}\nIn the following we discuss the distinction between the different phases in terms of expectation value of observables and correlation functions. \n\\iffalse\nThe energy is\n\\begin{eqnarray}\\label{eq:energy}\n E=-\\frac{\\partial}{\\partial \\beta}\\ln Z\n=\\frac{MN}{2}\\frac{1}{A_{1\\mathrm{BZ}}}\\frac{\\partial}{\\partial \\beta}\\int_{\\mathrm{1BZ}}\\mathrm{Re}[\\epsilon_{\\vec{q}}]~d^2q.\n\\end{eqnarray}\nNotice that the derivative has to be taken when $\\beta J_\\perp,\\beta h$ are fixed, so the energy does not include the $J_\\perp$ and $h$ terms, however, as we will show soon, the latter have zero expectation values and give no contribution to total energy. Entropy can be obtained by $S=(E-F)\/T$.\n\\fi\nThe general $n$-point correlation function is given by\n\\begin{eqnarray}\\label{eq:npoint}\n \\langle\\sigma_1\\sigma_2\\ldots\\sigma_n\\rangle=\\frac{1}{Z}\\mathrm{Tr}\\{\\sigma^{\\alpha_1}_1T_{12}\\sigma^{\\alpha_2}_2T_{23}\\ldots\\sigma^{\\alpha_n}_nT_n T^{M-k}\\}\\nonumber\\\\\n =\\frac{1}{\\Lambda^k_{\\mathrm{max}}}\\langle \\Lambda_{\\mathrm{max}}|\\sigma^{\\alpha_1}_1T_{12}\\sigma^{\\alpha_2}_2T_{23}\\ldots\\sigma^{\\alpha_n}_nT_n|\\Lambda_{\\mathrm{max}}\\rangle\n\\end{eqnarray}\nwhere $\\alpha_j=x,y,z$ if $\\sigma_j$ is on $x,y,z$-type plane, respectively, $T_{jj+1}$ is the transfer matrix between planes of $\\sigma_j$ and $\\sigma_{j+1}$~(suppose $\\sigma_j$ are arranged in increasing order in $t$-direction), $k=\\floor{m\/3}$ where $m$ is the total number of layers spanned by $\\sigma_1,\\sigma_2,\\ldots,\\sigma_n$.\n\nIt is easy to see that single spin expectation value vanishes $\\langle\\sigma\\rangle=\\langle \\Lambda_{\\mathrm{max}}|\\sigma^{\\alpha}|\\Lambda_{\\mathrm{max}}\\rangle=0$ since $\\sigma^\\alpha$ creates two vortices. Two point correlation $\\langle\\sigma_1\\sigma_2\\rangle$ also vanishes unless $\\sigma_1,\\sigma_2$ are on the same type of plane and are either on the same site or nearest neighbors when projected to the same plane, in this case the correlation is\n\\begin{eqnarray}\\label{eq:2point}\n \\langle\\sigma_1\\sigma_2\\rangle=\\sum_s\\left(\\frac{\\Lambda_{s}}{\\Lambda_{\\mathrm{max}}}\\right)^d\\langle \\Lambda_{\\mathrm{max}}|\\sigma^{\\alpha}_1|\\Lambda_{s}\\rangle\\langle \\Lambda_{s}|\\sigma^{\\alpha}_2|\\Lambda_{\\mathrm{max}}\\rangle,\n\\end{eqnarray}\nwhere the summation is over all excited states with 2 vortices created by $\\sigma^{\\alpha}_1$, $d$ is the vertical distance between $\\sigma_1,\\sigma_2$.\nThe four-point correlation function between two bonds is given by\n\\begin{equation}\\label{eq:4point}\n \\langle\\sigma_i\\sigma_j\\sigma_k\\sigma_l\\rangle_c=\\sum_s\\left(\\frac{\\Lambda_{s}}{\\Lambda_{\\mathrm{max}}}\\right)^d\\langle \\Lambda_{\\mathrm{max}}|i\\hat{c}_i\\hat{c}_j|\\Lambda_{s}\\rangle\\langle \\Lambda_{s}|ic_kc_l|\\Lambda_{\\mathrm{max}}\\rangle,\n\\end{equation}\nwhere the summation is over vortex-free excited states~(fermionic excitations).\n\nIn general, the non-vanishing condition for $n$-point function~\\eqref{eq:npoint} is that the operator product $\\prod_j \\sigma^{\\alpha_j}_j$ appears as a monomial in the expansion of the transfer matrix. \n\\fi\n\n\\section{Physical relevance of complex coupling constants }\\label{sec:justification}\nAlthough the complex coupling constants of Eq.~\\eqref{cond:exact} appear unphysical, this section argues that the model nevertheless gives insights into genuine physical systems.\n\nForemost, we expect the general strategy of this paper -- finding 3D classical models whose transfer matrices can be solved using techniques previously applied to solvable 2D quantum models -- to be a fruitful idea that may lead to a wealth of new solvable models, some of which may have real-valued energy. For example, Refs.~\\cite{Chapman2020characterizationof,Ogura2020geometric,elman2020free} have classified families of quantum spin models that can be solved by mapping to free fermions, and these provide a fertile source for new 3D solvable models. \n\nAs an example of this strategy, Sec.~\\ref{sec:physical_model} shows that the $A$-phase of our model can be realized in a model with real coupling constants. This provides a physical model showing the topological properties. As a speculative aside, we also note that this demonstrates that even models with complex-valued couplings may have the same universal physics as real-valued physical models, and thus the former may serve as windows into the latter. \n\nAdditionally, Sec.~\\ref{sec:quantum_amplitude} shows two different realizations of the partition function of our complex parameter Ising model in certain dynamical processes of a 3D quantum spin system. Both in principle allow the free energy of our model to be measured experimentally. They suggest that the statistical mechanics of Eqs.~(\\ref{eq:HIK},\\ref{eq:Z}) gives a solvable model of 3D DQPT~\\cite{Heyl_2018} that display topological features. %\n\n\\subsection{Realization of $A$-phase in a model with real energy}\\label{sec:physical_model}\n\nThe $A$-phase can be realized in a physical model with real energies, as we now show. Specifically, the model has a phase that reproduces the $A$-phase's topological properties, that contractible loops have expectation value $\\pm 1$ while noncontractible loops have expectation value $0$. \n \n\\begin{figure}\n\t\\center{\\includegraphics[width=0.7\\linewidth]{KTC4-3.png}}\n\t\\caption{\\label{fig:KTC4} The model Eq.~\\eqref{eq:H_KTC4} lies on a 3D cubic lattice where the classical spins sit on links in the $x$ and $y$ directions. There are four spin interactions $\\sigma_{v,1}\\sigma_{v,2}\\sigma_{v,3}\\sigma_{v,4}$ between spins around every lattice vertex $v$~(shown as blue diamond) and eight spin interactions $\\epsilon[\\sigma_{u(c)}, \\sigma_{l(c)}]$ around every elementary cube $c$~(shown as orange cube). Example of a contractible loop $L_1$ is shown as black square, and a noncontractible loop $L_2$ is shown as purple solid line.}\n\\end{figure}\nConsider a 3D square lattice where there is one classical Ising spin on each link in the $x$ and $y$ directions, but no spins live on the links in the $z$ direction, as shown in Fig.~\\ref{fig:KTC4}. The energy of a spin configuration $\\{\\sigma\\}$ is given by\n\\begin{equation}\\label{eq:H_KTC4}\nE[\\{\\sigma\\}]=-\\sum_v \\sigma_{v,1}\\sigma_{v,2}\\sigma_{v,3}\\sigma_{v,4}-\\sum_{c}\\epsilon[\\sigma_{\\{u(c)\\}}, \\sigma_{\\{l(c)\\}}],\n\\end{equation}\nwhere the first sum is over all vertices $v$, $\\sigma_{v,1},\\sigma_{v,2},\\sigma_{v,3},\\sigma_{v,4}$ denote the four spins linked to the vertex $v$, the second sum is over all cubes $c$, and $\\{u(c)\\},\\{l(c)\\}$ denote the upper and lower plaquettes of $c$, respectively. We use $\\sigma_{\\{p\\}}=(\\sigma_{p,1},\\sigma_{p,2},\\sigma_{p,3},\\sigma_{p,4})$ to denote the configurations of the four spins of the plaquette $p$. The energy of the cube $c$ is defined as $\\epsilon[\\sigma_{\\{u(c)\\}}, \\sigma_{\\{l(c)\\}}]=\\ln \\cosh(1)$ if $\\sigma_{\\{u(c)\\}}=\\sigma_{\\{l(c)\\}}$, $\\epsilon[\\sigma_{\\{u(c)\\}}, \\sigma_{\\{l(c)\\}}]=\\ln\\sinh(1)$ if $\\sigma_{\\{u(c)\\}}=-\\sigma_{\\{l(c)\\}}$ while $\\epsilon[\\sigma_{\\{u(c)\\}}, \\sigma_{\\{l(c)\\}}]=-\\infty$ otherwise. \n\nThe partition function is \n\\begin{equation}\\label{eq:Z_KTC4}\nZ=\\sum_{\\{\\sigma\\}}e^{- E[\\{\\sigma\\}]}=\\mathrm{Tr}[\\hat{T}^M],\n\\end{equation}\nwhere the transfer matrix $\\hat{T}$ is an operator acting on quantum spins lying on a 2D slice of the lattice, defined by\n\\begin{equation}\\label{eq:T_KTC4}\n\\hat{T}=\\exp\\left(\\sum_v \\hat{\\sigma}^z_{v,1}\\hat{\\sigma}^z_{v,2}\\hat{\\sigma}^z_{v,3}\\hat{\\sigma}^z_{v,4}+\\sum_{p}\\hat{\\sigma}^x_{p,1}\\hat{\\sigma}^x_{p,2}\\hat{\\sigma}^x_{p,3}\\hat{\\sigma}^x_{p,4}\\right),\n\\end{equation}\nwhich is simply $e^{-\\hat{H}}$ where $\\hat{H}$ is the Hamiltonian of Kitaev's toric code model. The principal eigenstates of $\\hat{T}$ are the 4-fold degenerate ground states of $\\hat{H}$. \n\nFig.~\\ref{fig:KTC4} shows the family of loop observables we are interested in. Using the same method as in Sec.~\\ref{sec:loop_observables}, these classical loop observables can be mapped to the conserved loop operators of the quantum toric code, and\nthe thermal expectation values of the former are mapped to quantum expectation values of the latter. Averaging over the four topologically degenerate principal eigenstates, we find that the expectation value of contractible loops is $+1$ while non-contractible loops have expectation value $0$. This reproduces the topological behavior of the $A$-phase of the Ising model presented in Sec.~\\ref{sec:loop_observables}.\n\n\\subsection{Realizing the partition function in quantum dynamics}\\label{sec:quantum_amplitude}\n\nAnother way in which classical statistical models with complex energy can be physically relevant is that the partition function $Z$ can be mapped to measurable quantities of certain~(unitary) quantum dynamical processes in 3D (not 2D) quantum systems. In this section we show two such constructions: Sec.~\\ref{sec:transition_amplitude} shows how to realize $Z$ as a transition amplitude, while Sec.~\\ref{sec:probe_spin_coherence} shows that $Z$ gives the quantum coherence of a probe spin-$1\/2$ coupled to the whole system. The phase transition we studied in our model is then mapped to a DQPT in these quantities.\n\n\\subsubsection{Interpreting the partition function as a transition amplitude}\\label{sec:transition_amplitude}\nConsider a 3D quantum spin system on the same lattice as Fig.~\\ref{fig:hc0}, and with a Hamiltonian given by Eq.~\\eqref{eq:HIK} with all $\\sigma_i$ replaced by $\\hat{\\sigma}_i^z$, and we will take all the parameters $J_x,J_y,J_z,J_\\perp,h$ to be real to guarantee hermiticity. The quantum transition amplitude between two arbitrary states is\n\\begin{equation}\n\t\\langle A|e^{-it\\hat{H}}|B\\rangle=\\sum_{\\{\\sigma\\}} e^{-it H[\\{\\sigma\\}]}\\langle A|\\{\\sigma\\}\\rangle\\langle\\{\\sigma\\}|B\\rangle,\n\\end{equation}\nwhere on the RHS we inserted a complete set of $\\hat{\\sigma}^z$ basis states. If the states $|A\\rangle,|B\\rangle$ are of the following form\n\\begin{equation}\\label{eq:initial_final_states}\n\t|A\\rangle=\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{X}}|\\psi(A_x)\\rangle_{ij}\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{Y}}|\\psi(A_y)\\rangle_{ij}\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{Z}}|\\psi(A_z)\\rangle_{ij},\n\\end{equation}\nwhere $\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{X}}$ is over all the red thick $x$-links in Fig.~\\ref{fig:hc0}, and similarly for $\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{Y}}$ and $\\bigotimes_{\\langle ij\\rangle\\in \\mathbf{Z}}$, and the local state on each link $\\langle ij\\rangle$ is defined as $|\\psi(A)\\rangle_{ij}=\\frac{1}{2\\sqrt{\\cosh 2\\mathrm{Re}(A)}}\\sum_{\\sigma_i,\\sigma_j}e^{A\\sigma_i\\sigma_j}|\\sigma_i,\\sigma_j\\rangle$. Note that Eq.~\\eqref{eq:initial_final_states} defines product states since the thick links $\\mathbf{X},\\mathbf{Y},\\mathbf{Z}$ are non-overlapping. Then we have\n\\begin{equation}\\label{eq:transition_amplitude}\n\t\\langle A|e^{-it\\hat{H}}|B\\rangle=\\mathrm{const.}\\times Z(K_x,K_y,K_z,it J_\\perp,it h),\n\\end{equation}\nwhere $K_j=A^*_j+B_j+itJ_j,j=x,y,z$. Therefore, when $tJ_\\perp\\equiv \\pi\/4~(\\mathrm{mod}~ \\pi\/2),t h\\equiv \\pi\/4~ (\\mathrm{mod} ~\\pi)$, the transition amplitude is given by the results we derived previously. %\n\nQuantum transition amplitudes, or closely related objects called dynamical partition functions $f(t)\\propto-\\ln \\langle A|e^{-it\\hat{H}}|B\\rangle$, are the central objects in the study of DQPTs~\\cite{heyl2013dynamical,Andraschko2014dynamical,Heyl2014dynamical,Vosk2014dynamical,Heyl2015dynamical,Schmitt2015dynamical,Heyl_2018}. In this literature, a dynamical phase transition typically referes to a singularity of the dynamical evolution of a physical quantity~[e.g. $f(t)$] at a critical time. In our model, the time is fixed at special values e.g. $t_0=\\pi\/(4J_\\perp)=\\pi\/(4h)$ to guarantee solvability, and the singularity occurs in $f(t_0)$ %\nas we tune the parameters $K_x,K_y,K_z$ across the phase boundary shown in Fig.~\\ref{fig:critical}. Although the situation is slightly different, the analogy is clear, and we also expect that if we fix $K_x,K_y,K_z$ to be exactly at the phase boundary, say $K_z=K_x+K_y$, and let the system evolve in time, then there will likely be a singularity in $f(t)$ at $t_0$, i.e. a DQPT in the usual sense. \n\nAlthough quantum transition amplitudes are much harder to measure experimentally compared to local observables, there are promising experimental setups~\\cite{Jurcevic2017direct,Tian2020observation} that measure this quantity in relatively small systems, and are capable of observing signatures of dynamical phase transition. \n\n\\subsubsection{Mapping the partition function to a probe spin coherence}\\label{sec:probe_spin_coherence}\nWe can also realize the partition function as a probe spin coherence, based on the idea of measuring Yang-Lee zeros in the classical Ising model~\\cite{wei2012lee,peng2015experimental}. To this end we couple a probe spin-$1\/2$ to the whole 3D (quantum) spin system~(bath) shown in Fig.~\\ref{fig:hc0}, with probe-bath interaction %\n\\begin{eqnarray}\\label{eq:probe-bath_interaction}\n\tH_I%\n\t&=& \\hat{\\tau}^z\\otimes\\left(-J_\\perp\\sum_{\\langle ij\\rangle\\in\\boldsymbol{\\perp}}\\hat{\\sigma}^z_{i}\\hat{\\sigma}^z_{j}+h\\sum_i\\hat{\\sigma}^z_i\\right)\\nonumber\\\\\n\t&=&\\frac{1}{2}\\hat{\\tau}^z \\hat{B}\n\\end{eqnarray}\nwhere $\\hat{\\tau}^z$ acts on the probe spin, and $J_\\perp$ and $h$ are real. The probe spin is initialized in a superposition state $(|\\uparrow\\rangle+|\\downarrow\\rangle)\/\\sqrt{2}$, and the system~(bath) is initially in equilibrium at temperature $T$ with only interactions in the horizontal $x,y,z$ links, described by the canonical ensemble in Eq.~\\eqref{eq:Z} with $J_\\perp=h=0$. When we turn on the probe-bath interaction in Eq.~\\eqref{eq:probe-bath_interaction}, the thermal\nfluctuation of the field $\\hat{B}$ induces decoherence of the probe spin~(due to a random phase $Bt$). The probe spin coherence, defined as the ensemble average\nof $e^{i\\hat{B} t}$, is mapped to~\\cite{wei2012lee}\n\\begin{equation}\\label{eq:probe_spin_decoherence}\n\tL(t)\\equiv\\langle e^{i\\hat{B} t}\\rangle=\\frac{Z(\\beta J_x,\\beta J_y, \\beta J_z, i J_\\perp t,i h t)}{Z(\\beta J_x,\\beta J_y, \\beta J_z,0,0)}.\n\\end{equation}\nTherefore, when $tJ_\\perp\\equiv \\pi\/4~(\\mathrm{mod}~ \\pi\/2)$ and $t h\\equiv \\pi\/4~ (\\mathrm{mod} ~\\pi)$, $L(t)$ is given by our exact solution in Sec.~\\ref{sec:solution}~[notice that the denominator of Eq.~\\eqref{eq:probe_spin_decoherence} can be calculated easily and has no singularity], and has a topological phase transition when the parameters $J_x,J_y,J_z$ are tuned across the phase boundary in Fig.~\\ref{fig:critical}. \nThis kind of probe spin coherence has been measured experimentally in an Ising model of 10 spins~\\cite{peng2015experimental}. \n\n\\section{Summary and Outlook}\\label{sec:summary}\nWe exactly solved a 3D classical Ising model on a special 3D lattice, which has some of its coupling constants fixed to imaginary values. The solution exploits the special structure of the transfer matrix, which can be mapped to free fermions using a method similar to the solution of Kitaev's honeycomb model. The analytic solution reveals two distinct phases, with a third order phase transition between them. The two phases can be distinguished by measuring the product of spins on certain loops, the expectation value of which is quantized to certain rational values~($0$, $1$, or $1\/3$), depending only on the phase and the topology of the loop. \nWe therefore see that the model not only gives insight into interacting many-body systems in 3D, but that the behavior it shows is particularly interesting: there are phases with topological properties, and a continuous phase transition between them. \n\nWe expect the topological character of the phases to be universal, as discussed in Sec.~\\ref{sec:loop_observables}. \nWe also expect universality in some other correlations we have not calculated in this paper. For example, the gapless $B$-phase has power-law decaying two-point correlations. For the gapped $B$-phase~(i.e. with the $\\kappa$-perturbation introduced in Sec.~\\ref{sec:phase_boundary}), if we put the system on a large cylinder~(with axis parallel to the $z$-direction), due to the existence of gapless chiral edge modes on the boundary of the 2D quantum system~(defined by the transfer matrix $\\hat{T}$), we expect that the Ising model has power-law decaying correlations on the cylinder boundary even though all two-point correlations in the bulk decay exponentially. We expect the universality in these power-law exponents~(i.e. remain the same when local perturbations are present).\n\nDespite the unphysical complex coupling constants, we described two connections to physical systems. First, the universal long-distance properties of the two phases and the phase transition may be reproduced in a physical 3D system. We demonstrated this by explicitly constructing another 3D classical statistical model with positive Boltzmann weights that has topological properties identical to the $A$-phase of our 3D Ising model. More speculatively, this suggests that physical systems may have the same universal behavior as models with complex couplings independent of whether the corresponding real-coupling models can be explicitly found or solved. \nWe are unsure if the $B$-phase can be realized in a physical classical system, but we expect this to be challenging if at all possible, since Ref.~\\cite{Ringel2017Nogothm} suggests the prevalence of sign-problems in a family of closely related phases. \nSecond, the partition function of our model can be realized in certain dynamical processes of a 3D quantum spin system, either as a transition amplitude or as a probe spin coherence, allowing the free energy to be experimentally measured in principle, and the phase transitions studied in our model are related to DQPTs in these 3D quantum systems. \n\nOur model may have other connections to real physical systems beyond the above two. First, when $K_x, K_y, K_z$ are purely imaginary, our transfer matrix $\\hat{T}$ in Eq.~\\eqref{eq:def_TM} becomes the unitary evolution operator of a periodically driven Kitaev model studied in Ref.~\\cite{FloquetKitaev2017}, so our technique of diagonalizing $\\hat{T}$ may be useful in studying certain properties of that system. Second, when $K_x, K_y,K_z\\to \\pm\\infty$, $\\hat{T}$ becomes a projection operator representing the sequential measurement of $\\hat{\\sigma}^z_{i}\\hat{\\sigma}^z_{j}, \\hat{\\sigma}^y_{i}\\hat{\\sigma}^y_{j}$, and $\\hat{\\sigma}^x_{i}\\hat{\\sigma}^x_{j}$ on all the $z$-, $y$-, and $x$-links, respectively, which is reminiscent of the measurement process of the honeycomb quantum memory code proposed in Ref.~\\cite{hastings2021dynamically}.\n\nOur results may also provide hints for constructing a genuinely 3D--i.e. one which does not factorize into decoupled 2D models--classical statistical model with positive Boltzmann weights and a continuous phase transition, a problem that has been studied for more than 60 years but never solved. As one possible direction, we note that our model can be straightforwardly generalized to a large family of solvable 3D classical statistical models, whose transfer matrix is similar to one of the generalized Kitaev models~\\cite{Yao2007Exact,Yang2007Mosaic,SI2008Anyonic,Mandal2009Exactly,Yao2009Algebraic,Wu2009Gamma,Ryu2009Three,Tikhonov2010Quantum,Lai2011SU2,Yao2011Fermionic,Barkeshli2015Generalized} that can also be solved by mapping to free fermions. As free-fermion solvable spin models have been systematically classified recently~\\cite{Chapman2020characterizationof,Ogura2020geometric,elman2020free}, it is natural to ask if one of them can be promoted to a transfer matrix that corresponds to a physical 3D classical statistical model.\n\n\n\\acknowledgements\nZ.W. is especially grateful to Zongping Gong who suggested the idea in Sec.~\\ref{sec:probe_spin_coherence}. We also thank Sarang Gopalakrishnan, Bhuvanesh Sundar, and Maxim Olchanyi for helpful discussions. This work was supported in part by the Welch Foundation~(C-1872) and the National Science Foundation~(PHY-1848304).\nK.H.'s contribution benefited from discussions at the KITP, which was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. %\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe recent growth of sensitive optical time-domain surveys has revealed and expanded exciting new classes of stellar explosions. These include superluminous supernovae, which can be up to 10--100 times more luminous than ordinary massive star explosions (e.g.~\\citealt{Quimby2011,nicholl2013,inserra2013,howell2013,decia2018,lunnan2018,quimby2018}; see \\citealt{Gal-Yam2019} for a recent review). Conventionally, the optical emission from most core-collapse supernovae is powered by the radioactive decay of $^{56}$Ni (Type Ib\/c) and by thermal energy generated via shock heating of the stellar envelope (Type IIL, IIp). However, the peak luminosities of SLSNe greatly exceed the luminosity expected from those conventional mechanisms, and the origin of the energy is still debated. \n\nA popular model for powering the time-dependent emission of SLSNe, particularly the hydrogen-poor Type I class (SLSN-I), involves energy input from a young central engine, such as a black hole or neutron star, formed in the explosion. For example, the accretion onto the compact object from bound debris of the explosion could power an outflow which heats the supernova ejecta from within (\\citealt{Quataert2012,woosley2012,margalit2016,moriya2018}). Alternatively, the central engine could be a strongly magnetized neutron star with a millisecond rotation period, whose rotationally powered wind provides a source of energetic particles which heat the supernova ejecta \\citep{kasen2010,woosley2010,dessart2012a,metzger2015a,sukhbold2016}. The magnetar\\footnote{To remain consistent with the SLSNe literature, the term magnetar is used throughout this paper. Magnetars generally have large dipole magnetic fields $B\\simeq$ \\SIrange{e13}{e15}{G} with a rotation period of a few seconds. In the case of the SLSN magnetar model, the radiation is extracted from the rotational energy of the young millisecond pulsars, but with large magnetic fields characteristic of magnetars.} model provides a good fit to the optical light curves of most SLSNe-I \\citep{inserra2013,nicholl2017d}. Furthermore, analyses of the nebular spectra of hydrogen-poor SLSNe \\citep{nicholl2019,jerkstrand2017} and Type-Ib SNe \\citep{milisavljevic2018} support the presence of a persistent central energy source, consistent with an energetic neutron star.\n\nThe details of how the magnetar would couple its energy to the ejecta are uncertain. Several models consider that the rotationally powered wind from a young pulsar inflates a nebula of relativistic electron\/positron pairs and energetic radiation behind the expanding ejecta \\citep{kotera2013,metzger2014b,murase2015}. At the wind termination shock, the pairs are heated and radiate X-rays and gamma rays with high efficiency via synchrotron and inverse-Compton processes. Photons which evade absorption via $\\gamma\\textrm{-}\\gamma$ pair creation in the nebula can be ``absorbed\" by the ejecta further out, thermalizing their energy and directly powering the supernova's optical emission (e.g.~\\citealt{metzger2014b, vurm2021}). \n\nThermalization of the nebular radiation will be most efficient at early times, when the column through the ejecta shell and ``compactness'' of the nebula are at their highest. At these times one would expect the optical light curve to faithfully track the energy input of the central engine. However, as the ejecta expand, the radiation field dilutes and the shell becomes increasingly transparent to high-energy and very-high-energy photons. The increasing transparency, and correspondingly decreasing thermalization efficiency, eventually causes the supernova's optical luminosity to drop below the rate of energy injection from the central engine \\citep{wang2015,chen2015}, with the remaining radiation escaping directly from the nebula as gamma rays or X-rays (the putative ``missing\" luminosity). \n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.9\\columnwidth]{Opacity_SN2015bn_vs_time.pdf}\n \\includegraphics[width=0.9\\columnwidth]{Opacity_SN2017egm_vs_time_weightlate_v2.pdf}\n \\caption{Optical depth at different photon energies as a function of time, calculated for ejecta properties (mass $M_{\\odot}$, mean velocity, etc) derived from observations of SN2015bn \\citep{nicholl2018a} and SN2017egm \\citep{nicholl2017c} shown in Table \\ref{tab:event_physical_params}. Top: SN2015bn. Bottom: SN2017egm. The horizontal dotted line represents $\\tau_{\\rm eff} = 1$. The cross-sections for photon-photon and photon-matter pair production opacities are taken from \\citet{zdziarski1989}. The solid lines correspond to target blackbody radiation temperature $T_{\\rm eff} = (L_{\\rm opt}\/4\\pi R^2)^{1\/4}$, where $L_{\\rm opt}$ and $R$ are the optical luminosity and ejecta radius, respectively. The dashed lines are computed with a temperature floor of $T = 4000$~K, to mimic the approximate spectrum in the nebular phase. Below ${\\sim}10$~GeV the opacity is dominated by photon-matter pair production at all times. Above $100$~GeV, pair production on the thermal target radiation field dominates up to a few years.}\n \\label{fig:tau}\n\\end{figure}\n\nAs the ejecta expand and the spin down luminosity weakens, the conditions for various processes responsible for photon energy loss change and impact the effective optical depth. Within a few months, the effective optical depth to high-energy (HE; 100 MeV to 100 GeV) photons emitted from the central engine nears unity, and at several hundred days it reaches unity for very-high-energy (VHE; 100 GeV \u2013 100 TeV) photons. \n\nFigure~\\ref{fig:tau} shows examples of the effective optical depth through the ejecta for photons of various energies as a function of time. They have been calculated using time-dependent properties for the supernova ejecta and radiation field motivated by the observations of SN2015bn and SN2017egm, both particularly well-studied SLSNe-I explored in \\cite{vurm2021}.\n\nThe dominant processes involved in the calculation of the gamma-ray optical depth include photon-matter and photon-photon interactions, particularly pair production on the nuclei and soft radiation fields in the ejecta. An accurate treatment considering the radiation transport is discussed in depth in \\citet{vurm2021}. The standard version of the magnetar model \\citep{kasen2010, woosley2010, nicholl2017d} does not consider this time-dependent calculation and relies on constant effective opacities to optical and high-energy photons. Figure~\\ref{fig:tau} provides a useful guiding timescale for when to consider gamma-ray emission at various energies, calculated with the model in \\citet{vurm2021} using the ejecta properties fit to the optical data in Table \\ref{tab:event_physical_params}.\n\nGiven its comparatively nearby distance at z=0.1136, SN2015bn is an excellent candidate event to test the magnetar hypothesis. The optical light curve shows a steepening from $\\propto t^{-2}$ decay to $\\propto t^{-4}$ around ${\\sim}200$ days \\citep{nicholl2018a}. This behavior is consistent with a leakage of high-energy radiation from a magnetar nebula \\citep{nicholl2018a}. A deep search in the ${\\sim}0.1-10$ keV X-ray band resulted in non-detections \\citep{bhirombhakdi2018}, eliminating the possibility that leakage from the nebula occurs in the softer X-ray bands.\n\n\\citet{margutti2018b} present a similar search for late-time X-ray emission from a larger sample of SLSNe-I, mostly resulting in upper limits; however, see \\citet{levan2013} for an X-ray detection of the SLSN-I SCP 06F6 that could still support the magnetar hypothesis. X-ray non-detections are not surprising, because the ejecta are likely to still be opaque in the $\\lesssim 10$ keV band due to photoelectric absorption in the hydrogen-poor ejecta \\citep{margalit2018a}. Intriguingly, \\citet{Eftekhari2019} detected radio emission from the location of the SLSN PTF10hgi at 7.5 years after the explosion and argued that the emission could be synchrotron emission from an engine-powered nebula.\n\nSome effort has been underway to search for nebular leakage in the gamma-ray band. \\citet{renault-tinacci2018} obtained upper limits on the $0.6-600$ GeV luminosities from SLSNe by a stacked analysis of 45 SLSNe with {\\it Fermi}-LAT. The majority of their sample were SLSNe-I, the most likely class to be powered by a central engine; however the results were dominated by a single, extremely close Type II event (SLSN-II), CSS140222. Hydrogen-rich SLSNe make up the Type II class (SLSN-II), which are suggested to be powered by the interaction of the circumstellar medium with the supernova ejecta. Nevertheless, even with CSS140222 included, the upper limits are at best marginally constraining on the inferred missing luminosity. \n\nIn this paper, the search is expanded to gamma-ray emission from SLSNe-I in the HE to VHE bands using the {\\it Fermi} Gamma-Ray Space Telescope and the ground-based VERITAS observatory. In particular, observations of SN2015bn and SN2017egm are presented here. SN2017egm is the closest SLSN-I to date in the Northern Hemisphere at z=0.0310 \\citep{nicholl2017c,bose2017}. Observations of young supernovae with gamma-ray telescopes have been few, with no detections so far. Some tantalizing candidates like iPTF14hls and SN 2004dj have been explored with {\\it Fermi}-LAT but are unconfirmed due to large localization regions overlapping with other gamma-ray candidates \\citep{yuan2018,xi2020}. MAGIC carried out observations of a Type I SN \\citep{ahnen2017a}. HESS observed a sample of core-collapse SNe \\citep{Abdalla2019}, and later obtained upper limits on SN 1987A \\citep{theh.e.s.s.collaboration2015}. Our observations are the first of superluminous supernovae. \n\nThroughout this paper, a flat $\\Lambda$CDM cosmology is used, with $H_0 = \\SI{67.7}{km.s^{-1}.Mpc^{-1}}$, $\\Omega_{M}=0.307$, and $\\Omega_{\\Lambda} = 0.6911$ \\citep{planckcollaboration2016}. The corresponding luminosity distances to SN2015bn and SN2017egm are \\SI{545.37}{\\mega\\pc} (z=0.1136) \\citep{nicholl2016} and \\SI{139.29}{\\mega\\pc} (z=0.0310) \\citep{bose2017}.\n\n\n\n\n\n\\section{Observations \\& Methods} \\label{sec:obs}\n\\label{sec:observations}\nThe superluminous supernovae SN2015bn and SN2017egm were observed with {\\it Fermi}-LAT and VERITAS during 2015--2016 and 2017--2020, respectively. SN2015bn is a SLSN-I explosion from 23 Dec 2014 (MJD 57014) and it peaked optically on 19 Mar 2015 (MJD 57100) \\cite{nicholl2016a}. SN2017egm is a SLSN-I explosion from 23 May 2017 (MJD 57896) and it peaked optically on 18 Jun 2017 (MJD 57922) \\cite{bose2017}. Some properties of the SLSNe are given in Table \\ref{tab:event_physical_params}. Details regarding the optical, {\\it Fermi}-LAT and VERITAS observations and the data-analysis methods are below. \n\n\n\n\\begin{table}[ht]\n \\begin{center}\n \\caption{Properties of the SLSNe considered in this paper. The quantities $P_{0}$, $B$, $M_{\\rm ej}$, $\\kappa$, $E_{\\rm SN}$, $v_{\\rm ej}$, $\\kappa_{\\gamma}$ and $M_{\\rm NS}$ were obtained from a best-fit to the UVOIR supernova light curves, with errors found in \\cite{nicholl2017c, nicholl2017d}. }\n \\begin{tabular}{cl|rr}\n \\multicolumn{1}{l}{Parameter} & {[}unit{]} & {SN2015bn} & {SN2017egm} \\\\ \\hline\n RA & $^{\\circ}$ & 173.4232 & 154.7734 \\\\\n Dec & $^{\\circ}$ & 0.725 & 46.454 \\\\\n z & - & 0.1136 & 0.0310 \\\\\n $t_{0}^{(a)}$ & MJD & 57014 & 57896 \\\\\n $t_{pk}^{(b)}$ & MJD & 57100 & 57922 \\\\ \\hline\n $P_0^{(c)}$ & ms & $2.50^{+0.29}_{-0.17}$ & $5.83^{+0.73}_{-0.70}$ \\\\\n $B^{(d)}$ & $10^{14}$ G & $0.26^{+0.07}_{-0.05}$ & $0.94^{+0.13}_{-0.16}$ \\\\\n $M_{ej}^{(e)}$ & $M_{\\odot}$ & $10.8^{+0.83}_{-1.34}$ & $2.99^{+0.30}_{-0.23}$ \\\\\n $\\kappa^{(f)}$ & cm$^{2}$g$^{-1}$ & $0.18^{+0.01}_{-0.02}$ & $0.12^{+0.04}_{-0.06}$ \\\\\n \n $v_{ej}^{(h)}$ & $10^{8}$ cm s$^{-1}$ & $5.68^{+0.16}_{-0.14}$ & $10.3^{+0.35}_{-0.27}$ \\\\\n $\\kappa_{\\gamma}^{(i)}$ & cm$^{2}$g$^{-1}$ & $0.008^{+0.01}_{-0.01}$ & $0.080^{+0.15}_{-0.06}$ \\\\\n $M_{\\rm NS}^{(j)}$ & $M_{\\odot}$ & $1.84^{+0.28}_{-0.23}$ & $1.57^{+0.25}_{-0.29}$ \n \\end{tabular}\n \\label{tab:event_physical_params} \n \\end{center}\n $^{(a)}$Epoch of explosion; $^{(b)}$Epoch of optical flux peak; $^{(c)}$Initial spin-period; $^{(d)}$magnetic field strength of magnetar; $^{(e)}$Total mass, $^{(f)}$effective opacity; $^{(g)}$kinetic energy; $^{(h)}$mean velocity of supernova ejecta; $^{(i)}$gamma-ray effective opacity; and $^{(j)}$neutron star mass.\n\\end{table}\n\n\\subsection{{\\it Fermi}-LAT} \\label{subsec:Fermi}\nThe Large Area Telescope (LAT) on board the {\\it Fermi} satellite has operated since 2008 \\citep{Atwood2009}. It is sensitive to photons between \\SI{{\\sim}20}{\\MeV} and \\SI{{\\sim}300}{\\GeV} and has ${\\sim}60\\degree$ field of view, enabling it to survey the entire sky in about three hours. \n\nThe data were analyzed using the publicly available {\\it Fermi}-LAT data with the \\texttt{Fermitools} suite of tools provided by the {\\it Fermi} Science Support Center (FSSC). Using the \\texttt{Fermipy} analysis package \\citep{wood2017}\n\\footnote{\\url{https:\/\/fermipy.readthedocs.io\/en\/latest\/} ; v0.19.0}, the data were prepared for a binned likelihood analysis in which a spatial spectral model is fit over the energy bins. The data were selected using the SOURCE class of events, which are optimized for point-source analysis, within a region of $15\\degree$ radius from the analysis target position. Due to the effect of the Earth, a $90\\degree$ zenith angle cut was applied to remove any external background events. The standard background models were applied to the test model, incorporating an isotropic background and a galactic diffuse emission model without any modifications. The standard 4FGL catalog was then queried for sources within the field of view and their default model parameters \\cite{abdollahi2020}. \n \nAdditional putative point sources were added to each field of view as needed to support convergence of the fit. These sources were added for all analysis time scales. This process continued until the distribution of test statistics for the field of view was Gaussian with standard deviation near 1 and mean centered at zero, and the residual maps were near uniformly zero without strong features. These conditions indicate the appropriate coverage of spectral sources within the analysis was reached and no putative sources are missing. The fitting process is performed in discrete energy bins while optimizing the spectral shape, but the distribution of test statistics is evaluated with the stacked data spanning the full energy range. With the improvements to {\\it Fermi}-LAT low-energy sensitivity in PASS8 reconstruction, the low energy bin covering $100-612$ MeV was also added. \n \nIn the case of both SN2015bn and SN2017egm, the data were fitted with a power-law spectral model, $N(E) = N_{0} E^{\\Gamma}$, with a free prefactor and a fixed photon index $\\Gamma$ of -2.0. From the fit, the reported flux upper limit was found using a 95\\% confidence level with the bounded Rolke method \\citep{rolke2005}. In all cases reported here, the upper limit reported is the integral energy flux, integrated over the energy ranges described for each case, which has units of \\si{MeV cm^{-2}.s^{-1}}. This flux is converted to luminosity with the adopted distance for each event.\n \n\nSN2015bn was observed from 23 Dec 2014 to 23 Mar 2018. This observation period begins after the explosion, and is binned in a few windows to account for the absorption of low-energy gamma rays by the ejecta at early times (Figure~\\ref{fig:tau}). The first $\\sim90$ days is observed to make sure there are no early emission during the expected absorption period. The data were thereafter binned in time intervals of six months to maximize observation depth and sensitivity to time dependent variation. SN2017egm was observed 23 May 2017 to 21 Aug 2020. Again, this period covers the 3.5 years from the discovery date, starting with $\\sim90$ days after the explosion and split into six 6-month bins thereafter. The 3.5 year observation period is selected to cover approximately 1000 days after the explosion. After this period, it is expected that the predicted luminosity will have decreased below the {\\it Fermi}-LAT detectable limit. \n\nSN2015bn is within 5\\textdegree\\ of the Sun each year in August, so a one-month time cut is applied to each relevant time bin (to cover a $\\sim15$\\textdegree\\ radius field of view). SN2017egm is not near the the path of the Sun, so this cut was not applied.\n\n\\subsection{VERITAS} \\label{subsec:veritas}\nThe Very Energetic Radiation Imaging Telescope Array System (VERITAS) is an imaging atmospheric cherenkov telescope (IACT) array at the Fred Lawrence Whipple Observatory (FLWO) in southern Arizona, USA \\citep{weekes2002, Holder2006}. It consists of four 12-m telescopes separated by approximately $100$ m, and the observatory is sensitive to photons within the energy range $\\backsim$\\SI{100}{GeV} to $\\backsim$\\SI{30}{TeV}. The instrument has an angular resolution (68\\% containment) of $\\backsim$0.1\\degr\\ at \\SI{1}{TeV}, an energy resolution of $\\backsim$15\\% at \\SI{1}{TeV}, and 3.5\\degr\\ field of view. \n\n\nVERITAS serendipitously observed SN2015bn for a total of 1.01 hours between 7 May 2015 and 22 May, 2015, approximately 135 days from explosion (49 days from the date of peak magnitude), as a part of an unrelated campaign. Another 1.7 hours were taken between 25 May 2016 and 30 May 2016. Data were taken in good weather and dark sky conditions. Since SN2015bn was not the target source, its sky position averages 1.4\\textdegree~from the center of the camera. \n\nVERITAS directly observed SN2017egm for 8.7 hours between 24 Mar 2019 and 5 Apr 2019, under dark sky conditions, as part of a Directors Discretionary Time (DDT) campaign, approximately 670 days from explosion. This target was triggered based on the predicted gamma-ray luminosity (see section \\ref{sec:MagnetarSpinDown} and appendix \\ref{sec:appendix} for a description) derived from the optical observation. Although it was almost two years after the explosion, the nearby distance yielded a gamma-ray luminosity prediction still within reach of VERITAS, making this an enticing target to follow up.\n\nThe SN2017egm data in this paper were taken using ``wobble\" pointing mode, where the source is offset from the center of the camera by $0.5\\degree$. This mode creates space for a radially symmetric off region to be used for background estimation in the same field of view, saving time from targeted background observations that contain the same data observing conditions. The data were processed with standard VERITAS calibration and reconstruction pipelines, and then cross-checked with a separate analysis chain \\citep{Maier2017,cogan2008}. \n \n \nUsing an Image Template Method (ITM) to improve event angular and energy reconstruction \\citep{christiansen2017}, analysis cuts are determined with a set of a priori data selection cuts optimized on sources with a moderate power-law index (from -2.5 to -3).\n\nUnfortunately, the large offset on SN2015bn due to the serendipitous observation precludes us from using ITM in the analysis, so in that case SN2015bn is analyzed without templates by calculating image moments directly from candidate images triggered by the camera \\citep{Maier2017,cogan2008}. In both cases, the signal and background counts are determined using the reflected region method. \n\n\nThe upper limit is calculated for both SN2015bn and SN2017egm. The bounded Rolke method for upper limit calculation is used, assuming a power law spectrum with index of -2.0 and 95\\% confidence level \\citep{rolke2005}. Since the calculation of the upper limit depends on the underlying spectral model, a range of power-law spectral indices from -2 to -3 was computed to estimate impact of the model dependence. In all cases reported here, the upper limit reported is the integral photon flux, integrated over the energy ranges described for each case, which has units of \\si{cm^{-2}.s^{-1}}. This flux is converted to integral energy flux using the same spectral model so that the luminosity can be computed with the adopted distance.\n\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics{SN2015bn_ebl_lc.pdf}\n \\caption{Light curves of SN2015bn spanning 30 to 1500 days after explosion. Curves shown include (1) the (thermal) supernova luminosity, $L_{\\rm opt}$, fit to UVOIR bolometric luminosity data (in red; \\citealt{nicholl2018a}) to obtain the magnetar parameters; (2) magnetar spin down luminosity, $L_{\\rm mag}$ (green dotted lined); and (3) predicted gamma-ray luminosity that escape the ejecta, $L_{\\gamma}$ (pink dot-dashed line; Equations \\ref{eq:Lmag}, \\ref{eq:trapped} and \\ref{eq:leaking}). Black bars show {\\it Fermi}-LAT upper limits reported for six 180 day bins starting ${\\sim}90$ days after explosion. The olive open box shows the VERITAS integral energy flux} upper limit taken ${\\sim}135$ days after the explosion, with EBL absorption correction applied. Upper limits on the 0.2-10 keV X-ray luminosity from {\\it Chandra} are from \\citet{bhirombhakdi2018} in green. Grey shaded regions labeled ``$\\tau_{\\gamma} <1$\" show the approximate time after which gamma rays of the indicated energy should escape ejecta, based on Figure \\ref{fig:tau}. A purple dot-dashed line shows the engine luminosity, $L_{\\mathrm{BH}}$ (Eq.~\\ref{eq:L_BH}), in an alternative model in which the supernova optical luminosity is powered by fall-back accretion onto a black hole. All upper limits denote the 95\\% confidence level.\n \\label{fig:SN2015bn_lc}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics{SN2017egm_lc_ebl_corrected.pdf}\n \\caption{The SN2017egm light curve spanning 10 to 1300 days after explosion, following the same format as Figure~\\ref{fig:SN2015bn_lc}. UVOIR data are shown in red \\citep{bose2017, nicholl2017c}. Integral energy flux upper limits from {\\it Fermi}-LAT are reported for six 180 day bins starting ${\\sim}90$ days after the explosion. Integral energy flux upper limits are shown for VERITAS data taken ${\\sim}670$ days after explosion, with EBL absorption correction applied. The maximum luminosity of the black hole accretion model $L_{\\mathrm{BH}}$ (Eq. \\ref{eq:L_BH}) is shown in purple. All upper limits denote the 95\\% confidence level.}\n \\label{fig:SN2017egm_lc}\n\\end{figure*}\n\n\\begin{table*}[ht]\n \\begin{center}\n \\caption{Results from VERITAS observations for both epochs of SN2015bn, and SN2017egm. Shown are the quality selected livetime, number of gamma-ray-like events in the on and off-source regions, the normalization, the observed excess of the gamma-rays and the statistical significance. The integral flux upper limit is shown for the given energy threshold, without EBL absorption correction, integrated up to \\SI{30}{\\TeV}.}\n \\label{tab:veritas_total_results}\n \\begin{tabular}{cl|rrr}\n \\multicolumn{1}{l}{Parameter} & {[}unit{]} & SN2015bn$_{1}$ & SN2015bn$_{2}$ & SN2017egm \\\\ \n \\hline\n \\hline\n Start (MJD) & [day] & 57149 & 57533 & 58566 \\\\\n End (MJD) & [day] & 57164 & 57538 & 58578 \\\\\n Livetime & [hour] & 1.0 & 1.8 & 8.7 \\\\\n On & [event] & 4 & 10 & 49\\\\\n Off & [event] & 179 & 188 & 596 \\\\\n $\\alpha^{(a)}$ & - & 0.0286 & 0.0299 & 0.0634 \\\\\n Excess & [event] & -1.1 & 4.4 & 11.2 \\\\\n Significance & [$\\sigma$] & -0.5 & 1.7 & 1.6 \\\\\n Flux UL & [$\\SI{e-13}{cm^{-2}~s^{-1}}$] & 28.5 & 27.8 & 10.2 \\\\\n $E_{\\rm threshold}$ & [GeV] & $>320$ & $>420$ & $>350$ \\\\ \n \\end{tabular}\n \\end{center}\n $^{(a)}$ Ratio of relative exposure for On and Off regions.\n\\end{table*}\n\n\\section{Results}\n\\label{sec:results}\n\nNo statistically significant detections were made of either SN2015bn or SN2017egm across the energy range 100 MeV to 30 TeV. Integral energy upper limits are reported for the energy ranges given for each instrument. Figure~\\ref{fig:SN2015bn_lc} and Figure~\\ref{fig:SN2017egm_lc} show the {\\it Fermi}-LAT and VERITAS upper limits in comparison to the supernova optical light curves and the theoretically-predicted escaping luminosity from the magnetar model. \n\\subsection{Optical}\\label{sec:optical_result}\nThe SN2015bn integrated ultraviolet-optical-infrared (UVOIR) light curve data are reproduced here from previous analyses \\citep{nicholl2016, nicholl2016a, nicholl2018a}. To produce these bolometric light curves, the multi-band optical data were interpolated and integrated at each epoch using the code \\texttt{superbol} \\citep{nicholl2018}. \n\nSimilarly, the SN2017egm UVOIR data are also reproduced here with \\texttt{superbol} \\citep{bose2017, nicholl2017c}.\n\n\\subsection{{\\it Fermi}-LAT}\nBoth SN2015bn and SN2017egm are not statistically significant sources in the first $\\sim90$ days or the subsequent 6-month bin starting 90 days after the explosion. These sources also remain undetected in any of the following 6-month bins, and in the multi-year data sets. \n\nThe evaluation of the integral energy flux upper limit for the {\\it Fermi}-LAT observations within each time bin was performed assuming a powerlaw spectral model with an index of -2. The model dependence of this calculation naturally impacts the interpretations in section \\ref{sec:discussion}, so the the fit was performed with indices 2, 2.5 and 3 to find the impact of the model on the final upper limit. An uncertainty of about $10\\%$ was found based on varying the index.\n\nSN2015bn is found to have test statistic (TS) of 0.06,\nwith 12 predicted events above the isotropic diffuse background $\\simeq \\num{4.8e4} $ events over the entire period. The flux upper limit is $\\SI{1.6e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}. In the first $\\sim90$ days after the explosion, where the gamma ray emission is not expected due to the high gamma-ray absorption (see Figure \\ref{fig:tau}), the flux upper limit is $\\SI{3.5e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}, with a TS of 0. For the first 6-month period, when the signal is most likely, the flux upper limit is $\\SI{1.9e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ for $TS \\simeq 0$, consistent with a non-detection. All of the following 6-month bins reported non-detections with $TS<2$. \n\nSN2017egm is found to have $TS=4.4$,\nwith 43 predicted events above the isotropic diffuse background $\\simeq \\num{5.9e4} $ events. The flux upper limit is $\\SI{1.2e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}. In the first $\\sim90$ days after the explosion, where the gamma ray emission is not expected due to the high gamma-ray absorption (see Figure \\ref{fig:tau}), the flux upper limit is $\\SI{3.2e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}, with a TS of 0. For the first 6-month period, when the signal is most likely, the flux upper limit is $\\SI{4.9e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ for $TS=\\num{10.1}$, consistent with a non-detection. All of the following 6-month bins reported non-detections with $TS<1$.\n\n\\subsection{VERITAS}\nTable \\ref{tab:veritas_total_results} reports the results from VERITAS observations of SN2015bn and SN2017egm. Each observation is consistent with a non-detection. The significance of each excess of observed events above background is below 2 standard deviations (sigma). The flux upper limits are also given, calculated by integrating above the threshold energy of the instrument.\n\nThe statistical significance of an excess is estimated using Equation 17 of Li \\& Ma \\citep{li1983}. SN2015bn has significance value of $-0.5 \\sigma$ in the first epoch observation. The integral flux upper limit from \\SIrange{0.32}{30}{\\TeV} for SN2015bn is \\SI{2.85e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{1.27e44}{\\erg.s^{-1}} at a redshift of 0.1136. Due to the serendipitous nature of the observation, SN2015bn is significantly off-axis, which lowers the instrument sensitivity at the energy threshold of \\SI{320}{\\GeV}. Additionally, a 10\\% systematic uncertainty is added to the flux normalization and reported energy threshold due to instrument degradation during the period of 2012-2015 \\cite{nievasrosillo2021}. This uncertainty is derived empirically from the observation of the Crab Nebula over the same period. During the second observation in 2016, SN2015bn was found to have a significance of 1.7. The integral flux upper limit from \\SIrange{0.42}{30}{\\TeV} for SN2015bn is \\SI{2.78e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{1.60e44}{\\erg.s^{-1}}. \n\nFor SN2017egm, the Li \\& Ma significance value is $0.2 \\sigma$ and an integral upper limit from \\SIrange{0.35}{30}{\\TeV} is \\SI{1.0238e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{3.54e42}{\\erg. s^{-1}} above the energy threshold of \\SI{350}{\\GeV} at redshift z=0.0310. The systematic correction due to instrument degradation during the period of 2012-2019 is applied automatically with the use of the throughput-calibrated analysis templates \\citep{nievasrosillo2021}. In the cases of both SN2015bn and SN2017egm, the impact of varying the power law model index parameter from -2 to -5 is about 10\\%, which is a negligible in the context of their respective light curves.\n\nVHE photons are absorbed by the extragalactic background light (EBL) throughout the universe, so the flux must be corrected to account for the missing photons. This absorption is energy and redshift dependent. Deabsorption is applied to the flux using the model of \\citet{Dominguez2011}. \nThe EBL deabsorption factor was convolved with the upper limit calculation, assuming the same spectral shape (a power law with the photon index of -2.0). \nThe deabsorbed integral photon upper limit for SN2015bn within the energy range \\SIrange{0.32}{30}{\\TeV}, is \\SI{3.36e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{1.49e44}{\\erg.s^{-1}}. For the second observation, the deabsorbed integral photon upper limit for SN2015bn within the energy range \\SIrange{0.42}{30}{\\TeV}, is \\SI{3.30e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{1.91e44}{\\erg.s^{-1}}. For SN2017egm, with a slightly smaller energy range \\SIrange{0.350}{30}{\\TeV}, the deabsorbed integral photon flux is \\SI{1.07e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{3.70e42}{\\erg.s^{-1}}. These EBL corrected values are plotted in Figure \\ref{fig:SN2015bn_lc} and Figure \\ref{fig:SN2017egm_lc}.\n\n\\section{Discussion}\n\\label{sec:discussion}\nThe source of the extra luminosity powering SLSNe-I may be found in the signature of its late time gamma-ray emission. This section explores the HE to VHE emission hundreds of days after the explosion. The following models with a gamma-ray emission component for the powering mechanism are discussed: 1) magnetar central engine (see section \\ref{sec:MagnetarSpinDown}), 2) black hole central engine (see section \\ref{sec:BlackHole}), and 3) circumstellar interaction (see section \\ref{sec:Circumstellar}).\n\n\\subsection{Magnetar Central Engine} \\label{sec:MagnetarSpinDown}\n\nThe most promising mechanism for powering SLSNe-I is the rotational energy input from a central magnetar. In this scenario, a young pulsar or magnetar inflates a nebula of relativistic particles, which radiate high-energy gamma rays and X-rays. This section initially explores a simple implementation of the magnetar model (see Appendix \\ref{sec:appendix} for full description), followed by a more complete model described in detail in \\citet{vurm2021} for both SN2015bn and SN2017egm. The application of this so-called self-consistent model is necessary to directly predict the energy-dependent luminosities within the energy ranges of the {\\it Fermi}-LAT and VERITAS observations, a major contribution that is not possible with simpler implementation described in the appendix.\n\nAt early times after the explosion (around and immediately after the maximum in the optical emission) the gamma rays are absorbed and thermalized by the expanding supernova ejecta. At these times, the luminosity and shape of the optical light curve can be used to constrain the parameters of the magnetar. In this model, the radiation of an input energy reservoir (the spin down luminosity of a rotating magnetar) diffuses through the ejecta following the analytical solution by \\citet{arnett1982} (equation \\ref{eq:trapped}). \n\nThe time evolution of the magnetar's spin-down luminosity can be modeled by assuming a rotating dipole magnetic field whose energy loss is dominated by emission of radiation in the gamma-ray and X-ray bands (see Appendix \\ref{sec:appendix} for details). \n\nThis luminosity depends on the magnetar initial spin period, surface dipole magnetic field strength, and neutron star mass, $L_{\\rm mag}(t, P_0, B, M_{\\rm NS})$ (equation \\ref{eq:Lmag}). The emitted radiation thermalizes as it diffuses through the ejecta. The conditions of the ejecta determine the optical and gamma-ray outputs, dominated by the values of the ejecta mass, ejecta velocity, and optical and gamma-ray opacities to form $L_{\\rm opt}(t, M_{\\rm ej}, v_{\\rm ej}, \\kappa,\\kappa_{\\gamma})$ (equation \\ref{eq:leaking}) and $L_{\\gamma}(t, M_{\\rm ej}, v_{\\rm ej}, \\kappa, \\kappa_{\\gamma})$ (equation \\ref{eq:escape}). \n\nFor SN2015bn and SN2017egm, the parameters for the magnetar and the supernova ejecta properties were found by fitting their integrated ultraviolet-optical-infrared (UVOIR) light curves, shown with red points in Figure~\\ref{fig:SN2015bn_lc} and Figure~\\ref{fig:SN2017egm_lc}. All fits were conducted using non-linear least squares minimization\\footnote{\\texttt{scipy.optimize.curve\\_fit}}.\nThe best-fit parameters with errors for the magnetar model are given in Table \\ref{tab:event_physical_params}. The redshifts and time of peak optical magnitude are shown in the table as listed in The Open Supernova Catalog \\citep{Guillochon2016}\\footnote{\\url{https:\/\/sne.space}}.\n\nThese parameters are consistent with the results of previous fits \\citep{nicholl2018a,nicholl2017c} that took into account both the optical spectral energy distribution and light curve using the open source code \\texttt{MOSFiT} \\footnote{\\url{https:\/\/mosfit.readthedocs.io\/en\/latest\/}}. \nThe relative statistical errors on these fit parameters may be optimistic at $\\sim10\\%$, and the systematic errors will still need to be incorporated for a better understanding the magnetar parameter space. The largest contributor to the magnetar power are the period and magnetic field values, which determine the overall magnitude of the luminosity. The ejecta mass and velocity determine the time to optical peak by the diffusion of the emission through the ejecta.\n\nA particularly important shortfall of this model is the constant effective opacity to both optical and gamma-ray photons, rather than a time-dependent treatment of the opacity. TeV gamma rays interact preferentially with optical photons, so at the time of the peak optical emission, $\\gamma\\gamma$ absorption by optical photons will be high, reducing any predicted gamma-ray emission by this model. Equation \\ref{eq:leaking} is a bolometric luminosity, so it does not take into account the energy and time dependent opacity, instead fitting a constant effective $\\kappa$ and $\\kappa_{\\gamma}$ to generate the time dependent optical depth. \n\nTherefore, Figure~\\ref{fig:tau} is used as a guide for when to expect $L_{\\gamma}$ to provide an appropriate estimate for the gamma-ray emission. The shaded regions in Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc} estimate the time periods when photons of the given energies can escape. It is important to reiterate that this model is energy independent, representing the bolometric luminosity not thermalized by the ejecta. This model cannot distinguish the emission between LAT and VERITAS energy bands since it does not consider the physical model of the nebula; the self-consistent model described by \\cite{vurm2021} and discussed below will be an attempt to do so explicitly.\n\nFollowing the methodology in Appendix \\ref{sec:appendix} with the magnetar parameters for each SLSN, $L_{\\rm mag}(t)$, $L_{\\rm opt}(t)$, and $L_{\\gamma}(t)$ were calculated and are shown in comparison to the gamma-ray limits in Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc}.\n\nFor SN2015bn (Figure \\ref{fig:SN2015bn_lc}), neither the {\\it Fermi}-LAT upper limits nor the VERITAS upper limit constrain the predicted escaping luminosity. \nSimilarly, for SN2017egm (Figure \\ref{fig:SN2017egm_lc}), both the VERITAS and {\\it Fermi}-LAT upper limits are not deep enough to constrain the predicted escaping luminosity. An important caveat to these upper limits is that the escaping luminosity may also be emitted at energies not explored here, such as hard X-rays or gamma-rays greater than \\SI{ 30}{\\TeV}.\n\nThe optimal time to observe with a pointed instrument sensitive at a particular photon energy results from a trade-off between the dropping ($\\propto t^{-2}$) magnetar luminosity and the rising transparency of the ejecta; predicting the optimal time post-peak to observe requires knowledge of the evolution of the optical spectrum. It is possible to accumulate enough optical data within a few weeks after the optical peak to fit the magnetar model for a reliable prediction of the gamma-ray luminosity. In the case of SN2017egm, the gamma-ray luminosity prediction was anchored by the late optical data points about 1 year after the explosion. This means that had the VERITAS observations been taken at that point (more than a year earlier than the original observation), they would have been deeply constraining to the magnetar model.\n\n\nGoing beyond these relatively model independent statements to compare to a more specific spectral energy distribution for the escaping magnetar nebula requires a detailed model for the nebula emission and its transport through the expanding supernova ejecta. Such a model offers preliminary support that a significant fraction of $L_{\\gamma}$ may come out in the VHE band \\citep{vurm2021}. In this case, the VHE limits on SN2015bn and SN2017egm do not strongly constrain the parameters of the magnetar model, such as the nebular magnetization.\n\n\nThe model of \\citet{vurm2021} self-consistently follows the evolution of high-energy electron\/positron pairs injected into the nebula by the magnetar wind and their interaction with the broadband radiation and magnetic fields. They found that the thermalization efficiency and the amount of gamma-ray leakage depends strongly on the nebular magnetization, $\\varepsilon_B$, i.e. the fraction of residual magnetic energy in the nebula relative to that injected by the magnetar.\n\nThe model is simulated for dimensionless $\\varepsilon_B$ values set between $10^{-6}$ and $10^{-2}$; the higher magnetizations lead to greater synchrotron efficiencies, which dominate within a few hundred days, and lead to the optical emission tracking the spin-down luminosity. Lowering the magnetization to $10^{-7}-10^{-6}$ for SLSN-I events like those in this work delays the transition to synchrotron-dominated thermalization, so that the predicted optical emission actually tracks the observed data.\n\nThe theoretical light curves and gamma-ray upper limits are shown in Figure ~\\ref{fig:Indrek}. \\citet{vurm2021} concluded that the predicted low magnetizations constrained by the optical data alone presents new challenges to the theoretical framework regarding the dissipation of the nebular magnetic field. This may invoke magnetic reconnection ahead of the wind termination shock or near the termination shock through forced reconnection of alternating field stripes described in \\citet{komissarov2013}, \\citet{lyubarsky2003}, \\citet{margalit2018b}. It is also possible that the true luminosity of the central engine decreases faster in time than the simpler $\\propto t^{-2}$ magnetic spin down, such that escaping VHE emission is not necessary to explain the model. These VHE upper limits do not rule out this model, and do not settle the challenges inferred by the low magnetization required to fit the optical data. Further observations are needed to probe the nebular magnetization and synchrotron efficiency, and deep VHE observations will contribute to these constraints.\n\nThe non-detection of x-rays for both events is consistent with the predictions of \\cite{margalit2018a} of a fully ionized ejecta. Even under the most optimistic conditions - an engine that puts 100\\% of its spin-down luminosity into ionizing photons of ideal energies - cannot reduce the opacity enough to allow x-rays to escape under the usual assumptions (e.g. spherically symmetric ejecta shell). \n\n\n\n\n\\begin{figure}[ht]\n \\includegraphics[width=1.1\\columnwidth]{SN2015bn_lc_ebl_model.pdf}\n \\includegraphics[width=1.1\\columnwidth]{SN2017egm_lc_ebl_model.pdf}\n \\caption{Model light curve for nebular magnetization (from \\cite{vurm2021}) for SN2015bn with $\\varepsilon_B = 10^{-7}$ (top panel) and SN2017egm with $\\varepsilon_B=10^{-6}$ (bottom panel). }\n \\label{fig:Indrek}\n\\end{figure}\n\n\\subsection{Black Hole Central Engine} \\label{sec:BlackHole}\n\nInstead of forming a neutron star like a magnetar, a SLSN-I might form a black hole, in which case the optical peak of the light curve could be powered by energy released from the fallback accretion of ejecta from the explosion (e.g.~\\citealt{dexter2013}). Even if a black hole does not form immediately, it could form at late times once the magnetar accretes enough fallback material \\citep{moriya2016a}. The main practical difference as compared to a magnetar in section \\ref{sec:MagnetarSpinDown} is that the black hole central engine power would be predicted to decay with the fall-back accretion rate $\\dot{M}_{\\rm fb} \\propto t^{-5\/3}$ instead of $\\propto t^{-2}$. Thus, in principle, for the same luminosity at the time of the optical maximum $t_{\\rm pk}$, the central engine output at times $t \\gg t_{\\rm pk}$ could be enhanced by a factor $\\propto (t\/t_{\\rm pk})^{1\/3} \\sim 2$ for $t \\sim 1$ year and $t_{\\rm pk} \\sim 1$ month, thus tightening our constraints.\n\nIn Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc}, a rough estimate of the maximal engine luminosity in the BH accretion scenario is shown, which is calculated as \n\n\\begin{align}\nL_{\\mathrm{BH}}=\\frac{2^{5\/3} L^{\\rm pk}_{\\rm opt}}{\\left(1+\\frac{t}{t_{\\rm pk}}\\right)^{5\/3}} \\label{eq:L_BH},\n\\end{align}\n\nwhere $L^{\\rm pk}_{\\rm opt}$ is the peak optical luminosity, scaled so that $L_{\\rm BH} = L_{\\rm opt}$ around the optical peak.\n\nOn the other hand, while gamma rays are naturally expected from the ultra-relativistic spin-down powered nebula of a magnetar, it is less clear this would be the case for a black hole engine. For instance, the majority of the power from a black hole engine could emerge in a mildly relativistic wind from the black hole accretion disk instead of an ultra-relativistic spin-down powered pulsar wind.\n\nAs seen in both Figure \\ref{fig:SN2015bn_lc} (SN2015bn) and Figure \\ref{fig:SN2017egm_lc} (SN2017egm), the gamma-ray emission in the black hole scenario is not constrained in the {\\it Fermi}-LAT and VERITAS energy bands.\n\n\n\\subsection{Circumstellar Interaction} \\label{sec:Circumstellar}\n\nAn alternative model for powering the light curve of SLSNe is to invoke the collision of the supernova ejecta with a slower expanding circumstellar shell or disk surrounding the progenitor at the time of the explosion (e.g.~\\citealt{smith2006,chevalier2011,Moriya2013a}). Features of this circumstellar model (CSM), such as the narrow hydrogen emission lines that indicate the interaction of a slow-moving gas, provide compelling evidence for this being a powering mechanism for many but not all of the hydrogen-rich class of SLSNe (SLSNe-II; e.g.~\\citealt{smith2007,nicholl2020}). \n\nShock interaction could in principle also power some hydrogen-poor SLSNe (SLSNe-I), particularly in cases where the circumstellar interaction is more deeply embedded and less directly visible (e.g.~\\citealt{sorokina2016,kozyreva2017}). \nThere is growing evidence for hydrogen-poor supernovae showing hydrogen features from the interaction in their late-time spectra \\citep{Milisavljevic2015,Yan2015,Yan2017,Chen2018,Kuncarayakti2018,Mauerhan2018}. The light echo from iPTF16eh \\citep{lunnan2018} implies a significant amount of hydrogen-poor circumstellar medium in a SLSN-I at ${\\sim}10^{17}$ cm. However, this material is too distant for the ejecta to reach by the time of maximum optical light and hence cannot be responsible for boosting the peak luminosity.\n\nIn principle, the gamma-ray observations of SLSNe can constrain shock models. In many cases, this may not work out since most of the emission from shock-heated plasma is either expected to: (1) come out in the X-ray band, as is well studied in other CSM-powered supernovae such as SNe IIn like SN 1998S \\citep{Pooley2002}, SN 2006jd \\citep{Chandra2012}, and SN 2010jl \\citep{Chandra2015}, and SNe Ib\/c \\citep{Chevalier2006}; or (2) be absorbed by the surrounding ejecta and reprocessed into the optical band. Thus, these VHE limits on SLSNe do not constrain the bulk of the shock power. \n\nHigher-energy radiation can be produced if the shocks accelerate a population of non-thermal relativistic particles which interact with ambient ions or the supernova optical emission to generate gamma rays (e.g. via the decay of $\\pi^{0}$ generated via hadronic interactions with matter and radiation; e.g., \\citealt{murase2011}). However, because shocks typically place a fraction $\\epsilon_{\\rm rel} \\lesssim 0.1$ of their total power into relativistic particles (or even less; \\citealt{steinberg2018,fang2019}), the predicted gamma-ray luminosities (matching the same level of optical emission as magnetar models) would be at least 10 times lower than $L_{\\gamma}$ predicted by the magnetar nebula scenario, thus rendering our VHE upper limits unconstraining on non-thermal emission from shocks on SN2015bn and SN2017egm. This is consistent with upper limits from the Type IIn SN 2010j from {\\it Fermi}-LAT, which \\citet{Murase2019} used to constrain $\\epsilon_{\\rm rel} \\lesssim 0.05-0.1$. \n\n\\section{Future Prospects}\n\\label{sec:future}\n\nThese results demonstrate that high-energy gamma-ray observations of SLSN-I are on the brink of enabling constraints on the light curves and even spectral energy distribution of magnetar models. Given the rarity of bright, nearby SLSN-I, and the need to take observations in the optimal window (when $L_{\\gamma}$ is near maximum), careful planning will be required to make progress going ahead \\citep{prajs2017,Quimby2011,mccrum2015}. The strategy outlined below will focus only on SLSN-I, as type II SLSN are likely to be powered by a mechanism that requires a different consideration of the temporal and spectral evolution of the gamma-ray emission.\n\nStandard arrays of IACTs provide an improved instantaneous sensitivity to gamma-ray emission over {\\it Fermi}-LAT due to $10^4$ to $10^5$ larger effective area, counterbalanced in part by the pointed nature of their observations. To propose a strategy, we firstly re-visited the characteristics of a large sample of observed SLSNe and performed a systematic study.\n\n\\citet{nicholl2017d} fit a sample of 38 SLSNe light curves using MOSFiT to obtain a distribution of magnetar model parameters. This sample is a selection of SLSNe with well observed events classified as Type-I with published data near the optical peak, forming a representative sample of good SLSNe-I for a population study. For each event in this sample, the following was calculated: the escaping gamma-ray luminosity $L_{\\gamma}$ following the procedure outlined in Appendix \\ref{sec:appendix} and the flux $F_{\\gamma} = L_{\\gamma}\/4\\pi D_{\\rm L}^{2}$ based on the source luminosity distance $D_{L}$. In performing this analysis, rather than fitting the value of $\\kappa_{\\gamma}$ individually to each optical light curve (as done in \\citealt{nicholl2017d}), the value $\\kappa_{\\gamma} = \\SI{0.01}{cm^{2} g^{-1}}$ is fixed in all events, based on the best-fit to SN2015bn (given its particularly high-quality late-time data, which provides the most leverage on $\\kappa_{\\gamma}$). \n\nThe results for $F_{\\gamma}(t)$ are shown in the top panel of Figure~\\ref{fig:distributions}. In the magnetar model, the predicted gamma-ray flux could emerge anywhere across the HE to VHE bands and hence it represents an upper limit on flux in the bands accessible to {\\it Fermi}-LAT and IACTs. The bottom two panels of Figure~\\ref{fig:distributions} show the distribution of the peak escaping flux $F_{\\rm \\gamma, max}$ and time of the peak flux relative to the explosion. For most SLSNe-I presented here, $F_{\\rm \\gamma, max}$ is well below the sensitivity of VERITAS and even the future Cherenkov Telescope Array (CTA) \\citep{thectaconsortium2019}. Also note that the characteristic timescale to achieve the peak gamma-ray flux is $\\approx 2-3$ months from the explosion. This timescale occurs approximately at the same time as when the optical depth of the ejecta to VHE emission falls below unity, when the VHE photons can escape (Figure~\\ref{fig:tau}). \n\n\\begin{figure}[ht!]\n \\centering\n \n \n \n \\includegraphics[width=\\columnwidth]{Combined_LCs_flux_dist.pdf}\n \\caption{Top: Escaping gamma-ray luminosity $L_{\\gamma}(t)$ for the sample of SLSNe fit by \\citet{nicholl2017d}. Five well studied SN are highlighted in blue, including SN2015bn. Overplotted are the VERITAS and CTA sensitivity curves for various exposures. Middle: Distribution of peak escaping gamma-ray flux $F_{\\rm \\gamma,max} = {\\rm max}[L_{\\gamma}]\/4\\pi D^{2}$, for the light curves from the top panel where $D$ is the distance to each source. Again, VERITAS and CTA sensitivities for different exposures are shown as vertical dashed lines. Bottom: Distributions of times since explosion to reach the maximum gamma-ray flux $F_{\\gamma,max}$ from $F_{\\gamma}$ above.} \n \n \\label{fig:distributions}\n\\end{figure}\n\nFigure~\\ref{fig:Optical_to_Escaping_Fluxes} shows $F_{\\rm \\gamma,600 d}$ as a function of the peak optical magnitude of the SLSNe-I from the same sample as in Figure~\\ref{fig:distributions}. The selection of fluxes at \\SI{600}{\\day} approximates the time when the effective opacity to 1 TeV photons reaches 1, based on Figure~\\ref{fig:tau}. The top axis also gives the all-sky rate of SLSNe-I above a given peak optical magnitude, which is estimated using the magnitude distribution of SLSNe-I and assuming they occur at a comoving volumetric rate of $ R(z)=19(1+z)^{3.28}\\SI{}{\\, Gpc^{-3}\\, yr^{-1}}$ following \\citet{nicholl2017b,lunnan2018,decia2018}. This estimation captures the general volumetric rate of events, but is unreliable for exceptionally bright events such as SN2017egm due to the small population for estimating the magnitude normalization. A bright event like SN2017egm may actually happen more often than once a century.\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics[width=.95\\textwidth]{Optical_to_Escaping_Fluxes_kap_1TeV.pdf}\n \\caption{\n Blue dots show the peak optical apparent magnitudes of a sample of SLSNe-I \\citep{nicholl2017d} as a function of their predicted maximum gamma-ray luminosity at 600 days after explosion ($F_{\\gamma, 600d}$). The top axis shows the approximate rate of events above the given peak optical magnitude, calculated using the method described in the main text. Peak maximum gamma-ray luminosities are calculated from fits of optical data with fixed $\\kappa_{\\gamma} = 0.01$ cm$^{2}$ g$^{-1}$. Integral sensitivities of various instruments are overplotted for different exposures. Solid lines: VERITAS 10 and 50 hour integral sensitivities above 220 GeV. Dotted lines: CTA (in development) 10 and 50 hour integral sensitivities above 125 GeV as estimated from 50 hour Monte Carlo simulations of the southern array \\citep{thectaconsortium2019} and extrapolated to 10 hours. Similar extrapolation is done for {\\it Fermi}-LAT from 10 years to 6 months \\citep{nolan2012} (dashed line). Proposed project AMEGO integral sensitivity above 100 MeV for 6 month observation window is also plotted (dash-dot line) \\citep{kierans2020}. \n }\n \\label{fig:Optical_to_Escaping_Fluxes}\n\\end{figure*}\n\nShown for comparison in Figure~\\ref{fig:Optical_to_Escaping_Fluxes} are the integral sensitivities of various gamma-ray instruments for different exposures. For IACT instruments such as VERITAS and the future CTA, sensitivity is defined as the minimum flux necessary to reach $5 \\sigma$ detection of a point-like source, requiring at least 10 excess gamma rays and the number of signal counts at least $5\\%$ of the number of background counts. For VERITAS, the sensitivity was calculated using observed Crab Nebula data to estimate the rates of signal and background photons with cuts optimized for a $\\Gamma = -2.5$ power-law spectrum, and then re-scaled for the appropriate observation time \\citep{Park2015}. For CTA, Monte Carlo simulations were used to derive angular resolution, background rates and energy dispersion features -- the instrument response functions (IRF) -- based on the Prod3b-v2 telescope configuration for the Southern site and its atmosphere \\citep{cherenkovtelescopearrayobservatory2016}. These IRFs are publicly available and were analyzed using the open-source CTOOLS\\footnote{\\url{http:\/\/cta.irap.omp.eu\/ctools\/}} \\citep{Knodlseder2016}. A power law spectral model was used to estimate the integral sensitivity above \\SIlist{0.125;1}{\\TeV} each for observations of \\SIlist{10;50}{\\hour} (see \\citealt{Fioretti2016} for further discussion on CTA integral sensitivity).\n\nBased on this systematic study, we propose the following observation strategy: 1) Receive automated public alert and Type I classification of SLSN from a survey instrument such as the Zwicky Transient Facility (ZTF). Classification is generally determined by identification of early spectral components such as OII absorption features. 2) During the multi-day rise and fall of bolometric optical light curve, fit the magnetar model ($L_{\\rm opt}$, yielding parameters for $L_{\\rm mag}$ and $L_{\\gamma}$) 3) Compare $L_{\\gamma}$ to the telescope sensitivity at the appropriate day when the effective $\\gamma$-$\\gamma$ opacity falls below ${\\sim1}$ for the telescope's sensitive energy range (see Figure~\\ref{fig:tau}). In the case of IACTs sensitive to energies above $\\SI{100}{\\GeV}$, the gamma rays will escape the magnetar a few hundred days after explosion, requiring a bright SLSN-I that will power gamma rays for as much as two years. \n\nEstimating ${\\sim}35\\%$ of all-sky visibility at VERITAS due to Sun, Moon, and seasonal weather cut, and above 60\\textdegree~ elevation, VERITAS is capable of detecting up to ${\\sim}0.4$ and ${\\sim}4$ SLSNe-I per year for \\SI{10}{h} and \\SI{50}{h} exposures, respectively. The next-generation CTA observatory will be able to detect as many as ${\\sim}8$ and ${\\sim}80$ events for \\SI{10}{hr} and \\SI{50}{h}, respectively, assuming a larger sky visibility fraction of ${\\sim}80\\%$ when both North and South arrays are included. On the other hand, SLSNe at greater distances also imply a stronger role of $\\gamma-\\gamma$ interactions on the EBL in suppressing the $\\gtrsim$ TeV emission, decreasing the observed integral flux by as much as 60 times at redshifts near 0.5 in the VERITAS energy range. \n\nFigure~\\ref{fig:TeV_gamma_flux_sens} shows the distribution of fluxes at \\SI{200}{\\day} and \\SI{600}{\\day} which are approximate average dates when the opacity to \\SI{100}{\\GeV} and \\SI{1}{\\TeV} photons falls below 1, respectively, and they are able to escape the ejecta. Accounting for this time delay for the opacity to drop, the expected rate of bright events drops by another 3 to 15 times. While past observations have not been followed up until this publication, the distribution of predicted gamma-ray fluxes hints that, particularly for \\SI{100}{\\GeV} photons, future SLSN-I will be observable with current and planned observatories. \n\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=\\columnwidth]{Escaping_Fluxes_kappa_gamma_dist.pdf}\n \\caption{ Distribution of gamma-ray luminosities $L_{\\gamma}$ at $t=\\SI{200}{\\day}$ (top) and $t=\\SI{600}{\\day}$ (bottom), when the optical depth for \\SI{100}{\\GeV} and \\SI{1}{\\TeV} photons drops below 1, calculated for a sample of 38 SLSNe \\citep{nicholl2017d}}\n \\label{fig:TeV_gamma_flux_sens}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{sec:conclusions}\nSLSN-I are potential gamma-ray emitters, and this paper provides the first upper limits at different times after the optical outburst for two good candidates. The reported upper limits approach the magnetar spin-down luminosity limit of SN2015bn and SN2017egm. While the expected gamma-ray luminosity in either the magnetar central-engine scenario or the shock-acceleration scenario is not constrained by these limits, a relativistic jet powered by fall-back accretion onto a black hole is disfavored in both cases. We explore prospects for obtaining improved VHE gamma-ray constraints in the future by current and planned IACTs. We estimate the Type-I SLSNe rate for VERITAS and CTA, considering observation constraints and the time delay due to the optical depth. For sufficiently nearby and bright SLSN-I, 0.4 and 4 events per year can be observed by VERITAS from 10-hr and 50-hr observation, respectively, and similarly rates of 8 and 80 events per year can be expected by CTA. \n\n\n\n\n\\acknowledgments \n{This research is supported by grants from the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, and by the Helmholtz Association in Germany. MN is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No.~948381) and by a Fellowship from the Alan Turing Institute. IV acknowledges support by the ETAg grant PRG1006 and by EU through the ERDF CoE grant TK133. VVD's work is supported by NSF grant 1911061 awarded to the University of Chicago (PI: Vikram Dwarkadas). We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument. \nThis research has made use of the CTA instrument response functions provided by the CTA Consortium and Observatory, see \\url{http:\/\/www.cta-observatory.org\/science\/cta-performance\/} (version prod3b-v2) for more details.\n}\n\n\\software{fermipy (v0.19), \\citep{wood2017},\nastropy \\citep{Robitaille2013,Price-Whelan2018},\nCTOOLs, \\citep{Knodlseder2016a}, superbol \\citep{nicholl2018} , EventDisplay \\citep{Maier2017}}, VEGAS \\citep{cogan2008}\n\n\\facilities{VERITAS, {\\it Fermi}-LAT}\n\n\\clearpage\n\n\n\\section{Introduction}\\label{sec:intro}\n\nThe recent growth of sensitive optical time-domain surveys has revealed and expanded exciting new classes of stellar explosions. These include superluminous supernovae, which can be up to 10--100 times more luminous than ordinary massive star explosions (e.g.~\\citealt{Quimby2011,howell2013,inserra2013,nicholl2013,decia2018,lunnan2018,quimby2018}; see \\citealt{Gal-Yam2019} for a recent review). Conventionally, the optical emission from most core-collapse supernovae is powered by the radioactive decay of $^{56}$Ni (Type Ib\/c) and by thermal energy generated via shock heating of the stellar envelope (Type IIL, IIp). However, the peak luminosities of SLSNe greatly exceed the luminosity expected from those conventional mechanisms, and the origin of the energy is still debated. \n\nA popular model for powering the time-dependent emission of SLSNe, particularly the hydrogen-poor Type I class (SLSN-I), involves energy input from a young central engine, such as a black hole or neutron star, formed in the explosion. For example, the accretion onto the compact object from bound debris of the explosion could power an outflow which heats the supernova ejecta from within (\\citealt{woosley2012,Quataert2012,margalit2016,moriya2018}). Alternatively, the central engine could be a strongly magnetized neutron star with a millisecond rotation period, whose rotationally powered wind provides a source of energetic particles that heat the supernova ejecta \\citep{kasen2010,woosley2010,dessart2012a,metzger2015a,sukhbold2016}. The magnetar\\footnote{To remain consistent with the SLSNe literature, the term magnetar is used throughout this paper. Magnetars generally have large dipole magnetic fields $B\\simeq$ \\SIrange{e13}{e15}{G} with a rotation period of a few seconds. In the case of the SLSN magnetar model, the radiation is extracted from the rotational energy of the young millisecond pulsars, but with large magnetic fields characteristic of magnetars.} model provides a good fit to the optical light curves of most SLSNe-I \\citep{inserra2013,nicholl2017d}. Furthermore, analyses of the nebular spectra of hydrogen-poor SLSNe \\citep{jerkstrand2017,nicholl2019} and Type Ib SNe \\citep{milisavljevic2018} support the presence of a persistent central energy source, consistent with an energetic neutron star.\n\nThe details of how the magnetar would couple its energy to the ejecta are uncertain. Several models consider that the rotationally powered wind from a young pulsar inflates a nebula of relativistic electron\/positron pairs and energetic radiation behind the expanding ejecta \\citep{kotera2013,metzger2014b,murase2015}. At the wind-termination shock, the pairs are heated and radiate X-rays and gamma rays with high efficiency via synchrotron and inverse-Compton processes. Photons that evade absorption via $\\gamma\\textrm{-}\\gamma$ pair creation in the nebula can be ``absorbed\" by the ejecta further out, thermalizing their energy and directly powering the supernova's optical emission (e.g.~\\citealt{metzger2014b, vurm2021}). \n\nThermalization of the nebular radiation will be most efficient at early times, when the column through the ejecta shell and ``compactness'' of the nebula are at their highest. At these times one would expect the optical light curve to faithfully track the energy input of the central engine. However, as the ejecta expand, the radiation field dilutes and the shell becomes increasingly transparent to high-energy and very-high-energy photons. The increasing transparency, and correspondingly decreasing thermalization efficiency, eventually causes the supernova's optical luminosity to drop below the rate of energy injection from the central engine \\citep{chen2015,wang2015}, with the remaining radiation escaping directly from the nebula as gamma rays or X-rays (the putative ``missing\" luminosity). \n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.9\\columnwidth]{Opacity_SN2015bn_vs_time.pdf}\n \\includegraphics[width=0.9\\columnwidth]{Opacity_SN2017egm_vs_time_weightlate_v2.pdf}\n \\caption{Optical depth at different photon energies as a function of time, calculated for ejecta properties (mass $M_{\\odot}$, mean velocity, etc.) derived from observations of SN2015bn \\citep{nicholl2018a} and SN2017egm \\citep{nicholl2017c} shown in Table \\ref{tab:event_physical_params}. Top: SN2015bn. Bottom: SN2017egm. The horizontal dotted line represents $\\tau_{\\rm eff} = 1$. The cross-sections for photon-photon and photon-matter pair production opacities are taken from \\citet{zdziarski1989}. The solid lines correspond to target blackbody radiation temperature $T_{\\rm eff} = (L_{\\rm opt}\/4\\pi R^2)^{1\/4}$, where $L_{\\rm opt}$ and $R$ are the optical luminosity and ejecta radius, respectively. The dashed lines are computed with a temperature floor of $T = 4000$~K, to mimic the approximate spectrum in the nebular phase. Below ${\\sim}10$~GeV the opacity is dominated by photon-matter pair production at all times. Above $100$~GeV, pair production on the thermal target radiation field dominates up to a few years.}\n \\label{fig:tau}\n\\end{figure}\n\nAs the ejecta expand and the spin-down luminosity weakens, the conditions for various processes responsible for photon energy loss change and impact the effective optical depth. Within a few months, the effective optical depth to high-energy (HE; 100 MeV to 100 GeV) photons emitted from the central engine nears unity and, at several hundred days it reaches unity for very-high-energy (VHE; 100 GeV \u2013 100 TeV) photons. \n\nFigure~\\ref{fig:tau} shows examples of the effective optical depth through the ejecta for photons of various energies as a function of time. They have been calculated using time-dependent properties for the supernova ejecta and radiation field motivated by the observations of SN2015bn and SN2017egm, both particularly well-studied SLSNe-I explored in \\cite{vurm2021}.\n\nThe dominant processes involved in the calculation of the gamma-ray optical depth include photon-matter and photon-photon interactions, particularly pair production on the nuclei and soft radiation fields in the ejecta. An accurate treatment considering the radiation transport is discussed in depth in \\citet{vurm2021}. The standard version of the magnetar model \\citep{kasen2010, woosley2010, nicholl2017d} does not consider this time-dependent calculation and relies on constant effective opacities to optical and high-energy photons. Figure~\\ref{fig:tau} provides a useful guiding timescale for when to consider gamma-ray emission at various energies, calculated with the model in \\citet{vurm2021} using the ejecta properties fit to the optical data in Table \\ref{tab:event_physical_params}.\n\nGiven its comparatively nearby distance at z=0.1136, SN2015bn is an excellent candidate event to test the magnetar hypothesis. The optical light curve shows a steepening from $\\propto t^{-2}$ decay to $\\propto t^{-4}$ around ${\\sim}200$ days \\citep{nicholl2018a}. This behavior is consistent with a leakage of high-energy radiation from a magnetar nebula \\citep{nicholl2018a}. A deep search in the ${\\sim}0.1-10$ keV X-ray band resulted in nondetections \\citep{bhirombhakdi2018}, eliminating the possibility that leakage from the nebula occurs in the softer X-ray bands.\n\n\\citet{margutti2018b} present a similar search for late-time X-ray emission from a larger sample of SLSNe-I, mostly resulting in upper limits; however, see \\citet{levan2013} for an X-ray detection of the SLSN-I SCP 06F6 that could still support the magnetar hypothesis. X-ray nondetections are not surprising, because the ejecta are likely to still be opaque in the $\\lesssim 10$ keV band due to photoelectric absorption in the hydrogen-poor ejecta \\citep{margalit2018a}. Intriguingly, \\citet{Eftekhari2019} detected radio emission from the location of the SLSN PTF10hgi at 7.5 yr after the explosion and argued that the emission could be synchrotron emission from an engine-powered nebula.\n\nSome effort has been underway to search for nebular leakage in the gamma-ray band. \\citet{renault-tinacci2018} obtained upper limits on the $0.6-600$ GeV luminosities from SLSNe by a stacked analysis of 45 SLSNe with {\\it Fermi}-LAT. The majority of their sample were SLSNe-I, the most likely class to be powered by a central engine; however the results were dominated by a single, extremely close Type II event (SLSN-II), CSS140222. Hydrogen-rich SLSNe make up the Type II class (SLSN-II), which are suggested to be powered by the interaction of the circumstellar medium with the supernova ejecta. Nevertheless, even with CSS140222 included, the upper limits are at best marginally constraining on the inferred missing luminosity. \n\nIn this paper, the search is expanded to gamma-ray emission from SLSNe-I in the HE to VHE bands using the {\\it Fermi} Gamma-Ray Space Telescope and the ground-based VERITAS observatory. In particular, observations of SN2015bn and SN2017egm are presented here. SN2017egm is the closest SLSN-I to date in the Northern Hemisphere at z=0.0310 \\citep{bose2017,nicholl2017c}. Observations of young supernovae with gamma-ray telescopes have been few, with no detections so far. Some tantalizing candidates like iPTF14hls and SN 2004dj have been explored with {\\it Fermi}-LAT but are unconfirmed due to large localization regions overlapping with other gamma-ray candidates \\citep{yuan2018,xi2020}. MAGIC carried out observations of a Type I SN \\citep{ahnen2017a}. HESS observed a sample of core-collapse SNe \\citep{Abdalla2019}, and later obtained upper limits on SN 1987A \\citep{theh.e.s.s.collaboration2015}. Our observations are the first of superluminous supernovae. \n\nThroughout this paper, a flat $\\Lambda$CDM cosmology is used, with $H_0 = \\SI{67.7}{km.s^{-1}.Mpc^{-1}}$, $\\Omega_{M}=0.307$, and $\\Omega_{\\Lambda} = 0.6911$ \\citep{planckcollaboration2016}. The corresponding luminosity distances to SN2015bn and SN2017egm are \\SI{545.37}{\\mega\\pc} (z=0.1136) \\citep{nicholl2016} and \\SI{139.29}{\\mega\\pc} (z=0.0310) \\citep{bose2017}.\n\n\n\\section{Observations and Methods} \\label{sec:obs}\n\\label{sec:observations}\nThe superluminous supernovae SN2015bn and SN2017egm were observed with {\\it Fermi}-LAT and VERITAS during 2015--2016 and 2017--2020, respectively. SN2015bn is a SLSN-I explosion from 23 Dec 2014 (MJD 57014) and it peaked optically on 19 Mar 2015 (MJD 57100) \\cite{nicholl2016a}. SN2017egm is a SLSN-I explosion from 23 May 2017 (MJD 57896) and it peaked optically on 18 Jun 2017 (MJD 57922) \\cite{bose2017}. Some properties of the SLSNe are given in Table \\ref{tab:event_physical_params}. Details regarding the optical, {\\it Fermi}-LAT and VERITAS observations and the data-analysis methods are below. \n\n\n\\begin{table}[ht]\n \\begin{center}\n \\caption{Properties of the SLSNe considered in this paper. }\n \\begin{tabular}{cl|rr}\n \\multicolumn{1}{l}{Parameter} & {(}Unit{)} & {SN2015bn} & {SN2017egm} \\\\ \\hline\n R.A. & $^{\\circ}$ & 173.4232 & 154.7734 \\\\\n decl & $^{\\circ}$ & 0.725 & 46.454 \\\\\n z & - & 0.1136 & 0.0310 \\\\\n $t_{0}^{(a)}$ & MJD & 57014 & 57896 \\\\\n $t_{pk}^{(b)}$ & MJD & 57100 & 57922 \\\\ \\hline\n $P_0^{(c)}$ & ms & $2.50^{+0.29}_{-0.17}$ & $5.83^{+0.73}_{-0.70}$ \\\\\n $B^{(d)}$ & $10^{14}$ G & $0.26^{+0.07}_{-0.05}$ & $0.94^{+0.13}_{-0.16}$ \\\\\n $M_{ej}^{(e)}$ & $M_{\\odot}$ & $10.8^{+0.83}_{-1.34}$ & $2.99^{+0.30}_{-0.23}$ \\\\\n $\\kappa^{(f)}$ & cm$^{2}$ g$^{-1}$ & $0.18^{+0.01}_{-0.02}$ & $0.12^{+0.04}_{-0.06}$ \\\\\n \n $v_{ej}^{(h)}$ & $10^{8}$ cm s$^{-1}$ & $5.68^{+0.16}_{-0.14}$ & $10.3^{+0.35}_{-0.27}$ \\\\\n $\\kappa_{\\gamma}^{(i)}$ & cm$^{2}$ g$^{-1}$ & $0.008^{+0.01}_{-0.01}$ & $0.080^{+0.15}_{-0.06}$ \\\\\n $M_{\\rm NS}^{(j)}$ & $M_{\\odot}$ & $1.84^{+0.28}_{-0.23}$ & $1.57^{+0.25}_{-0.29}$ \n \\end{tabular}\n \\label{tab:event_physical_params} \n \\end{center}\n Notes. The quantities $P_{0}$, $B$, $M_{\\rm ej}$, $\\kappa$, $E_{\\rm SN}$, $v_{\\rm ej}$, $\\kappa_{\\gamma}$ and $M_{\\rm NS}$ were obtained from a best-fit to the UVOIR supernova light curves, with errors found in \\cite{nicholl2017c, nicholl2017d}. \n $^{(a)}$Epoch of explosion; $^{(b)}$Epoch of optical flux peak; $^{(c)}$Initial spin-period; $^{(d)}$Magnetic field strength of magnetar; $^{(e)}$Total mass, $^{(f)}$Effective opacity; $^{(g)}$Kinetic energy; $^{(h)}$Mean velocity of supernova ejecta; $^{(i)}$Gamma-ray effective opacity and $^{(j)}$Neutron star mass.\n\\end{table}\n\n\\subsection{{\\it Fermi}-LAT} \\label{subsec:Fermi}\nThe Large Area Telescope (LAT) on board the {\\it Fermi} satellite has operated since 2008 \\citep{Atwood2009}. It is sensitive to photons between \\SI{{\\sim}20}{\\MeV} and \\SI{{\\sim}300}{\\GeV} and has an ${\\sim}60\\degree$ field of view, enabling it to survey the entire sky in about 3 hours. \n\nThe data were analyzed using the publicly available {\\it Fermi}-LAT data with the \\texttt{Fermitools} suite of tools provided by the {\\it Fermi} Science Support Center (FSSC). Using the \\texttt{Fermipy} analysis package \\citep{wood2017}\n\\footnote{\\url{https:\/\/fermipy.readthedocs.io\/en\/latest\/} ; v0.19.0}, the data were prepared for a binned likelihood analysis in which a spatial spectral model is fit over the energy bins. The data were selected using the SOURCE class of events, which are optimized for point-source analysis, within a region of $15\\degree$ radius from the analysis target position. Due to the effect of the Earth, a $90\\degree$ zenith angle cut was applied to remove any external background events. The standard background models were applied to the test model, incorporating an isotropic background and a galactic diffuse emission model without any modifications. The standard 4FGL catalog was then queried for sources within the field of view and their default model parameters \\cite{abdollahi2020}. \n \nAdditional putative point sources were added to each field of view as needed to support convergence of the fit. These sources were added for all analysis time scales. This process continued until the distribution of test statistics for the field of view was Gaussian with standard deviation near 1 and mean centered at 0, and the residual maps were near uniformly 0 without strong features. These conditions indicate the appropriate coverage of spectral sources within the analysis was reached and no putative sources are missing. The fitting process is performed in discrete energy bins while optimizing the spectral shape, but the distribution of test statistics is evaluated with the stacked data spanning the full energy range. With the improvements to {\\it Fermi}-LAT low-energy sensitivity in PASS8 reconstruction, the low-energy bin covering $100-612$ MeV was also added. \n \nIn the case of both SN2015bn and SN2017egm, the data were fitted with a power-law spectral model, $N(E) = N_{0} E^{\\Gamma}$, with a free prefactor and a fixed photon index $\\Gamma$ of -2.0. From the fit, the reported flux upper limit was found using a 95\\% confidence level with the bounded Rolke method \\citep{rolke2005}. In all cases reported here, the upper limit reported is the integral energy flux, integrated over the energy ranges described for each case, which has units of \\si{MeV cm^{-2}.s^{-1}}. This flux is converted to luminosity with the adopted distance for each event.\n \n\nSN2015bn was observed from 2014 December 23 to 2018 Mar 23. This observation period begins after the explosion, and is binned in a few windows to account for the absorption of low-energy gamma rays by the ejecta at early times (Figure~\\ref{fig:tau}). The first $\\sim90$ days is observed to make sure there are no early emission during the expected absorption period. The data were thereafter binned in time intervals of six months to maximize observation depth and sensitivity to time-dependent variation. SN2017egm was observed 2017 May 23 to 2020 Aug 21. Again, this period covers the 3.5 yr from the discovery date, starting with $\\sim90$ days after the explosion and split into six 6 month bins thereafter. The 3.5 yr observation period is selected to cover approximately 1000 days after the explosion. After this period, it is expected that the predicted luminosity will have decreased below the {\\it Fermi}-LAT detectable limit. \n\nSN2015bn is within 5\\textdegree\\ of the Sun each year in August, so a one-month time cut is applied to each relevant time bin (to cover an $\\sim15$\\textdegree\\ radius field of view). SN2017egm is not near the the path of the Sun, so this cut was not applied.\n\n\\subsection{VERITAS} \\label{subsec:veritas}\nThe Very Energetic Radiation Imaging Telescope Array System (VERITAS) is an imaging atmospheric cherenkov telescope (IACT) array at the Fred Lawrence Whipple Observatory (FLWO) in southern Arizona, USA \\citep{weekes2002, Holder2006}. It consists of four 12 m telescopes separated by approximately $100$ m, and the observatory is sensitive to photons within the energy range $\\backsim$\\SI{100}{GeV} to $\\backsim$\\SI{30}{TeV}. The instrument has an angular resolution (68\\% containment) of $\\backsim$0.1\\degr\\ at \\SI{1}{TeV}, an energy resolution of $\\backsim$15\\% at \\SI{1}{TeV}, and 3.5\\degr\\ field of view. \n\n\nVERITAS serendipitously observed SN2015bn for a total of 1.01 hr between 2015 May 7 and 2015 May 22, approximately 135 days from explosion (49 days from the date of peak magnitude), as a part of an unrelated campaign. Another 1.7 hr were taken between 2016 May 25 and 30. Data were taken in good weather and dark sky conditions. Since SN2015bn was not the target source, its sky position averages 1.4\\textdegree~from the center of the camera. \n\nVERITAS directly observed SN2017egm for 8.7 hr between 2019 Mar 24 and 2019 Apr 5, under dark sky conditions, as part of a Directors Discretionary Time (DDT) campaign, approximately 670 days from explosion. This target was triggered based on the predicted gamma-ray luminosity (see section \\ref{sec:MagnetarSpinDown} and appendix \\ref{sec:appendix} for a description) derived from the optical observation. Although it was almost two years after the explosion, the nearby distance yielded a gamma-ray luminosity prediction still within reach of VERITAS, making this an enticing target to follow up.\n\nThe SN2017egm data in this paper were taken using ``wobble\" pointing mode, where the source is offset from the center of the camera by $0.5\\degree$. This mode creates space for a radially symmetric off region to be used for background estimation in the same field of view, saving time from targeted background observations that contain the same data observing conditions. The data were processed with standard VERITAS calibration and reconstruction pipelines, and then cross-checked with a separate analysis chain \\citep{cogan2008,Maier2017}. \n \nUsing an Image Template Method (ITM) to improve event angular and energy reconstruction \\citep{christiansen2017}, analysis cuts are determined with a set of a priori data-selection cuts optimized on sources with a moderate power-law index (from -2.5 to -3).\n\nUnfortunately, the large offset on SN2015bn due to the serendipitous observation precludes us from using ITM in the analysis, so in that case SN2015bn is analyzed without templates by calculating image moments directly from candidate images triggered by the camera \\citep{cogan2008,Maier2017}. In both cases, the signal and background counts are determined using the reflected region method. \n\n\nThe upper limit is calculated for both SN2015bn and SN2017egm. The bounded Rolke method for upper limit calculation is used, assuming a power law spectrum with index of -2.0 and 95\\% confidence level \\citep{rolke2005}. Since the calculation of the upper limit depends on the underlying spectral model, a range of power-law spectral indices from -2 to -3 was computed to estimate impact of the model dependence. In all cases reported here, the upper limit reported is the integral photon flux, integrated over the energy ranges described for each case, which has units of \\si{cm^{-2}.s^{-1}}. This flux is converted to integral energy flux using the same spectral model so that the luminosity can be computed with the adopted distance.\n\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics{SN2015bn_ebl_lc.pdf}\n \\caption{Light curves of SN2015bn spanning 30-1500 days after explosion. Curves shown include (1) the (thermal) supernova luminosity, $L_{\\rm opt}$, fit to UVOIR bolometric luminosity data (in red; \\citealt{nicholl2018a}) to obtain the magnetar parameters; (2) magnetar spin-down luminosity, $L_{\\rm mag}$ (green dotted lined); and (3) predicted gamma-ray luminosity that escape the ejecta, $L_{\\gamma}$ (pink dotted-dashed line; Equations \\ref{eq:Lmag}, \\ref{eq:trapped} and \\ref{eq:leaking}). Black bars show {\\it Fermi}-LAT upper limits reported for six 180 day bins starting ${\\sim}90$ days after explosion. The olive open box shows the VERITAS integral energy flux} upper limit taken ${\\sim}135$ days after the explosion, with EBL absorption correction applied. Upper limits on the 0.2-10 keV X-ray luminosity from {\\it Chandra} are from \\citet{bhirombhakdi2018} in green. Gray shaded regions labeled ``$\\tau_{\\gamma} <1$\" show the approximate time after which gamma rays of the indicated energy should escape ejecta, based on Figure \\ref{fig:tau}. A purple dotted-dashed line shows the engine luminosity, $L_{\\mathrm{BH}}$ (Eq.~\\ref{eq:L_BH}), in an alternative model in which the supernova optical luminosity is powered by fallback accretion onto a black hole. All upper limits denote the 95\\% confidence level.\n \\label{fig:SN2015bn_lc}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics{SN2017egm_lc_ebl_corrected.pdf}\n \\caption{The SN2017egm light curve spanning 10-1300 days after explosion, following the same format as Figure~\\ref{fig:SN2015bn_lc}. UVOIR data are shown in red \\citep{bose2017, nicholl2017c}. Integral energy flux upper limits from {\\it Fermi}-LAT are reported for six 180 day bins starting ${\\sim}90$ days after the explosion. Integral energy flux upper limits are shown for VERITAS data taken ${\\sim}670$ days after explosion, with EBL absorption correction applied. The maximum luminosity of the black hole accretion model $L_{\\mathrm{BH}}$ (Eq. \\ref{eq:L_BH}) is shown in purple. All upper limits denote the 95\\% confidence level.}\n \\label{fig:SN2017egm_lc}\n\\end{figure*}\n\n\\begin{table*}[ht]\n \\begin{center}\n \\caption{Results from VERITAS observations for both epochs of SN2015bn, and SN2017egm. }\n \\label{tab:veritas_total_results}\n \\begin{tabular}{cl|rrr}\n \\multicolumn{1}{l}{Parameter} & {(}Unit{)} & SN2015bn$_{1}$ & SN2015bn$_{2}$ & SN2017egm \\\\ \n \\hline\n \\hline\n Start (MJD) & [day] & 57149 & 57533 & 58566 \\\\\n End (MJD) & [day] & 57164 & 57538 & 58578 \\\\\n Live time & [hr] & 1.0 & 1.8 & 8.7 \\\\\n On & [event] & 4 & 10 & 49\\\\\n Off & [event] & 179 & 188 & 596 \\\\\n $\\alpha^{(a)}$ & - & 0.0286 & 0.0299 & 0.0634 \\\\\n Excess & [event] & -1.1 & 4.4 & 11.2 \\\\\n Significance & [$\\sigma$] & -0.5 & 1.7 & 1.6 \\\\\n Flux UL & [$\\SI{e-13}{cm^{-2}~s^{-1}}$] & 28.5 & 27.8 & 10.2 \\\\\n $E_{\\rm threshold}$ & [GeV] & $>320$ & $>420$ & $>350$ \\\\ \n \\end{tabular}\n \\end{center}\n Notes. Shown are the quality selected livetime, number of gamma-ray-like events in the on and off-source regions, the normalization, the observed excess of the gamma-rays and the statistical significance. The integral flux upper limit is shown for the given energy threshold, without EBL absorption correction, integrated up to \\SI{30}{\\TeV}.\n $^{(a)}$ Ratio of relative exposure for on and off regions.\n\\end{table*}\n\n\\section{Results}\n\\label{sec:results}\n\nNo statistically significant detections were made of either SN2015bn or SN2017egm across the energy range 100 MeV to 30 TeV. Integral energy upper limits are reported for the energy ranges given for each instrument. Figure~\\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc} show the {\\it Fermi}-LAT and VERITAS upper limits in comparison to the supernova optical light curves and the theoretically predicted escaping luminosity from the magnetar model. \n\n\\subsection{Optical}\\label{sec:optical_result}\nThe SN2015bn integrated ultraviolet-optical-infrared (UVOIR) light-curve data are reproduced here from previous analyses \\citep{nicholl2016, nicholl2016a, nicholl2018a}. To produce these bolometric light curves, the multiband optical data were interpolated and integrated at each epoch using the code \\texttt{superbol} \\citep{nicholl2018}. \n\nSimilarly, the SN2017egm UVOIR data are also reproduced here with \\texttt{superbol} \\citep{bose2017, nicholl2017c}.\n\n\\subsection{{\\it Fermi}-LAT}\nBoth SN2015bn and SN2017egm are not statistically significant sources in the first $\\sim90$ days or the subsequent 6 month bin starting 90 days after the explosion. These sources also remain undetected in any of the following 6 month bins, and in the multiyear data sets. \n\nThe evaluation of the integral energy flux upper limit for the {\\it Fermi}-LAT observations within each time bin was performed assuming a power-law spectral model with an index of -2. The model dependence of this calculation naturally impacts the interpretations in section \\ref{sec:discussion}, so the the fit was performed with indices 2, 2.5 and 3 to find the impact of the model on the final upper limit. An uncertainty of about $10\\%$ was found based on varying the index.\n\nSN2015bn is found to have test statistic (TS) of 0.06,\nwith 12 predicted events above the isotropic diffuse background $\\simeq \\num{4.8e4} $ events over the entire period. The flux upper limit is $\\SI{1.6e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}. In the first $\\sim90$ days after the explosion, where the gamma ray emission is not expected due to the high gamma-ray absorption (see Figure \\ref{fig:tau}), the flux upper limit is $\\SI{3.5e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}, with a TS of 0. For the first 6-month period, when the signal is most likely, the flux upper limit is $\\SI{1.9e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ for $TS \\simeq 0$, consistent with a nondetection. All of the following 6 month bins reported nondetections with $TS<2$. \n\nSN2017egm is found to have $TS=4.4$,\nwith 43 predicted events above the isotropic diffuse background $\\simeq \\num{5.9e4} $ events. The flux upper limit is $\\SI{1.2e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}. In the first $\\sim90$ days after the explosion, where the gamma ray emission is not expected due to the high gamma-ray absorption (see Figure \\ref{fig:tau}), the flux upper limit is $\\SI{3.2e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ over the energy range \\SI{100}{\\MeV} to \\SI{500}{\\GeV}, with a TS of 0. For the first 6-month period, when the signal is most likely, the flux upper limit is $\\SI{4.9e-6}{\\MeV.\\cm^{-2}.\\s^{-1}}$ for $TS=\\num{10.1}$, consistent with a non-detection. All of the following 6-month bins reported non-detections with $TS<1$.\n\n\\subsection{VERITAS}\nTable \\ref{tab:veritas_total_results} reports the results from VERITAS observations of SN2015bn and SN2017egm. Each observation is consistent with a nondetection. The significance of each excess of observed events above background is below 2 standard deviations ($\\sigma$). The flux upper limits are also given, calculated by integrating above the threshold energy of the instrument.\n\nThe statistical significance of an excess is estimated using Equation 17 of Li \\& Ma \\citep{li1983}. SN2015bn has significance value of $-0.5 \\sigma$ in the first epoch observation. The integral flux upper limit from \\SIrange{0.32}{30}{\\TeV} for SN2015bn is \\SI{2.85e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{1.27e44}{\\erg.s^{-1}} at a redshift of 0.1136. Due to the serendipitous nature of the observation, SN2015bn is significantly off axis, which lowers the instrument sensitivity at the energy threshold of \\SI{320}{\\GeV}. Additionally, a 10\\% systematic uncertainty is added to the flux normalization and reported energy threshold due to instrument degradation during the period of 2012-2015 \\cite{nievasrosillo2021}. This uncertainty is derived empirically from the observation of the Crab Nebula over the same period. During the second observation in 2016, SN2015bn was found to have a significance of 1.7. The integral flux upper limit from \\SIrange{0.42}{30}{\\TeV} for SN2015bn is \\SI{2.78e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{1.60e44}{\\erg.s^{-1}}. \n\nFor SN2017egm, the Li \\& Ma significance value is $0.2 \\sigma$ and an integral upper limit from \\SIrange{0.35}{30}{\\TeV} is \\SI{1.0238e-12}{cm^{-2}.s^{-1}}, which corresponds to an upper limit on the luminosity of \\SI{3.54e42}{\\erg. s^{-1}} above the energy threshold of \\SI{350}{\\GeV} at redshift z=0.0310. The systematic correction due to instrument degradation during the period of 2012-2019 is applied automatically with the use of the throughput-calibrated analysis templates \\citep{nievasrosillo2021}. In the cases of both SN2015bn and SN2017egm, the impact of varying the power-law model index parameter from -2 to -5 is about 10\\%, which is a negligible in the context of their respective light curves.\n\nVHE photons are absorbed by the extragalactic background light (EBL) throughout the universe, so the flux must be corrected to account for the missing photons. This absorption is energy and redshift dependent. Deabsorption is applied to the flux using the model of \\citet{Dominguez2011}. \nThe EBL deabsorption factor was convolved with the upper limit calculation, assuming the same spectral shape (a power law with the photon index of -2.0). \nThe deabsorbed integral photon upper limit for SN2015bn within the energy range \\SIrange{0.32}{30}{\\TeV}, is \\SI{3.36e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{1.49e44}{\\erg.s^{-1}}. For the second observation, the deabsorbed integral photon upper limit for SN2015bn within the energy range \\SIrange{0.42}{30}{\\TeV}, is \\SI{3.30e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{1.91e44}{\\erg.s^{-1}}. For SN2017egm, with a slightly smaller energy range \\SIrange{0.350}{30}{\\TeV}, the deabsorbed integral photon flux is \\SI{1.07e-12}{cm^{-2}.s^{-1}}, which corresponds to a luminosity upper limit of \\SI{3.70e42}{\\erg.s^{-1}}. These EBL-corrected values are plotted in Figure \\ref{fig:SN2015bn_lc} and Figure \\ref{fig:SN2017egm_lc}.\n\n\\section{Discussion}\n\\label{sec:discussion}\nThe source of the extra luminosity powering SLSNe-I may be found in the signature of its late-time gamma-ray emission. This section explores the HE to VHE emission hundreds of days after the explosion. The following models with a gamma-ray emission component for the powering mechanism are discussed: (1) magnetar central engine (see section \\ref{sec:MagnetarSpinDown}), (2) black hole central engine (see section \\ref{sec:BlackHole}), and (3) circumstellar interaction (see section \\ref{sec:Circumstellar}).\n\n\\subsection{Magnetar Central Engine} \\label{sec:MagnetarSpinDown}\n\nThe most promising mechanism for powering SLSNe-I is the rotational energy input from a central magnetar. In this scenario, a young pulsar or magnetar inflates a nebula of relativistic particles, which radiate high-energy gamma rays and X-rays. This section initially explores a simple implementation of the magnetar model (see Appendix \\ref{sec:appendix} for full description), followed by a more complete model described in detail in \\citet{vurm2021} for both SN2015bn and SN2017egm. The application of this so-called self-consistent model is necessary to directly predict the energy-dependent luminosities within the energy ranges of the {\\it Fermi}-LAT and VERITAS observations, a major contribution that is not possible with simpler implementation described in the appendix.\n\nAt early times after the explosion (around and immediately after the maximum in the optical emission) the gamma rays are absorbed and thermalized by the expanding supernova ejecta. At these times, the luminosity and shape of the optical light curve can be used to constrain the parameters of the magnetar. In this model, the radiation of an input energy reservoir (the spin-down luminosity of a rotating magnetar) diffuses through the ejecta following the analytical solution by \\citet{arnett1982} (equation \\ref{eq:trapped}). \n\nThe time evolution of the magnetar's spin-down luminosity can be modeled by assuming a rotating dipole magnetic field whose energy loss is dominated by emission of radiation in the gamma-ray and X-ray bands (see Appendix \\ref{sec:appendix} for details). \n\nThis luminosity depends on the magnetar initial spin period, surface dipole magnetic field strength, and neutron star mass, $L_{\\rm mag}(t, P_0, B, M_{\\rm NS})$ (equation \\ref{eq:Lmag}). The emitted radiation thermalizes as it diffuses through the ejecta. The conditions of the ejecta determine the optical and gamma-ray outputs, dominated by the values of the ejecta mass, ejecta velocity, and optical and gamma-ray opacities to form $L_{\\rm opt}(t, M_{\\rm ej}, v_{\\rm ej}, \\kappa,\\kappa_{\\gamma})$ (equation \\ref{eq:leaking}) and $L_{\\gamma}(t, M_{\\rm ej}, v_{\\rm ej}, \\kappa, \\kappa_{\\gamma})$ (equation \\ref{eq:escape}). \n\nFor SN2015bn and SN2017egm, the parameters for the magnetar and the supernova ejecta properties were found by fitting their integrated ultraviolet-optical-infrared (UVOIR) light curves, shown with red points in Figures~\\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc}. All fits were conducted using nonlinear least squares minimization\\footnote{\\texttt{scipy.optimize.curve\\_fit}}.\nThe best-fit parameters with errors for the magnetar model are given in Table \\ref{tab:event_physical_params}. The redshifts and time of peak optical magnitude are shown in the table as listed in The Open Supernova Catalog \\citep{Guillochon2016}\\footnote{\\url{https:\/\/sne.space}}.\n\nThese parameters are consistent with the results of previous fits \\citep{nicholl2018a,nicholl2017c} that took into account both the optical spectral energy distribution and light curve using the open-source code \\texttt{MOSFiT} \\footnote{\\url{https:\/\/mosfit.readthedocs.io\/en\/latest\/}}. \nThe relative statistical errors on these fit parameters may be optimistic at $\\sim10\\%$, and the systematic errors will still need to be incorporated for a better understanding the magnetar parameter space. The largest contributor to the magnetar power are the period and magnetic field values, which determine the overall magnitude of the luminosity. The ejecta mass and velocity determine the time to optical peak by the diffusion of the emission through the ejecta.\n\nA particularly important shortfall of this model is the constant effective opacity to both optical and gamma-ray photons, rather than a time-dependent treatment of the opacity. TeV gamma rays interact preferentially with optical photons, so at the time of the peak optical emission, $\\gamma\\gamma$ absorption by optical photons will be high, reducing any predicted gamma-ray emission by this model. Equation \\ref{eq:leaking} is a bolometric luminosity, so it does not take into account the energy and time-dependent opacity, instead fitting a constant effective $\\kappa$ and $\\kappa_{\\gamma}$ to generate the time-dependent optical depth. \n\nTherefore, Figure~\\ref{fig:tau} is used as a guide for when to expect $L_{\\gamma}$ to provide an appropriate estimate for the gamma-ray emission. The shaded regions in Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc} estimate the time periods when photons of the given energies can escape. It is important to reiterate that this model is energy independent, representing the bolometric luminosity not thermalized by the ejecta. This model cannot distinguish the emission between LAT and VERITAS energy bands since it does not consider the physical model of the nebula; the self-consistent model described by \\cite{vurm2021} and discussed below will be an attempt to do so explicitly.\n\nFollowing the methodology in Appendix \\ref{sec:appendix} with the magnetar parameters for each SLSN, $L_{\\rm mag}(t)$, $L_{\\rm opt}(t)$, and $L_{\\gamma}(t)$ were calculated and are shown in comparison to the gamma-ray limits in Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc}.\n\nFor SN2015bn (Figure \\ref{fig:SN2015bn_lc}), neither the {\\it Fermi}-LAT upper limits nor the VERITAS upper limit constrain the predicted escaping luminosity. \nSimilarly, for SN2017egm (Figure \\ref{fig:SN2017egm_lc}), both the VERITAS and {\\it Fermi}-LAT upper limits are not deep enough to constrain the predicted escaping luminosity. An important caveat to these upper limits is that the escaping luminosity may also be emitted at energies not explored here, such as hard X-rays or gamma-rays greater than \\SI{ 30}{\\TeV}.\n\nThe optimal time to observe with a pointed instrument sensitive at a particular photon energy results from a trade-off between the dropping ($\\propto t^{-2}$) magnetar luminosity and the rising transparency of the ejecta; predicting the optimal time post-peak to observe requires knowledge of the evolution of the optical spectrum. It is possible to accumulate enough optical data within a few weeks after the optical peak to fit the magnetar model for a reliable prediction of the gamma-ray luminosity. In the case of SN2017egm, the gamma-ray luminosity prediction was anchored by the late optical data points about 1 yr after the explosion. This means that had the VERITAS observations been taken at that point (more than a year earlier than the original observation), they would have been deeply constraining to the magnetar model.\n\n\nGoing beyond these relatively model-independent statements to compare to a more specific spectral energy distribution for the escaping magnetar nebula requires a detailed model for the nebula emission and its transport through the expanding supernova ejecta. Such a model offers preliminary support that a significant fraction of $L_{\\gamma}$ may come out in the VHE band \\citep{vurm2021}. In this case, the VHE limits on SN2015bn and SN2017egm do not strongly constrain the parameters of the magnetar model, such as the nebular magnetization.\n\n\nThe model of \\citet{vurm2021} self-consistently follows the evolution of high-energy electron\/positron pairs injected into the nebula by the magnetar wind and their interaction with the broadband radiation and magnetic fields. They found that the thermalization efficiency and the amount of gamma-ray leakage depends strongly on the nebular magnetization, $\\varepsilon_B$, i.e. the fraction of residual magnetic energy in the nebula relative to that injected by the magnetar.\n\nThe model is simulated for dimensionless $\\varepsilon_B$ values set between $10^{-6}$ and $10^{-2}$; the higher magnetizations lead to greater synchrotron efficiencies, which dominate within a few hundred days, and lead to the optical emission tracking the spin-down luminosity. Lowering the magnetization to $10^{-7}-10^{-6}$ for SLSN-I events like those in this work delays the transition to synchrotron-dominated thermalization, so that the predicted optical emission actually tracks the observed data. \n\nThe theoretical light curves and gamma-ray upper limits are shown in Figure ~\\ref{fig:Indrek}. \\citet{vurm2021} concluded that the predicted low magnetizations constrained by the optical data alone presents new challenges to the theoretical framework regarding the dissipation of the nebular magnetic field. This may invoke magnetic reconnection ahead of the wind-termination shock or near the termination shock through forced reconnection of alternating field stripes described in \\citet{komissarov2013}, \\citet{lyubarsky2003}, \\citet{margalit2018b}. It is also possible that the true luminosity of the central engine decreases faster in time than the simpler $\\propto t^{-2}$ magnetic spin down, such that escaping VHE emission is not necessary to explain the model. These VHE upper limits do not rule out this model, and do not settle the challenges inferred by the low magnetization required to fit the optical data. Further observations are needed to probe the nebular magnetization and synchrotron efficiency, and deep VHE observations will contribute to these constraints.\n\nThe nondetection of X-rays for both events is consistent with the predictions of \\cite{margalit2018a} of a fully ionized ejecta. Even under the most optimistic conditions - an engine that puts 100\\% of its spin-down luminosity into ionizing photons of ideal energies - cannot reduce the opacity enough to allow X-rays to escape under the usual assumptions (e.g. spherically symmetric ejecta shell). \n\n\\begin{figure}[ht]\n \\includegraphics[width=1.1\\columnwidth]{SN2015bn_lc_ebl_model.pdf}\n \\includegraphics[width=1.1\\columnwidth]{SN2017egm_lc_ebl_model.pdf}\n \\caption{Model light curve for nebular magnetization (from \\cite{vurm2021}) for SN2015bn with $\\varepsilon_B = 10^{-7}$ (top panel) and SN2017egm with $\\varepsilon_B=10^{-6}$ (bottom panel). }\n \\label{fig:Indrek}\n\\end{figure}\n\n\\subsection{Black Hole Central Engine} \\label{sec:BlackHole}\n\nInstead of forming a neutron star like a magnetar, a SLSN-I might form a black hole, in which case the optical peak of the light curve could be powered by energy released from the fallback accretion of ejecta from the explosion (e.g.~\\citealt{dexter2013}). Even if a black hole does not form immediately, it could form at late times once the magnetar accretes enough fallback material \\citep{moriya2016a}. The main practical difference as compared to a magnetar in section \\ref{sec:MagnetarSpinDown} is that the black hole central-engine power would be predicted to decay with the fallback accretion rate $\\dot{M}_{\\rm fb} \\propto t^{-5\/3}$ instead of $\\propto t^{-2}$. Thus, in principle, for the same luminosity at the time of the optical maximum $t_{\\rm pk}$, the central-engine output at times $t \\gg t_{\\rm pk}$ could be enhanced by a factor $\\propto (t\/t_{\\rm pk})^{1\/3} \\sim 2$ for $t \\sim 1$ yr and $t_{\\rm pk} \\sim 1$ month, thus tightening our constraints.\n\nIn Figures \\ref{fig:SN2015bn_lc} and \\ref{fig:SN2017egm_lc}, a rough estimate of the maximal engine luminosity in the BH accretion scenario is shown, which is calculated as \n\n\\begin{align}\nL_{\\mathrm{BH}}=\\frac{2^{5\/3} L^{\\rm pk}_{\\rm opt}}{\\left(1+\\frac{t}{t_{\\rm pk}}\\right)^{5\/3}} \\label{eq:L_BH},\n\\end{align}\n\nwhere $L^{\\rm pk}_{\\rm opt}$ is the peak optical luminosity, scaled so that $L_{\\rm BH} = L_{\\rm opt}$ around the optical peak.\n\nOn the other hand, while gamma rays are naturally expected from the ultra-relativistic spin-down-powered nebula of a magnetar, it is less clear that this would be the case for a black hole engine. For instance, the majority of the power from a black hole engine could emerge in a mildly relativistic wind from the black hole accretion disk instead of an ultra-relativistic spin-down-powered pulsar wind.\n\nAs seen in both Figure \\ref{fig:SN2015bn_lc} (SN2015bn) and Figure \\ref{fig:SN2017egm_lc} (SN2017egm), the gamma-ray emission in the black hole scenario is not constrained in the {\\it Fermi}-LAT and VERITAS energy bands.\n\n\\subsection{Circumstellar Interaction} \\label{sec:Circumstellar}\n\nAn alternative model for powering the light curve of SLSNe is to invoke the collision of the supernova ejecta with a slower expanding circumstellar shell or disk surrounding the progenitor at the time of the explosion (e.g.~\\citealt{smith2006,chevalier2011,Moriya2013a}). Features of this circumstellar model (CSM), such as the narrow hydrogen emission lines that indicate the interaction of a slow-moving gas, provide compelling evidence for this being a powering mechanism for many but not all of the hydrogen-rich class of SLSNe (SLSNe-II; e.g.~\\citealt{smith2007,nicholl2020}). \n\nShock interaction could in principle also power some hydrogen-poor SLSNe (SLSNe-I), particularly in cases where the circumstellar interaction is more deeply embedded and less directly visible (e.g.~\\citealt{sorokina2016,kozyreva2017}). \nThere is growing evidence for hydrogen-poor supernovae showing hydrogen features from the interaction in their late-time spectra \\citep{Milisavljevic2015,Yan2015,Yan2017,Chen2018,Kuncarayakti2018,Mauerhan2018}. The light echo from iPTF16eh \\citep{lunnan2018} implies a significant amount of hydrogen-poor circumstellar medium in a SLSN-I at ${\\sim}10^{17}$ cm. However, this material is too distant for the ejecta to reach by the time of maximum optical light and hence cannot be responsible for boosting the peak luminosity.\n\nIn principle, the gamma-ray observations of SLSNe can constrain shock models. In many cases, this may not work out since most of the emission from shock-heated plasma is expected to either (1) come out in the X-ray band, as is well studied in other CSM-powered supernovae such as SNe IIn like SN 1998S \\citep{Pooley2002}, SN 2006jd \\citep{Chandra2012}, SN 2010jl \\citep{Chandra2015}, and SNe Ib\/c \\citep{Chevalier2006} or (2) be absorbed by the surrounding ejecta and reprocessed into the optical band. Thus, these VHE limits on SLSNe do not constrain the bulk of the shock power. \n\nHigher-energy radiation can be produced if the shocks accelerate a population of nonthermal relativistic particles that interact with ambient ions or the supernova optical emission to generate gamma rays (e.g. via the decay of $\\pi^{0}$ generated via hadronic interactions with matter and radiation; e.g., \\citealt{murase2011}). However, because shocks typically place a fraction $\\epsilon_{\\rm rel} \\lesssim 0.1$ of their total power into relativistic particles (or even less; \\citealt{steinberg2018,fang2019}), the predicted gamma-ray luminosities (matching the same level of optical emission as magnetar models) would be at least 10 times lower than $L_{\\gamma}$ predicted by the magnetar nebula scenario, thus rendering our VHE upper limits unconstraining on non-thermal emission from shocks on SN2015bn and SN2017egm. This is consistent with upper limits from the Type IIn SN 2010j from {\\it Fermi}-LAT, which \\citet{Murase2019} used to constrain $\\epsilon_{\\rm rel} \\lesssim 0.05-0.1$. \n\n\\section{Future Prospects}\n\\label{sec:future}\n\nThese results demonstrate that high-energy gamma-ray observations of SLSN-I are on the brink of enabling constraints on the light curves and even spectral energy distribution of magnetar models. Given the rarity of bright, nearby SLSN-I, and the need to take observations in the optimal window (when $L_{\\gamma}$ is near maximum), careful planning will be required to make progress going ahead \\citep{Quimby2011,mccrum2015,prajs2017}. The strategy outlined below will focus only on SLSN-I, as type II SLSN are likely to be powered by a mechanism that requires a different consideration of the temporal and spectral evolution of the gamma-ray emission.\n\nStandard arrays of IACTs provide an improved instantaneous sensitivity to gamma-ray emission over {\\it Fermi}-LAT due to $10^4$ to $10^5$ larger effective area, counterbalanced in part by the pointed nature of their observations. To propose a strategy, we firstly revisited the characteristics of a large sample of observed SLSNe and performed a systematic study.\n\n\\citet{nicholl2017d} fit a sample of 38 SLSNe light curves using MOSFiT to obtain a distribution of magnetar model parameters. This sample is a selection of SLSNe with well-observed events classified as Type I with published data near the optical peak, forming a representative sample of good SLSNe-I for a population study. For each event in this sample, the following was calculated: the escaping gamma-ray luminosity $L_{\\gamma}$ following the procedure outlined in Appendix \\ref{sec:appendix} and the flux $F_{\\gamma} = L_{\\gamma}\/4\\pi D_{\\rm L}^{2}$ based on the source luminosity distance $D_{L}$. In performing this analysis, rather than fitting the value of $\\kappa_{\\gamma}$ individually to each optical light curve (as done in \\citealt{nicholl2017d}), the value $\\kappa_{\\gamma} = \\SI{0.01}{cm^{2} g^{-1}}$ is fixed in all events, based on the best-fit to SN2015bn (given its particularly high-quality late-time data, which provides the most leverage on $\\kappa_{\\gamma}$). \n\nThe results for $F_{\\gamma}(t)$ are shown in the top panel of Figure~\\ref{fig:distributions}. In the magnetar model, the predicted gamma-ray flux could emerge anywhere across the HE to VHE bands; hence, it represents an upper limit on flux in the bands accessible to {\\it Fermi}-LAT and IACTs. The bottom two panels of Figure~\\ref{fig:distributions} show the distribution of the peak escaping flux $F_{\\rm \\gamma, max}$ and time of the peak flux relative to the explosion. For most SLSNe-I presented here, $F_{\\rm \\gamma, max}$ is well below the sensitivity of VERITAS and even the future Cherenkov Telescope Array (CTA) \\citep{thectaconsortium2019}. Also note that the characteristic timescale to achieve the peak gamma-ray flux is $\\approx 2-3$ months from the explosion. This timescale occurs approximately at the same time as when the optical depth of the ejecta to VHE emission falls below unity, when the VHE photons can escape (Figure~\\ref{fig:tau}). \n\n\\begin{figure}[ht!]\n \\centering\n \\includegraphics[width=\\columnwidth]{Combined_LCs_flux_dist.pdf}\n \\caption{Top: escaping gamma-ray luminosity $L_{\\gamma}(t)$ for the sample of SLSNe fit by \\citet{nicholl2017d}. Five well-studied SN are highlighted in blue, including SN2015bn. Overplotted are the VERITAS and CTA sensitivity curves for various exposures. Middle: distribution of peak escaping gamma-ray flux $F_{\\rm \\gamma,max} = {\\rm max}[L_{\\gamma}]\/4\\pi D^{2}$, for the light curves from the top panel where $D$ is the distance to each source. Again, VERITAS and CTA sensitivities for different exposures are shown as vertical dashed lines. Bottom: distributions of times since explosion to reach the maximum gamma-ray flux $F_{\\gamma,max}$ from $F_{\\gamma}$ above.} \n \\label{fig:distributions}\n\\end{figure}\n\nFigure~\\ref{fig:Optical_to_Escaping_Fluxes} shows $F_{\\rm \\gamma,600 d}$ as a function of the peak optical magnitude of the SLSNe-I from the same sample as in Figure~\\ref{fig:distributions}. The selection of fluxes at \\SI{600}{\\day} approximates the time when the effective opacity to 1 TeV photons reaches 1, based on Figure~\\ref{fig:tau}. The top axis also gives the all-sky rate of SLSNe-I above a given peak optical magnitude, which is estimated using the magnitude distribution of SLSNe-I and assuming they occur at a comoving volumetric rate of $ R(z)=19(1+z)^{3.28}\\SI{}{\\, Gpc^{-3}\\, yr^{-1}}$ following \\citet{nicholl2017b,lunnan2018,decia2018}. This estimation captures the general volumetric rate of events, but is unreliable for exceptionally bright events such as SN2017egm due to the small population for estimating the magnitude normalization. A bright event like SN2017egm may actually happen more often than once a century.\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics[width=.95\\textwidth]{Optical_to_Escaping_Fluxes_kap_1TeV.pdf}\n \\caption{\n Blue dots show the peak optical apparent magnitudes of a sample of SLSNe-I \\citep{nicholl2017d} as a function of their predicted maximum gamma-ray luminosity at 600 days after explosion ($F_{\\gamma, 600d}$). The top axis shows the approximate rate of events above the given peak optical magnitude, calculated using the method described in the main text. Peak maximum gamma-ray luminosities are calculated from fits of optical data with fixed $\\kappa_{\\gamma} = 0.01$ cm$^{2}$ g$^{-1}$. Integral sensitivities of various instruments are overplotted for different exposures. Solid lines: VERITAS 10 and 50 hr integral sensitivities above 220 GeV. Dotted lines: CTA (in development) 10 and 50 hr integral sensitivities above 125 GeV as estimated from 50 hr Monte Carlo simulations of the southern array \\citep{thectaconsortium2019} and extrapolated to 10 hours. Similar extrapolation is done for {\\it Fermi}-LAT from 10 yr to 6 months \\citep{nolan2012} (dashed line). Proposed project AMEGO integral sensitivity above 100 MeV for 6 month observation window is also plotted (dashed-dotted line) \\citep{kierans2020}. \n }\n \\label{fig:Optical_to_Escaping_Fluxes}\n\\end{figure*}\n\nShown for comparison in Figure~\\ref{fig:Optical_to_Escaping_Fluxes} are the integral sensitivities of various gamma-ray instruments for different exposures. For IACT instruments such as VERITAS and the future CTA, sensitivity is defined as the minimum flux necessary to reach $5 \\sigma$ detection of a point-like source, requiring at least 10 excess gamma rays and the number of signal counts at least $5\\%$ of the number of background counts. For VERITAS, the sensitivity was calculated using observed Crab Nebula data to estimate the rates of signal and background photons with cuts optimized for a $\\Gamma = -2.5$ power-law spectrum, and then rescaled for the appropriate observation time \\citep{Park2015}. For CTA, Monte Carlo simulations were used to derive angular resolution, background rates and energy dispersion features -- the instrument response functions (IRF) -- based on the Prod3b-v2 telescope configuration for the Southern site and its atmosphere \\citep{cherenkovtelescopearrayobservatory2016}. These IRFs are publicly available and were analyzed using the open-source CTOOLS\\footnote{\\url{http:\/\/cta.irap.omp.eu\/ctools\/}} \\citep{Knodlseder2016}. A power-law spectral model was used to estimate the integral sensitivity above \\SIlist{0.125;1}{\\TeV} each for observations of \\SIlist{10;50}{\\hour} (see \\citealt{Fioretti2016} for further discussion on CTA integral sensitivity).\n\nBased on this systematic study, we propose the following observation strategy: (1) Receive automated public alert and Type I classification of SLSN from a survey instrument such as the Zwicky Transient Facility (ZTF). Classification is generally determined by identification of early spectral components such as $O_{II}$ absorption features. (2) During the multiday rise and fall of bolometric optical light curve, fit the magnetar model ($L_{\\rm opt}$, yielding parameters for $L_{\\rm mag}$ and $L_{\\gamma}$) (3) Compare $L_{\\gamma}$ to the telescope sensitivity at the appropriate day when the effective $\\gamma$-$\\gamma$ opacity falls below ${\\sim1}$ for the telescope's sensitive energy range (see Figure~\\ref{fig:tau}). In the case of IACTs sensitive to energies above $\\SI{100}{\\GeV}$, the gamma rays will escape the magnetar a few hundred days after explosion, requiring a bright SLSN-I that will power gamma rays for as much as two years. \n\nEstimating ${\\sim}35\\%$ of all-sky visibility at VERITAS due to Sun, Moon, and seasonal weather cut, and above 60\\textdegree~ elevation, VERITAS is capable of detecting up to ${\\sim}0.4$ and ${\\sim}4$ SLSNe-I per year for \\SI{10}{h} and \\SI{50}{h} exposures, respectively. The next-generation CTA observatory will be able to detect as many as ${\\sim}8$ and ${\\sim}80$ events for \\SI{10}{hr} and \\SI{50}{h}, respectively, assuming a larger sky visibility fraction of ${\\sim}80\\%$ when both North and South arrays are included. On the other hand, SLSNe at greater distances also imply a stronger role of $\\gamma-\\gamma$ interactions on the EBL in suppressing the $\\gtrsim$ TeV emission, decreasing the observed integral flux by as much as 60 times at redshifts near 0.5 in the VERITAS energy range. \n\nFigure~\\ref{fig:TeV_gamma_flux_sens} shows the distribution of fluxes at \\SI{200}{\\day} and \\SI{600}{\\day} which are approximate average dates when the opacity to \\SI{100}{\\GeV} and \\SI{1}{\\TeV} photons falls below 1, respectively, and they are able to escape the ejecta. Accounting for this time delay for the opacity to drop, the expected rate of bright events drops by another 3 to 15 times. While past observations have not been followed up until this publication, the distribution of predicted gamma-ray fluxes hints that, particularly for \\SI{100}{\\GeV} photons, future SLSN-I will be observable with current and planned observatories. \n\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=\\columnwidth]{Escaping_Fluxes_kappa_gamma_dist.pdf}\n \\caption{ Distribution of gamma-ray luminosities $L_{\\gamma}$ at $t=\\SI{200}{\\day}$ (top) and $t=\\SI{600}{\\day}$ (bottom), when the optical depth for \\SI{100}{\\GeV} and \\SI{1}{\\TeV} photons drops below 1, calculated for a sample of 38 SLSNe \\citep{nicholl2017d}}\n \\label{fig:TeV_gamma_flux_sens}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{sec:conclusions}\nSLSN-I are potential gamma-ray emitters, and this paper provides the first upper limits at different times after the optical outburst for two good candidates. The reported upper limits approach the magnetar spin-down luminosity limit of SN2015bn and SN2017egm. While the expected gamma-ray luminosity in either the magnetar central-engine scenario or the shock-acceleration scenario is not constrained by these limits, a relativistic jet powered by fallback accretion onto a black hole is disfavored in both cases. We explore prospects for obtaining improved VHE gamma-ray constraints in the future by current and planned IACTs. We estimate the Type I SLSNe rate for VERITAS and CTA, considering observation constraints and the time delay due to the optical depth. For sufficiently nearby and bright SLSN-I, 0.4 and 4 events per year can be observed by VERITAS from 10 hr and 50 hr observation, respectively, and similarly rates of 8 and 80 events per year can be expected by CTA. \n\n\n\\acknowledgments \n{This research is supported by grants from the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, and by the Helmholtz Association in Germany. M.N. is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No.~948381) and by a Fellowship from the Alan Turing Institute. I.V. acknowledges support by the ETAg grant PRG1006 and by EU through the ERDF CoE grant TK133. V.V.D.'s work is supported by NSF grant 1911061 awarded to the University of Chicago (PI: Vikram Dwarkadas). We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument. \nThis research has made use of the CTA instrument response functions provided by the CTA Consortium and Observatory, see \\url{http:\/\/www.cta-observatory.org\/science\/cta-performance\/} (version prod3b-v2) for more details.\n}\n\n\\software{fermipy (v0.19), \\citep{wood2017},\nastropy \\citep{Robitaille2013,Price-Whelan2018},\nCTOOLs, \\citep{Knodlseder2016a}, superbol \\citep{nicholl2018} , EventDisplay \\citep{Maier2017}}, VEGAS \\citep{cogan2008}\n\n\\facilities{VERITAS, {\\it Fermi}-LAT}\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\nThe spread of misinformation is increasingly being recognized as an enormous problem. In recent times, misinformation has been reported to have grave consequences such as causing accidents \\cite{ma2016detecting}, while fake news around election times have reportedly reached millions of people \\cite{allcott2017social} causing concerns as to whether they might have influenced the electoral outcome. {\\it Post-Truth} was recognized as the Oxford Dictionary Word of the Year in 2016\\footnote{https:\/\/en.oxforddictionaries.com\/word-of-the-year\/word-of-the-year-2016}. These have spawned an extensive interest in the data analytics community in devising techniques to detect fake news in social and online media leveraging content, temporal and structural features (e.g., \\cite{kwon2013prominent}). A large majority of research efforts on misinformation detection has focused on the political domain within microblogging environments (e.g., \\cite{zhao2015enquiring,ma2016detecting,ma2017detect,qazvinian2011rumor,zubiaga2016learning,castillo2013predicting,zhang2017detecting}) where structural (e.g., the user network) and temporal propagation information (e.g., re-tweets in Twitter) are available in plenty. \n\nFake news and misinformation within the health domain have been increasingly recognized as a problem of immense significance. As a New York Times article suggests, {\\it `Fake news threatens our democracy. Fake medical news threatens our lives'} \\footnote{https:\/\/www.nytimes.com\/2018\/12\/16\/opinion\/statin-side-effects-cancer.html}. Fake health news is markedly different from fake news in politics or event-based contexts on at least two major counts; first, they originate in online websites with limited potential for dense and vivid digital footprints unlike social media channels, and secondly, the core point is conveyed through long and nuanced textual narratives. Perhaps in order to aid their spread, the core misinformation is often intertwined with trustworthy information. They may also be observed to make use of an abundance of anecdotes, conceivably to appeal to the readers' own experiences or self-conscious emotions (defined in \\cite{tracy2004putting}). This makes health misinformation detection a challenge more relevant to NLP than other fields of data analytics. \n\nWe target detection of health fake news within quasi-conventional online media sources which contain information in the form of articles, with content generation performed by a limited set of people responsible for it. We observe that the misinformation in these sources is typically of the kind where scientific claims or content from social media are exaggerated or distilled either knowingly or maliciously (to attract eyeballs). Some example headlines and excerpts from health fake news articles we crawled are shown in Table~\\ref{tab:examples}; these illustrate, besides other factors, the profusion of trustworthy information within them and the abundantly emotion-oriented narrative they employ. Such sources resemble newspaper websites in that consumers are passive readers whose consumption of the content happens outside social media platforms. This makes fake news detection a challenging problem in this realm since techniques are primarily left to work with just the article content - as against within social media where structural and temporal data offer ample clues - in order to determine their veracity.\n\n\\begin{table*}[!htb]\n\\caption{Examples of health fake news headlines and excerpts from them}\n\\label{tab:examples} \n\\centering\n\\resizebox{0.9\\linewidth}{!}{%\n\\begin{tabular}{p{15cm}}\n\\hline\\noalign{\\smallskip}\n\\textbf{Wi-Fi: A Silent Killer That Kills Us Slowly!} \\\\\nWiFi is the name of a popular wireless networking technology that uses radio waves to provide wireless high-speed Internet and network connections. People can browse the vast area of internet through this wireless device. A common misconception is that the term Wi-Fi is short for ``wireless fidelity'', however this is not the case. WiFi is simply a trademarked phrase that means IEEE 802.11x. The first thing people should examine is the way a device is connected to the router without cables. Well, wireless devices like cell phones, tablets, and laptops, emit WLAN signals (electromagnetic waves) in order to connect to the router. However, the loop of these signals harms our health in a number of ways. The British Health Agency conducted a study which showed that routers endanger our health and the growth of both, people and plants.\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\textbf{Russian Scientist Captures Soul Leaving Body; Quantifies Chakras} \\\\\nIt uses a small electrical current that is connected to the fingertips and takes less than a millisecond to send signals from. When these electric charges are pulsed through the body, our bodies naturally respond with a kind of `electron cloud' made up of light photons. Korotkov also used a type of Kirlian photography to show the exact moment someone's soul left their body at the time of death! He says there is a blue life force you can see leaving the body. He says the navel and the head are the first parts of us to lose their life force and the heart and groin are the last. In other cases, he's noted that the soul of people who have had violent or unexpected deaths can manifest in a state of confusion and their consciousness doesn't actually know that they have died.\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\textbf{Revolutionary juice that can burn stomach fat while sleeping} \\\\\nHaving excess belly fat poses a serious threat to your health. Fat around the midsection is a strong risk factor for heart disease, type 2 diabetes, and even some types of cancers. Pineapple-celery duo is an ideal choice for those wanting to shed the fat deposits around the stomach area due to the presence of enzymes that stimulate the fat burning hormones. All you need to do is drink this incredible burn-fat sleeping drink and refrain from eating too much sugar and starch foods during the day.\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}}\n\\end{table*}\n\n\\subsection{Our Contribution}\n\\label{sec:1.1}\n\nIn this paper, we consider the utility of the affective character of article content for the task of health fake news identification, a novel direction of inquiry though related to the backdrop of fake news identification approaches that target exploiting satire and stance \\cite{rubin2016fake,chopra2017towards}. We posit that fake and legitimate health news articles espouse different emotional characters that may be effectively utilized to improve fake news identification. We develop a simple method to amplify emotion information within documents by leveraging emotion lexicons, and empirically illustrate that such amplification helps significantly in improving the accuracy of health fake news identification within both supervised and unsupervised settings. Our emotion-enrichment method is intentionally of simple design in order to illustrate the generality of the point that emotion cognizance improves health fake news detection. While the influence of emotions on persuasion has been discussed in recent studies \\cite{vosoughi2018spread,majeed2017want}, our work provides the first focused data-driven analysis and quantification of the relationship between emotions and health fake news. Through illustrating that there are significant differences in the emotional character of fake and legitimate news in the health domain in that exaggerating the emotional content aids techniques that would differentiate them, our work sets the stage for further inquiry into identifying the nature of the differences in the emotional content. \n\nThe objective of our study is motivated by the need to illustrate the generality of the point that emotion cognizance improves fake news detection (as indicated or informally observed in various studies e.g., \\cite{vosoughi2018spread,majeed2017want}). Accordingly, we devise a methodology to leverage external emotion lexicons to derive emotion-enriched textual documents. Our empirical study in using these emotion-enriched documents for supervised and unsupervised fake news identification tasks establish that emotion cognizance improves the accuracy of fake news identification. This study is orthogonal but complementary to efforts that rely heavily on non-content features (e.g., \\cite{wu2018tracing}). \n\n\\section{Related Work}\n\\label{sec:rel}\n\nOur particular task, that of understanding the prevalence of emotions and its utility in detecting fake news in the health domain, has not been subject to much attention from the scholarly community. Herein, we survey two streams of related work very pertinent to our task, that of general fake news detection, and secondly, those relating to the analysis of emotions in fake news. \n\n\\subsection{Fake News Detection}\n\\label{sec:2.1}\n\nOwing to the emergence of much recent interest in the task of fake news detection, there have been many publications on this topic in the last few years. A representative and a non-comprehensive snapshot of work in the area appears in Table~\\ref{tab:lit}. As may be seen therein, most efforts have focused on detecting misinformation within microblogging platforms using content, network (e.g., user network) and temporal (e.g., re-tweets in Twitter) features within the platform itself \\cite{anoop2019leveraging}; some of them, notably \\cite{wu2018tracing}, target scenarios where the candidate article itself resides outside the microblogging platform, but the classification task is largely dependent on information within. An emerging trend, as exemplified by \\cite{ma2017detect,wu2018tracing}, has been to focus on how information propagates within the microblogging platform, to distinguish between misinformation and legitimate ones. Unsupervised misinformation detection techniques \\cite{zhang2017detecting,zhang2016distance} start with the premise that misinformation is rare and of differing character from the large majority, and use techniques that resemble outlier detection methods in flavor. Of particular interest is a recent work \\cite{guo2019exploiting} that targets to exploit emotions for fake news detection within microblogging platforms. This makes extensive usage of the {\\it publisher emotions}, the emotions expressed in the content, and {\\it social emotions}, the emotions expressed in responses, in order to improve upon the state-of-the-art in fake news detection accuracies. To contrast with this stream of work on fake news detection, it may be noted that our focus is on the health domain where information is usually in the form of long textual narratives, with limited information on the responses, temporal propagation and author\/spreader\/reader network structure available for the technique to make a veracity decision. \n\n\\begin{table}[!htb]\n\\caption{Overview of Related Literature}\n\\label{tab:lit} \n\\centering\n\\begin{tabular}{llccc}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{Work} & \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Target\\\\ Domain\\end{tabular}} & \\multicolumn{3}{c}{Features used} \\\\ \\cline{3-5} \\noalign{\\smallskip} \n & & Content & Network & Temporal \\\\ \n\\hline \\noalign{\\smallskip}\n\\multicolumn{5}{c}{Task Setting: Supervised} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\t\\cite{kwon2013prominent} & Twitter & \\ding{51} & \\ding{51} & \\ding{51} \\\\\n \\cite{zubiaga2017exploiting} & Twitter & \\ding{51} & \\ding{51} & \\ding{51} \\\\\n \\cite{qazvinian2011rumor} & Twitter & \\ding{51} & \\ding{51} & \\ding{51} \\\\\n \\cite{wu2018tracing} & Twitter & \\ding{51} & \\ding{51} & \\ding{51} \\\\\n \\cite{ma2016detecting} & Twitter & \\ding{51} & \\ding{55} & \\ding{51} \\\\\n \\cite{zhao2015enquiring} & Twitter & \\ding{51} & \\ding{55} & \\ding{51} \\\\\n \\cite{ma2017detect} & Twitter & \\ding{51} & \\ding{55} & \\ding{51} \\\\\n \\cite{guo2019exploiting} & Weibo & \\ding{51} & \\ding{51} & \\ding{51} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n \\multicolumn{5}{c}{Task Setting: Unsupervised} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip} \n \\cite{zhang2017detecting}& Weibo & \\ding{51} & \\ding{55} & \\ding{51} \\\\\n \\cite{zhang2016distance} & Weibo & \\ding{51} & \\ding{55} & \\ding{51} \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Emotions and Fake News}\n\\label{sec:2.2}\n\nFake news is generally crafted with the intent to mislead, and thus narratives powered with strong emotion content may be naturally expected within them. \\cite{bakir2018fake} analyze fake news vis-a-vis emotions and argue that what is most significant about the contemporary fake news furore is what it portends: the use of personally and emotionally targeted news produced by journalism referring to what they call as ``empathic media''. They further go on to suggest that the commercial and political phenomenon of empathically optimised automated fake news is on the near-horizon, and is a challenge needing significant attention from the scholarly community. A recent study, \\cite{paschen2019investigating}, conducts an empirical analysis on 150 real and 150 fake news articles from the political domain, and report finding significantly more negative emotions in the titles of the latter. Apart from being distinctly different in terms of domain, our focus being health (vs. politics for them), we also significantly differ from them in the intent of the research; our work is focused not on identifying the tell-tale emotional signatures of real vis-a-vis fake news, but on providing empirical evidence that there are differences in emotional content which may be exploited through simple mechanisms such as word-addition-based text transformations. In particular, our focus is on establishing that there are differences, and we keep identification of the nature of differences outside the scope of our present investigation. A recent tutorial survey on fake news in social media, \\cite{shu2019detecting}, also places significant emphasis on the importance of emotional information within the context of fake news detection. \n\n\\subsection{Our Work in Context} \n\\label{sec:2.3}\n\nTo put our work in context, we note that the affective character of the content has not been a focus of health fake news detection so far, to our best knowledge. Our effort is orthogonal but complementary to most work described above in that we provide evidence that emotion cognizance in general, and our emotion-enriched data representations in particular, are likely to be of much use in supervised and unsupervised fake news identification for the health domain. As observed earlier, identifying the nature of emotional differences between fake and real news in the health domain is outside the scope of our work, but would evidently lead to interesting follow-on work. \n\n\\section{Emotionizing Text}\n\\label{sec:3}\nThe intent in this paper is to provide evidence that the affective character of fake news and legitimate articles differ in a way that such differences can be leveraged to improve the task of fake news identification. First, we outline our methodology to leverage an external emotion lexicon to build emotion amplified (i.e., {\\it emotionized}) text representations. The methodology is designed to be very simple to describe and implement, so any gains out of emotionized text derived from the method can be attributed to emotion-enrichment in general and not to some nuances of the details, as could be the case if the transformation method were to involve sophisticated steps. The empirical analysis of our emotionized representations {\\it vis-a-vis} raw text for fake news identification will be detailed in the next section. \n\n\\subsection{The Task}\n\\label{sec:3.1}\nThe task of emotionizing is to leverage an emotion lexicon $\\mathcal{L}$ to transform a text document $D$ to an emotionized document $D'$. We would like $D'$ also to be similar in format to $D$ in being a sequence of words so that it can be fed into any standard text processing pipeline; retaining the document format in the output, it may be noted, is critical for the uptake of the method. In short:\n\\begin{center}\n$\nD, \\mathcal{L} \\xrightarrow[]{Emotionization} D'\n$\n\\end{center}\nWithout loss of generality, we expect that the emotion lexicon $\\mathcal{L}$ would comprise of many 3-tuples, e.g., $[w, e, s]$, each of which indicate the affinity of a word $w$ to an emotion $e$, along with the intensity quantified as a score $s \\in [0,1]$. An example entry could be $[unlucky, sadness, 0.7]$ indicating that the word {\\it unlucky} is associated with the {\\it sadness} emotion with an intensity of 0.7. \n\n\\begin{table*}[!htb]\n\\caption{Emotionized Health Fake News Excerpts (added emotion labels in bold)}\n\\label{tab:emotionize_example} \n\\centering\n\\resizebox{0.9\\linewidth}{!}{%\n\\begin{tabular}{p{15cm}}\n\\hline\\noalign{\\smallskip}\nWi-Fi: A Silent Killer {\\bf fear} That Kills {\\bf fear} Us Slowly! \\\\\nWiFi is the name of a popular wireless networking technology that uses radio waves to provide wireless high-speed Internet and network connections. People can browse the vast area of internet through this wireless device. A common misconception {\\bf fear} is that the term Wi-Fi is short for ``wireless fidelity'', however this is not the case. WiFi is simply a trademarked phrase that means IEEE 802.11x. The first thing people should examine is the way a device is connected to the router without cables. Well, wireless devices like cell phones, tablets, and laptops, emit WLAN signals (electromagnetic waves) in order to connect to the router. However, the loop of these signals harms {\\bf fear} our health in a number of ways. The British Health Agency conducted a study which showed that routers endanger {\\bf fear} our health and the growth {\\bf joy} of both, people and plants.\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nRussian Scientist Captures Soul Leaving {\\bf sadness} Body; Quantifies Chakras \\\\\nIt uses a small electrical current that is connected to the fingertips and takes less than a millisecond to send signals from. When these electric charges are pulsed through the body, our bodies naturally respond with a kind of `electron cloud' made up of light {\\bf joy} photons. Korotkov also used a type of Kirlian photography to show the exact moment someone's soul left their body at the time of death {\\bf sadness}! He says there is a blue life force you can see leaving {\\bf sadness} the body. He says the navel and the head are the first parts of us to lose {\\bf sadness} their life force and the heart and groin are the last. In other cases, he's noted that the soul of people who have had violent {\\bf anger} or unexpected deaths {\\bf sandess} can manifest in a state of confusion and their consciousness doesn't actually know that they have died {\\bf sadness}.\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nRevolutionary juice that can burn stomach fat while sleeping \\\\\nHaving excess belly fat poses a serious threat {\\bf anger} to your health. Fat around the midsection is a strong risk {\\bf fear} factor for heart disease {\\bf fear}, type 2 diabetes, and even some types of cancers {\\bf sadness}. Pineapple-celery duo is an ideal choice for those wanting to shed the fat deposits around the stomach area due to the presence of enzymes that stimulate the fat burning hormones. All you need to do is drink this incredible burn-fat sleeping drink and refrain from eating too much sugar and starch foods {\\bf joy} during the day.\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}}\n\\end{table*}\n\n\\subsection{Methodology}\n\\label{sec:3.2}\nInspired by recent methods leveraging lexical neighborhoods to derive word \\cite{mikolov2013distributed} and document \\cite{le2014distributed} embeddings, we design our emotionization methodology as one that alters the neighborhood of highly emotional words in $D$ by adding emotion labels. As illustrated in Algorithm~\\ref{algorithm}, we sift through each word in $D$ in order, outputting that word followed by its associated emotion from the lexicon $\\mathcal{L}$ into $D'$, as long as the word emotion association in the lexicon is stronger than a pre-defined threshold $\\tau$. In cases where the word is not associated with any emotion with a score greater than $\\tau$, no emotion label is output into $D'$. In summary, $D'$ is an `enlarged' version of $D$ where every word in $D$ that is strongly associated with an emotion additionally being followed by the emotion label. This ingestion of `artificial' words is similar in spirit to {\\it sprinkling} topic labels to enhance text classification \\cite{DBLP:conf\/acl\/HingmireC14}, where appending topic labels to document is the focus.\n\n\\begin{algorithm}[!htb]\n\\small\n\\SetKwInOut{Input}{input}\n\\SetKwInOut{Output}{output}\n\\SetKwInOut{Initialization}{Initialize}\n\\Input{Document $D$, Emotion-Lexicon $\\mathcal{L}$}\n\\Output{Emotionized Document $D'$}\n\\Parameter{Threshold $\\tau$}\n\\BlankLine\n\\caption{Emotionization \\label{algorithm}}\n\\renewcommand{\\nl}{\\let\\nl\\oldnl} Let $D = [w_1, w_2, \\ldots, w_n]$ \\;\ninitialize $D'$ to be empty \\;\n\\For{$(i = 1;\\ i \\leq n;\\ i++)$}{\n write $w_i$ as the next word in $D'$ \\;\n \\uIf{$(\\exists [w_i,e,s] \\in \\mathcal{L} \\wedge s \\geq \\tau)$}{\n write $e$ as the next word in $D'$ \\;\n}\n}\noutput $D'$\n\\end{algorithm}\n\\noindent\nA sample of article excerpts and their emotionized versions appear in Table~\\ref{tab:emotionize_example}. \n\n\\section{Empirical Study}\n\\label{sec:4}\nGiven our focus on evaluating the effectiveness of emotionized text representations over raw representations, we consider a variety of unsupervised and supervised methods (in lieu of evaluating on a particular state-of-the-art method) in the interest of generality. Data-driven fake news identification, much like any analytics task, uses a corpus of documents to learn a statistical model that is intended to be able to tell apart fake news from legitimate articles. Our empirical evaluation is centered on the following observation: {\\it for the same analytics model learned over different data representations, differences in effectiveness (e.g., classification or clustering accuracy) over the target task can intuitively be attributed to the data representation}. In short, if our emotionized text consistently yields better classification\/clustering models over those learned over raw text, emotion cognizance and amplification may be judged to influence fake news identification positively. We first describe our dataset, followed by the empirical study settings and their corresponding results. \n\n\\subsection{Dataset and Emotion Lexicon}\n\\label{sec:4.1}\n\nWith most fake news datasets being focused on microblogging websites in the political domain making them less suitable for content-focused misinformation identification tasks as warranted by the domain of health, we curated a new dataset of fake and legitimate news articles within the topic of {\\it health and well being} which will be publicly released upon publication to aid future research. For legitimate news, we crawled $500$ health and well-being articles from reputable sources such as CNN, NYTimes, New Indian Express and many others. For fake news, we crawled $500$ articles on similar topics from well-reported misinformation websites such as BeforeItsNews, Nephef, MadWorldNews, and many others. These were manually verified for category suitability. The detailed dataset statistics is shown in Table \\ref{tab:dataset}. \n\nFor the lexicon, we use the NRC Intensity Emotion Lexicon \\cite{mohammad2017word} which has data in the 3-tuple form outlined earlier. For simplicity, we filter the lexicon to retain only one entry per word, choosing the emotion entry with which the word has the highest intensity; this filtered out around 22\\% of entries in our lexicon. This filtering entails that each word in $D$ can only introduce up to one extra token in $D'$; the emotionization using the filtered corpus was seen to lengthen documents by an average of 2\\%, a very modest increase in document size. To put it in perspective, only around one in fifty words triggered the lexicon label attachment step. Interestingly, there was only a slight difference in the lengthening of document across the classes; while fake news documents were seen to be enlarged by $2.2\\%$ on average, legitimate news articles recorded an average lengthening by $1.8\\%$. For e.g., out of 1923 word sense entries that satisfy the threshold $\\tau = 0.6$, our filter-out-non-best heuristic filtered out 424 entries (i.e., 22\\%); thus, only slightly more than one-fifth of entries were affected. This heuristic to filter out all-but-one entry per word was motivated by the need to ensure that document structures be not altered much (by the introduction of too many lexicon words), so assumptions made by the downstream data representation learning procedure such as document well-formedness are not particularly disadvantaged. \n\n\\begin{table*}[!htb]\n\\caption{Dataset Details}\n\\label{tab:dataset} \n\\centering\n\\begin{tabular}{lccccc}\n\\hline\n\\noalign{\\smallskip}\n\\multicolumn{1}{c}{Dataset}& Class & Documents & Average Words & Average Sentences & Total Words \\\\ \\hline\n\\noalign{\\smallskip}\n\\multirow{2}{*}{Health and Well Being} & Real & 500 & 724 & 31 & 362117 \\\\ \n\\noalign{\\smallskip}\n& Fake & 500 & 578 & 28 & 289477 \\\\ \n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Supervised Setting - Conventional Classifiers}\n\\label{sec:4.2}\nLet $\\mathcal{D} = \\{ \\ldots, D, \\ldots \\}$ be the corpus of all news articles, and $\\mathcal{D}' = \\{ \\ldots, D', \\ldots \\}$ be the corresponding emotionized corpus. Each document is labeled as either fake or not (0\/1). With word\/document embeddings gaining increasing popularity, we use the DBOW doc2vec model\\footnote{https:\/\/radimrehurek.com\/gensim\/models\/doc2vec.html} to build vectors over each of the above corpora separately, yielding two datasets of vectors, correspondingly called $\\mathcal{V}$ and $\\mathcal{V}'$. While the document embeddings are learnt over the corpora ($\\mathcal{D}$ or $\\mathcal{D}'$), the output comprises one vector for each document in the corpus that the learning is performed over. The doc2vec model uses an internal parameter $d$, the dimensionality of the embedding space, i.e., the length of the vectors in $\\mathcal{V}$ or $\\mathcal{V}'$. \n\nEach of these vector datasets are separately used to train a conventional classifier using train and test splits within them. By conventional classifier, we mean a model such as random forests, kNN, SVM, Naive Bayes, Decision Tree or AdaBoost. The classification model learns to predict a class label (one of {\\it fake} or {\\it real}) given a d-dimensional embedding vector. We use multiple train\/test splits for generalizability of results where the chosen dataset (either $\\mathcal{V}$ or $\\mathcal{V}'$) is partitioned into $k$ random splits (we use $k=10$); these lead to $k$ separate experiments with $k$ models learnt, each model learnt by excluding one of $k$ splits, and evaluated over their corresponding held-out split. The accuracies obtained by $k$ separate experiments are then simply averaged to obtain a single accuracy score for the chosen dataset ($Acc(\\mathcal{D})$ and $Acc(\\mathcal{D}')$ respectively). The quantum of improvement achieved, i.e., $Acc(\\mathcal{D}') - Acc(\\mathcal{D})$ is illustrative of the improvement brought in by emotion cognizance. \n\n\\subsection{Supervised Setting - Neural Networks}\n\nNeural network models such as LSTMs and CNNs are designed to work with vector sequences, one for each word in the document, rather than a single embedding for the document. This allows them to identify and leverage any existence of sequential patterns or localized patterns respectively, in order to utilize for the classification task. These models, especially LSTMs, have become very popular for building text processing pipelines, making them pertinent for a text data oriented study such as ours. \n\nAdapting from the experimental settings for the conventional classifiers in Section~\\ref{sec:4.2}, we learn LSTM and CNN classifiers with learnable word embeddings where each word would have a length of either $100$ or $300$. Unlike in Section~\\ref{sec:4.2} where the document embeddings are learnt separately and then used in a classifier, this model interleaves training of the classifier and learning of the embeddings, so the word embeddings are also trained, in the process, to benefit the task. The overall evaluation framework remains the same as before, with the classifier-embedding combo being learnt separately for $\\mathcal{D}$ and $\\mathcal{D}'$, and the quantum by which $Acc(\\mathcal{D}')$ surpasses $Acc(\\mathcal{D})$ used as an indication of the improvement brought about by the emotionization. \n\n\n\\subsubsection{Results and Discussion}\nTable~\\ref{tab:classification} lists the classification results of a variety of standard classifiers as well as those based on CNN and LSTM, across two values of $d$ and various values of $\\tau$. $d$ is overloaded for convenience in representing results; while it indicates the dimensionality of the document vector for the conventional classifiers, it indicates the dimensionality of the word vectors for the CNN and LSTM classifiers. Classification {\\it models learned over the emotionized text are seen to be consistently more effective for the classification task}, as exemplified by the higher values achieved by $Acc(\\mathcal{D}')$ over $Acc(\\mathcal{D})$ (highest values in each row are indicated in bold). While gains are observed across a wide spectrum of values of $\\tau$, the gains are seen to peak around $\\tau \\approx 0.6$. Lower values of $\\tau$ allow words of low emotion intensity to influence $D'$ while setting it to a very high value would add very few labels to $D'$ (at the extreme, using $\\tau=1.0$ would mean $D=D'$). Thus the observed peakiness is along expected lines, with $\\tau \\approx 0.6$ achieving a middle ground between the extremes. The quantum of gains achieved, i.e., $|Acc(\\mathcal{D}')-Acc(\\mathcal{D})|$, is seen to be significant, sometimes even bringing $Acc(\\mathcal{D}')$ very close to the upper bound of $1.0$; this establishes that emotionized text is much more suitable for supervised misinformation identification. It is further notable that the highest accuracy is achieved by AdaBoost as against the CNN and LSTM models; this may be due to the lexical distortions brought about addition of emotion labels limiting the emotionization gains in the LSTM and CNN classifiers that attempt to make use of the word sequences explicitly.\n\n\\begin{table}[!htb]\n\\caption{Classification Results (NB = Naive Bayes, RF = Random Forest, DT = Decision Tree, AB = AdaBoost)}\n\\label{tab:classification} \n\\centering\n\\resizebox{1\\linewidth}{!}{%\n\\begin{tabular}{lcccccc}\n\\hline\\noalign{\\smallskip}\n \\multirow{2}{*}{{\\small Method}} & \\multirow{2}{*}{$Acc(\\mathcal{D})$} & \\multicolumn{5}{c}{$Acc(\\mathcal{D}')$} \\\\ \\noalign{\\smallskip} \\cline{3-7} \\noalign{\\smallskip}\n & & $\\tau=0.0$ & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n \\multicolumn{7}{c}{Classification Results for $d = 100$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n NB & 0.770 & 0.780 & 0.780 & 0.785 & {\\bf 0.790} & 0.775 \\\\\n KNN & 0.750 & 0.750 & 0.755 & 0.760 & {\\bf 0.925} & 0.750 \\\\\n SVM & 0.500 & 0.650 & 0.750 & 0.750 & {\\bf 0.900} & 0.700 \\\\\n RF & 0.630 & 0.710 & 0.700 & 0.720 & {\\bf 0.840} & 0.805 \\\\\n DT & 0.680 & 0.690 & 0.700 & 0.780 & {\\bf 0.940} & 0.785 \\\\\n AB & 0.550 & 0.570 & 0.700 & 0.710 & {\\bf 0.965} & 0.825 \\\\\n CNN & 0.870 & 0.880 & 0.900 & 0.880 & {\\bf 0.910} & 0.880 \\\\\n LSTM & 0.905 & 0.900 & 0.910 & 0.910 & {\\bf 0.920} & {\\bf 0.920} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n \\multicolumn{7}{c}{Classification Results for $d = 300$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n NB & 0.770 & 0.800 & 0.810 & 0.790 & {\\bf 0.830} & 0.780 \\\\\n KNN & 0.720 & 0.740 & 0.750 & 0.760 & {\\bf 0.910} & 0.745 \\\\\n SVM & 0.600 & 0.670 & 0.720 & 0.740 & {\\bf 0.890} & 0.720 \\\\\n RF & 0.650 & 0.700 & 0.730 & 0.715 & {\\bf 0.820} & 0.750 \\\\\n DT & 0.600 & 0.650 & 0.730 & 0.780 & {\\bf 0.905} & 0.750 \\\\\n AB & 0.550 & 0.550 & 0.720 & 0.810 & {\\bf 0.945} & 0.750 \\\\\n CNN & 0.912 & 0.910 & {\\bf 0.927} & 0.920 & 0.920 & 0.910 \\\\\n LSTM & 0.900 & 0.902 & 0.900 & 0.902 & {\\bf 0.907} & 0.900 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}}\n\\end{table}\n\n\\subsection{Unsupervised Setting}\n\\label{sec:4.3}\nThe corresponding evaluation for the unsupervised setting involves clustering both $\\mathcal{V}$ and $\\mathcal{V}'$ (Ref. Sec.~\\ref{sec:4.2}) using the same method and profiling the clustering against the labels on the clustering purity measure\\footnote{https:\/\/nlp.stanford.edu\/IR-book\/html\/htmledition\/evaluation-of-clustering-1.html}; as may be obvious, the labels are used only for evaluating the clustering, clustering being an unsupervised learning method. We used K-Means \\cite{macqueen1967some} and DBSCAN \\cite{ester1996density} clustering methods, two very popular clustering methods that come from distinct families. K-Means uses a top-down approach to discover clusters, estimating cluster centroids and memberships at the dataset level, followed by iteratively refining them. DBSCAN, on the other hand, uses a more bottom-up approach, forming clusters and enlarging them by adding proximal data points progressively. Another aspect of difference is that K-Means allows the user to specify the number of clusters desired in the output, whereas DBSCAN has a substantively different mechanism, based on neighborhood density. For K-Means we measured purities, averaged over 1000 random initializations, across varying values of $k$ (desired number of output clusters); it may be noted that purity is expected to increase with $k$ with finer clustering granularities leading to better purities (at the extreme, each document in its own cluster would yield a purity of $1.0$). For DBSCAN we measured purities across varying values of $ms$ (minimum samples to form a cluster); the {\\it ms} parameter is the handle available to the user within the DBSCAN framework to indirectly control the granularity of the clustering (i.e., the number of clusters in the output). Analogous to the $Acc(.)$ measurements in classification, the quantum of purity improvements achieved by the emotionized text, i.e., $Pur(\\mathcal{D}')-Pur(\\mathcal{D})$, indicate any improved effectiveness of emotionized representations.\n\n\\subsubsection{Results and Discussion} Table~\\ref{tab:clustering} lists the clustering results in a format similar to that of the classification study. With the unsupervised setting posing a harder task, the quantum of improvements ($Pur(\\mathcal{D}')-Pur(\\mathcal{D})$) achieved by emotionization is correspondingly lower. We believe the cause of low accuracy is because most conventional combinations of document representation and clustering algorithm are suited to generate topically coherent clusters, and thus fare poorly on a substantially different task of fakeness identification. However, the trends are consistent with the earlier observations in that emotionization has a positive effect, with gains peaking around $\\tau \\approx 0.6$. \n\n\\begin{table}[!htb]\n\\caption{Clustering Results}\n\\label{tab:clustering} \n\\centering\n\\resizebox{1\\linewidth}{!}{%\n\\begin{tabular}{lcccccc}\n\\hline\\noalign{\\smallskip}\n \\multirow{2}{*}{} & \\multirow{2}{*}{$Pur(\\mathcal{D})$} & \\multicolumn{5}{c}{$Pur(\\mathcal{D}')$} \\\\ \\noalign{\\smallskip} \\cline{3-7} \\noalign{\\smallskip}\n & & $\\tau=0.0$ & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n k & \\multicolumn{6}{c}{K-Means Clustering Results for $d=100$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n 2 & 0.523 & 0.524 & 0.523 & 0.523 & {\\bf 0.561} & 0.529 \\\\\n 4 & 0.781 & 0.780 & 0.786 & 0.793 & {\\bf 0.816} & 0.793 \\\\\n 7 & 0.850 & 0.857 & 0.852 & 0.851 & {\\bf 0.869} & 0.856 \\\\\n 10 & 0.853 & 0.851 & 0.851 & 0.851 & {\\bf 0.877} & 0.857 \\\\\n 15 & 0.852 & 0.853 & 0.851 & 0.851 & {\\bf 0.878} & 0.858 \\\\\n 20 & 0.852 & 0.852 & 0.850 & 0.851 & {\\bf 0.887} & 0.857 \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n k & \\multicolumn{6}{c}{K-Means Clustering Results for $d=300$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n 2 & 0.513 & 0.520 & 0.520 & 0.520 & {\\bf 0.555} & 0.520 \\\\\n 4 & 0.771 & 0.778 & 0.781 & 0.789 & {\\bf 0.815} & 0.785 \\\\\n 7 & 0.840 & 0.840 & 0.850 & 0.849 & {\\bf 0.869} & 0.846 \\\\\n 10 & 0.850 & 0.850 & 0.850 & 0.850 & {\\bf 0.871} & 0.851 \\\\\n 15 & 0.851 & 0.853 & 0.851 & 0.851 & {\\bf 0.875} & 0.852 \\\\\n 20 & 0.850 & 0.852 & 0.850 & 0.850 & {\\bf 0.880} & 0.850 \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n ms & \\multicolumn{6}{c}{DBSCAN Clustering Results for $d=100$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n 20 & 0.610 & 0.620 & 0.620 & 0.620 & {\\bf 0.650} & 0.619 \\\\\n 40 & 0.627 & 0.655 & 0.645 & 0.581 & {\\bf 0.665} & 0.650 \\\\\n 60 & 0.716 & 0.721 & 0.720 & 0.725 & {\\bf 0.725} & 0.725 \\\\\n 80 & 0.851 & 0.850 & 0.851 & 0.856 & {\\bf 0.860} & 0.856 \\\\\n 100 & 0.845 & 0.841 & 0.848 & 0.847 & {\\bf 0.860} & 0.840 \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n ms & \\multicolumn{6}{c}{DBSCAN Clustering Results for $d=300$} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n 20 & 0.610 & 0.615 & 0.610 & 0.610 & {\\bf 0.635} & 0.620 \\\\\n 40 & 0.635 & 0.663 & 0.665 & 0.669 & {\\bf 0.670} & 0.655 \\\\\n 60 & 0.675 & 0.701 & 0.705 & 0.710 & {\\bf 0.715} & 0.700 \\\\\n 80 & 0.780 & 0.810 & 0.819 & 0.820 & {\\bf 0.825} & 0.808 \\\\\n 100 & 0.755 & 0.800 & 0.800 & 0.800 & {\\bf 0.805} & 0.800 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}}\n\\end{table}\n\n\\section{Conclusions}\n\\label{sec:con}\nIn this paper, we considered the utility of the affective character of news articles for the task of misinformation detection in the health domain. We illustrated that amplifying the emotions within a news story (and in a sense, uplift their importance) helps downstream algorithms, supervised and unsupervised, to identify health fake news better. In a way, our results indicate that fake and real news differ in the nature of emotional information within them, so exaggerating the emotional information within both stretch them further apart in any representation, helping to distinguish them from each other. In particular, our simple method to emotionize text using external emotion intensity lexicons were seen to yield text representations that were empirically seen to be much more suited for the task of identifying health fake news. In the interest of making a broader point establishing the utility of affective information for the task, we empirically evaluated the representations over a wide variety of supervised and unsupervised techniques and methods over varying parameter settings, across which consistent and significant gains were observed. This firmly establishes the utility of emotion information in improving health fake news identification. \n\n\\subsection{Future Work} \n\\label{sec:5.1}\nAs a next step, we are considering developing emotion-aware end-to-end methods for supervised and unsupervised health fake news identification. Secondly, we are considering the use of lexicons learned from data \\cite{bandhakavi2014generating} which may be better suited for fake news identification in niche domains. Third, we are exploring the usage of the affective content of responses to social media posts. \n\n\n\n\n\n\n\n\\bibliographystyle{spmpsci} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}