diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhuhx" "b/data_all_eng_slimpj/shuffled/split2/finalzzhuhx" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhuhx" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nLet $(M,J)$ be a complex manifold of real dimension $2n$, $n \\geq 2$, equipped with a Hermitian metric $g$ with associated fundamental $2$-form $\\omega=g(J\\cdot,\\cdot)$. Its \\emph{Lee form}, defined by $\\theta=-d^*\\omega \\circ J$, is the unique $1$-form satisfying $d\\omega^{n-1}=\\theta \\wedge \\omega^{n-1}$.\n\nA fundamental class of Hermitian metrics is provided by \\emph{K\\\"ahler} metrics, satisfying $d\\omega=0$. In literature, many generalizations of the K\\\"ahler condition have been introduced: two of them are the \\emph{balanced} (or \\emph{semi-K\\\"ahler}) condition, characterized by $d^*\\omega=0$ (or equivalently $\\theta=0$ or $d\\omega^{n-1}=0$) and the \\emph{locally conformally K\\\"ahler} (LCK) condition, namely $(M,J)$ admits an open cover $\\{U_i\\}$ and smooth maps $f_i \\in C^{\\infty}(U_i)$ such that $e^{-f_i}g\\rvert_{U_i}$ is a K\\\"ahler metric on $(U_i,J\\rvert_{U_i})$, where $g$ denotes the LCK metric.\nThe LCK condition is equivalently characterized by \nthe conditions $d\\omega=\\frac{1}{n-1} \\theta \\wedge \\omega$, $d\\theta=0$. If $\\theta$ is parallel with respect to the Levi-Civita connection, the LCK metric is called \\emph{Vaisman}. For general results about LCK metrics, we refer the reader to \\cite{DO, Orn, OV, AO1}.\n\nA further weakening of both the balanced and the LCK conditions is given by the \\emph{locally conformally balanced} (LCB) condition, whose definition is analogous to the one for the LCK condition and which is equivalently defined by $d\\theta=0$. LCB metrics have been studied, for instance, in \\cite{AU, AU1, FT1, LY, OOS, Oti, Shi, Shi1, Yang}. When $n=2$, balanced metrics are K\\\"ahler and LCB metrics are LCK. \n\nRecall also that a Hermitian metric is called \\emph{strong K\\\"ahler with torsion} (SKT, also known as \\emph{pluriclosed}) if $\\partial \\overline\\partial \\omega=0$ or, equivalently, if the torsion of the associated \\emph{Bismut connection} vanishes. The Bismut connection $\\nabla^B$ of a Hermitian manifold $(M,J,g)$ is the unique linear connection on $M$ having totally skew-symmetric torsion and satisfying $\\nabla^Bg=0$, $\\nabla^BJ=0$ (see \\cite{Bismut,Gau}). Its associated \\emph{Bismut-Ricci form} $\\rho^B$ is the $2$-form locally defined by\n\\[\n\\rho^B(X,Y)=-\\frac{1}{2} \\sum_{i=1}^{2n} g(R^B(X,Y)f_i,Jf_i),\\quad X,Y \\in \\Gamma_{\\text{loc}}(TM),\n\\]\nwhere $\\{f_1,\\ldots,f_{2n}\\}$ is a local $g$-orthonormal frame and $R^{B}(X,Y)=[\\nabla^B_X,\\nabla^B_Y]-\\nabla^B_{[X,Y]}$ denotes the curvature of $\\nabla^B$.\n\nA \\emph{hypercomplex structure} on a smooth $4m$-dimensional manifold $M$ is given by a triple of (integrable) complex structures $(I_1,I_2,I_3)$ satisfying $I_1I_2I_3=-\\text{Id}_{TM}$. A Riemannian metric on $M$ is called \\emph{(locally conformally) hyperk\\\"ahler} (LCHK) if it is (locally conformally) K\\\"ahler with respect to the three complex structures and the three induced Lee forms coincide. Hypercomplex and hyperk\\\"ahler structures on Lie groups where studied for instance in \\cite{Bar}, where four-dimensional Lie groups admitting left-invariant hypercomplex structures are classified, and \\cite{BDF}, where, in particular, it is shown that left-invariant hyperk\\\"ahler metrics on Lie groups are flat.\n\nWe are interested in the case where $M$ is a simply connected almost abelian Lie group $G$ or a compact almost abelian \\emph{solvmanifold}, namely a quotient $\\Gamma \\backslash G$, with $G$ a simply connected almost abelian Lie group and $\\Gamma$ a lattice of $G$, i.e., a discrete subgroup of $G$. A connected (solvable) Lie group $G$ is called \\emph{almost abelian} if it admits an abelian normal subgroup of codimension one, or equivalently if the Lie algebra $\\mathfrak{g}$ of $G$ admits an abelian ideal $\\mathfrak{n}$ of codimension one, so that $\\mathfrak{g}$ is isomorphic to the semi-direct product $\\mathbb{R}^k \\rtimes_D \\mathbb{R}$ for some $D \\in \\mathfrak{gl}_k$. If $\\mathfrak{g}$ is non-nilpotent, such an ideal is unique and coincides with the nilradical of $\\mathfrak{g}$.\n\nA left-invariant Hermitian structure $(J,g)$ on $G$ or $\\Gamma \\backslash G$ descends to a structure on the Lie algebra $\\mathfrak{g}$ of $G$, so that one can speak of Hermitian structures on $\\mathfrak{g}$.\nWhen $\\mathfrak{g}$ is almost abelian of real dimension $2n$, as shown in \\cite{LV}, these can be fully characterized in terms of the matrix associated with $\\text{ad}_{e_{2n}}\\rvert_{\\mathfrak{n}}$ with respect to some fixed unitary basis $\\left\\{ e_1,\\ldots, e_{2n} \\right\\}$ \\emph{adapted} to the splitting $\\mathfrak{g}= J\\mathfrak{k} \\oplus \\mathfrak{n}_1 \\oplus \\mathfrak{k}$, where $\\mathfrak{k} \\coloneqq \\mathfrak{n}^{\\perp_g}$ and $\\mathfrak{n}_1 \\coloneqq \\mathfrak{n} \\cap J\\mathfrak{n}$, and such that $Je_i = e_{2n+1-i}$, $i=1,\\ldots,n$.\n\nK\\\"ahler, SKT, balanced and LCK almost abelian Lie algebras were studied in terms of the data $(a,v,A)$ in \\cite{LV,AL,FP,FP1,AO}. Six-dimensional almost abelian Lie algebras admitting SKT structures were classified in \\cite{FP}, and in \\cite{FS} the result was extended to a wider class of two-step solvable Lie algebras. For the classification of six-dimensional almost abelian Lie algebras carrying balanced structures, see \\cite{FP1}.\n\nIn Section \\ref{sec_lcb} we characterize LCB almost abelian Lie algebras in terms of the aforementioned algebraic data and in terms of the behaviour of the associated Bismut-Ricci form. \n\nIn the following section, we classify six-dimensional almost abelian Lie algebras admitting LCK structures and those admitting LCB structures, building on the classification of six-dimensional almost abelian Lie algebras admitting complex structures in \\cite{FP}, and remark which of the corresponding Lie groups admit compact quotients by lattices. \n\nIn \\cite{OOS}, the authors investigate the existence of two different types of special Hermitian metrics on a fixed compact complex nilmanifold (namely, the quotient of a simply connected nilpotent Lie group by a lattice): in Section \\ref{sec_comp}, we consider analogous questions for almost abelian solvmanifolds, highlighting similarities and differences with respect to the nilpotent setting.\n\nFinally, in Section \\ref{LCHK} we study LCHK structures on almost abelian Lie algebras, giving a classification result in every dimension.\n\n\\smallskip\n\\emph{Acknowledgements}. The author would like to thank Anna Fino for suggesting the subject of this paper and for many useful comments and discussions. The author is also grateful to an anonymous referee for useful comments. The author was supported by GNSAGA of INdAM.\n\n\\section{Locally conformally balanced metrics} \\label{sec_lcb}\n\nLet $\\mathfrak{g}$ be a $2n$-dimensional almost abelian Lie algebra with a fixed abelian ideal $\\mathfrak{n}$ of codimension one. Assume $(J,g)$ is a Hermitian structure on $\\mathfrak{g}$ and denote by $\\mathfrak{n}_1 \\coloneqq \\mathfrak{n} \\cap J\\mathfrak{n}$ the maximal $J$-invariant subspace of $\\mathfrak{n}$, which does not depend on the metric $g$.\nThen, as shown in \\cite{LV}, with respect to a unitary basis $\\{e_1,\\ldots,e_{2n}\\}$ for $\\mathfrak{g}$ such that $\\mathfrak{n}=\\text{span}\\left$, $\\mathfrak{n}_1=\\text{span}\\left$, $Je_i=e_{2n+1-i}$, $i=1,\\ldots,n$, the matrix $B$ associated with $\\text{ad}_{e_{2n}}\\rvert_{\\mathfrak{n}}$ is of the form\n\\begin{equation} \\label{B}\n\tB=\\begin{pmatrix} a & 0 \\\\ v & A \\end{pmatrix}, \\quad a \\in \\mathbb{R},\\,v \\in \\mathfrak{n}_1,\\, A \\in \\mathfrak{gl}(\\mathfrak{n}_1,J_1),\n\\end{equation}\nwhere $J_1 \\coloneqq J \\rvert_{\\mathfrak{n}_1}$ and $\\mathfrak{gl}(\\mathfrak{n}_1,J_1)$ denotes endomorphisms of $\\mathfrak{n}_1$ commuting with $J_1$. We denote $\\mathfrak{k}\\coloneqq \\mathfrak{n}^{\\perp_g}=\\mathbb{R} e_{2n}$ and we say that the basis $\\{e_1,\\ldots,e_{2n}\\}$ is \\emph{adapted} to the splitting $\\mathfrak{g} = J\\mathfrak{k} \\oplus \\mathfrak{n}_1 \\oplus \\mathfrak{k}$.\nThe algebraic data $(a,v,A)$ fully characterizes the Hermitian structure $(J,g)$ and we we shall often denote the resulting Hermitian almost abelian Lie algebra by $(\\mathfrak{g}(a,v,A),J,g)$.\n\nBefore studying the LCB condition, we recall the known characterizations for special Hermitian almost abelian Lie algebras.\n\n\\begin{proposition}\n\t\\label{Herm-alm-ab}\nA Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ is\n\\begin{itemize}\n\t\\item \\makebox[1.35cm]{K\\\"ahler,\\hfill} if $v=0$, $A \\in \\mathfrak{u}(\\mathfrak{n}_1,J_1,g)$ (see \\cite{LV}),\n\t\\item {\\makebox[1.35cm]{LCK,\\hfill} if $v=0$, $A\\in \\mathbb{R}\\text{\\normalfont Id}_{\\mathfrak{n}_1}\\oplus \\mathfrak{u}(\\mathfrak{n}_1,J_1,g)$ or $n=2$, $A=0$ (see \\cite{AO}),}\n\t\\item {\\makebox[1.35cm]{balanced,\\hfill} if $v=0$, $\\operatorname{tr} A=0$ (see \\cite{FP1}),}\n\t\\item {\\makebox[1.35cm]{SKT,\\hfill} if $[A,A^t]=0$ and the eigenvalues of $A$ have real part $-\\frac{a}{2}$ or $0$ (see \\cite{AL}),}\n\\end{itemize}\nwhere $\\mathfrak{u}(\\mathfrak{n}_1,J_1,g)=\\mathfrak{so}(\\mathfrak{n}_1,g) \\cap \\mathfrak{gl}(\\mathfrak{n}_1,J_1)$.\n\\end{proposition}\n\nWe also recall that, in terms of an adapted unitary basis, the Lee form of a Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ is given by\n\\begin{equation} \\label{leeform}\n\\theta=(Jv)^{\\flat} - (\\operatorname{tr} A)e^{2n},\n\\end{equation}\nwhere the isomorphism $(\\cdot)^\\flat \\colon \\mathfrak{g} \\to \\mathfrak{g}^*$ is defined by $X^\\flat\\coloneqq g(X,\\cdot)$, $X \\in \\mathfrak{g}$. See \\cite{FP1} for details.\n\nWe are ready to prove the analogous characterization for LCB structures.\n\\begin{theorem} \\label{LCB_avA}\nA Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ is LCB if and only if \\mbox{$A^tv=0$}.\n\\end{theorem}\n\\begin{proof}\nObserve that, given any $1$-form $\\alpha \\in \\mathfrak{g}^*$, since\n$d\\alpha(X,e_{2n})=\\alpha([e_{2n},X])$, $X \\in \\mathfrak{g}$, one has\n\\[\nd\\alpha=(\\text{ad}_{e_{2n}}^* \\alpha) \\wedge e^{2n} = (a\\alpha(e_1)+\\alpha(v))\\,e^1 \\wedge e^{2n} + A^*(\\alpha\\rvert_{\\mathfrak{n}_1}),\n\\]\nwith respect to the fixed adapted unitary basis $\\{e_1,\\ldots,e_{2n}\\}$.\nThen the exterior derivative of the Lee form \\eqref{leeform} satisfies\n\\[\nd\\theta=g(v,Jv)\\,e^1 \\wedge e^{2n} + (A^tJv)^\\flat \\wedge e^{2n}=(A^tJv)^\\flat \\wedge e^{2n},\n\\]\nwhere $A^t \\in \\mathfrak{gl}(\\mathfrak{n}_1)$ is defined by $A^tX \\coloneqq (A^*(X^\\flat))^\\sharp$, $X \\in \\mathfrak{n}_1$, $(\\cdot)^\\sharp$ denoting the inverse of $(\\cdot)^\\flat$. Then $d\\theta$ vanishes if and only if $A^tJv=0$. $J_1$ commutes with $A$ and we have $J_1^t=-J_1$, so $J_1$ commutes with $A^t$ as well. The previous condition then reads $JA^tv=0$, which is equivalent to $A^tv=0$. \n\\end{proof}\n\nWe note that the condition $A^tv=0$ is equivalent to $g(v,AX)=0$ for all $X \\in \\mathfrak{n}_1$. In particular, when $v \\neq 0$, it implies $v \\notin \\operatorname{im} A$, so that $\\text{rank}(v\\lvert A)=\\text{rank}(A)+1$, where $v \\lvert A$ denotes the matrix obtained by juxtaposing $v$ and $A$.\n\nIn \\cite{AL}, the authors determined a formula for the Bismut-Ricci form of a Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$, obtaining\n\\begin{equation}\\label{rhoB}\n\\rho^B=-\\left(a^2-\\tfrac{1}{2}a\\operatorname{tr}A+\\lVert v \\rVert^2\\right)\\, e^1 \\wedge e^{2n} - (A^tv)^{\\flat} \\wedge e^{2n},\n\\end{equation}\nin terms of the fixed adapted unitary basis $\\{e_1,\\ldots,e_{2n}\\}$ (cf. also \\cite{FP}).\n\nThe next result is a straightforward consequence of Theorem \\ref{LCB_avA} and formula \\eqref{rhoB}.\n\\begin{proposition}\nA Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ is LCB if and only if $\\rho^B$ is of type $(1,1)$ (namely, $J\\rho^B=\\rho^B$), or equivalently if $\\rho^B(X,Y)=0$ for every $X \\in \\mathfrak{n}_1$, $Y \\in \\mathfrak{g}$.\n\\end{proposition}\n\n\\section{Classification in dimension six}\n\nWe now focus on the six-dimensional case, with the goal of classifying almost abelian Lie algebras admitting LCB structures.\nAs recalled in the introduction, LCB structures generalize K\\\"ahler, balanced and LCK structures. Six-dimensional almost abelian Lie algebras carrying K\\\"ahler structures and balanced structures were classified in \\cite{FP} and \\cite{FP1} respectively. Therefore, before considering strictly LCB structures, we focus on the LCK condition.\n\nIn the following, we denote a Lie algebra via its structure equations: for example, the notation\n\\[\n\\mathfrak{g}_{4}=(f^{16},f^{26},f^{36},f^{46},0,0)\n\\] \nmeans that the Lie algebra $\\mathfrak{g}_{4}$ is determined by a fixed basis $\\{f_1,\\ldots,f_6\\}$ whose dual coframe $\\{f^1,\\ldots,f^6\\}$ satisfies $df^1=f^{16}$, $df^2=f^{26}$, $df^3=f^{36}$, $df^4=f^{46}$, $df^5=df^6=0$, where $f^{ij}$ is a shorthand for the wedge product $f^i \\wedge f^j$.\n\nIn \\cite{Saw}, it was proven that a nilpotent Lie algebra admits an LCK structure if and only if it is isomorphic to $\\mathfrak{h}_{2n+1} \\oplus \\mathbb{R}$, for some $n \\geq 1$, where\n\\[\n\\mathfrak{h}_{2n+1}=\\left(0,\\ldots,0,\\sum_{i=1}^{n} f^{2i-1} \\wedge f^{2i}\\right)\n\\]\ndenotes the $2n+1$-dimensional real Heisenberg algebra. In particular, the four-dimensional $\\mathfrak{h}_3 \\oplus \\mathbb{R} = (0,0,0,f^{12})$ is the only one which is also almost abelian and, by \\cite[Remark 3.4 (ii)]{AO} and \\cite[Remarks 2.1, 2.3]{AngO}, one of the only two almost abelian Lie algebras admitting non-K\\\"ahler Vaisman metrics, up to isomorphism, the other one being $\\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$, where $\\mathfrak{aff}_2 =(0,f^{12})$ denotes the two-dimensional real affine Lie algebra. In fact, every Hermitian metric on $\\mathfrak{h}_3 \\oplus \\mathbb{R}$ and $\\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$ is Vaisman.\n\n\\begin{theorem} \\label{LCK_class}\nLet $\\mathfrak{g}$ be a six-dimensional almost abelian Lie algebra. Then $\\mathfrak{g}$ admits an LCK structure $(J,g)$, but no K\\\"ahler structures, if and only if it is isomorphic to one of the following:\n\\begin{itemize}\n\t\\setlength{\\itemindent}{-1em}\n\t\\item[] $\\mathfrak{g}_{1}\\cm{=\\mathfrak{k}_1^{p,p}}=(f^{16},pf^{26},pf^{36},pf^{46},pf^{56},0)$, \\, $p \\neq 0$,\\smallskip\n\t\\item[] $\\mathfrak{g}_{2}\\cm{=\\mathfrak{k}_{8}^{p,q,q}}=(pf^{16},qf^{26},qf^{36},qf^{46}+f^{56},-f^{46}+qf^{56},0)$, \\, $pq \\neq 0$, \\smallskip\n\t\\item[] $\\mathfrak{g}_{3}\\cm{=\\mathfrak{k}_{11}^{p,q,q,s}}=(pf^{16},qf^{26}+f^{36},-f^{26}+qf^{36},qf^{46}+rf^{56},-rf^{46}+qf^{56},0)$, \\, $pq \\neq 0$, $r \\neq 0$,\\smallskip\n\t\\item[] $\\mathfrak{g}_{4}\\cm{=\\mathfrak{k}_{20}^1}=(f^{16},f^{26},f^{36},f^{46},0,0)$,\\smallskip\n\t\\item[] $\\mathfrak{g}_{5}\\cm{=\\mathfrak{k}_{22}^{1,r}}=(f^{16},f^{26},f^{36}+rf^{46},-rf^{36}+f^{46},0,0)$, \\, $r \\neq 0$, \\smallskip\n\t\\item[] $\\mathfrak{g}_{6}\\cm{=\\mathfrak{k}_{25}^{p,p,r}}=(pf^{16}+f^{26},-f^{16}+pf^{26},pf^{36}+rf^{46},-rf^{36}+pf^{46},0,0)$, \\, $pr \\neq 0$.\n\\end{itemize}\nAmong these, only the indecomposable Lie algebras $\\mathfrak{g}_{1}^{p=-\\frac{1}{4}}$\\cm{$\\mathfrak{k}_1^{-\\frac{1}{4},-\\frac{1}{4}}$}, $\\mathfrak{g}_{2}^{p=-4q}$ \\cm{$\\mathfrak{k}_{8}^{p,-\\frac{1}{4}p,-\\frac{1}{4}p}$} and $\\mathfrak{g}_{3}^{p=-4q}$ \\cm{$\\mathfrak{k}_{11}^{p,-\\frac{1}{4}p,-\\frac{1}{4}p,s}$} are unimodular. None of the corresponding Lie groups admit compact quotients by lattices, by \\cite[Theorem 3.7]{AO}. \\end{theorem} \n\\begin{proof}\nLet $(J,g)$ be an LCK structure on $\\mathfrak{g}$. Let $\\{e_1,\\ldots,e_6\\}$ be a unitary basis of $(\\mathfrak{g},J,g)$ adapted to the splitting $\\mathfrak{g}=J\\mathfrak{k} \\oplus \\mathfrak{n}_1 \\oplus \\mathfrak{k}$, so that, by \\cite{AO}, the matrix $B$ associated with $\\text{ad}_{e_6}\\rvert_{\\mathfrak{n}}$ is of the form \\eqref{B}, with\n\\begin{equation} \\label{LCK}\nv=0, \\quad A= \\lambda\\,\\text{Id}_{\\mathfrak{n}_1} + U,\\; \\lambda \\in \\mathbb{R},\\,U \\in \\mathfrak{u}(\\mathfrak{n}_1,J_1,g).\n\\end{equation}\nSince $U$ is traceless, one must have $\\lambda=\\frac{\\operatorname{tr} A}{4}$.\nFollowing \\cite[Theorem 3.2]{FP}, up to taking a different basis $\\{e_2,\\ldots,e_5\\}$ for $\\mathfrak{n}_1$ and rescaling $e_6$, the fact that $A$ commutes with $J_1$ forces $A$ to be represented by a real $4 \\times 4$ matrix of one of the following types:\n\\begin{equation} \\label{A}\nA_1=\\left( \\begin{smallmatrix} p&0&0&0 \\\\ 0&p&0&0 \\\\ 0&0&q&0 \\\\ 0&0&0&q \\end{smallmatrix} \\right)\\!\\!, \\,\nA_2=\\left( \\begin{smallmatrix} p&1&0&0 \\\\ -1&p&0&0 \\\\ 0&0&q&0 \\\\ 0&0&0&q \\end{smallmatrix} \\right)\\!\\!, \\,\nA_3=\\left( \\begin{smallmatrix} p&1&0&0 \\\\ -1&p&0&0 \\\\ 0&0&q&r \\\\ 0&0&-r&q \\end{smallmatrix} \\right)\\!\\!, \\,\nA_4=\\left( \\begin{smallmatrix} p&1&0&0 \\\\ 0&p&0&0 \\\\ 0&0&p&1 \\\\ 0&0&0&p \\end{smallmatrix} \\right)\\!\\!, \\,\nA_5=\\left( \\begin{smallmatrix} p&1&-1&0 \\\\ -1&p&0&-1 \\\\ 0&0&p&1 \\\\ 0&0&-1&p \\end{smallmatrix} \\right)\\!\\!,\n\\end{equation}\n$p,q,r \\in \\mathbb{R}$, with $r \\neq 0$ to avoid redundancy.\nAll we need to do is determine which matrices $A_i$ in \\eqref{A} can be decomposed as $\\lambda \\operatorname{Id} + U$ for some $\\lambda \\in \\mathbb{R}$, $U \\in \\mathfrak{u}(\\mathfrak{n}_1,J_1,g)$. For each $i=1,2,3,4,5$, consider the matrix $U_i=A_i - \\frac{\\operatorname{tr} A}{4}\\operatorname{Id}$: for $i=4,5$, $U_i$ is never complex-diagonalizable (namely, diagonalizable as a complex matrix), so it cannot be skew-symmetric with respect to any metric; for $i=1,2,3$, the requirement that all the eigenvalues of $U_i$ should be pure imaginary imposes $p=q$, so that one is left with\n\\[\nU_1=\\left( \\begin{smallmatrix} 0&0&0&0 \\\\ 0&0&0&0 \\\\ 0&0&0&0 \\\\ 0&0&0&0 \\end{smallmatrix} \\right)\\!, \\quad\nU_2=\\left( \\begin{smallmatrix} 0&1&0&0 \\\\ -1&0&0&0 \\\\ 0&0&0&0 \\\\ 0&0&0&0 \\end{smallmatrix} \\right)\\!, \\quad\nU_3=\\left( \\begin{smallmatrix} 0&1&0&0 \\\\ -1&0&0&0 \\\\ 0&0&0&r \\\\ 0&0&-r&0 \\end{smallmatrix} \\right)\\!,\n\\]\nall of which are skew-symmetric with respect to the standard metric and commute with\n\\[\nJ_1=\\left( \\begin{smallmatrix} 0&-1&0&0\\\\1&0&0&0 \\\\ 0&0&0&-1 \\\\ 0&0&1&0 \\end{smallmatrix} \\right).\n\\]\nCompleting the corresponding $A_i$ to the full matrix\n\\[\nB=\\begin{pmatrix} a & 0 \\\\ 0 & A_i \\end{pmatrix}\n\\]\nrepresenting $\\text{ad}_{e_6}\\rvert_{\\mathfrak{n}}$ and assuming $\\operatorname{tr} A_i \\neq 0$ to discard the K\\\"ahler cases,\none can easily see which algebras can be obtained:\n\\begin{itemize}\n\\item[] $A_1$ yields $\\mathfrak{g}_{1}$ and $\\mathfrak{g}_{4}$,\n\\item[] $A_2$ yields $\\mathfrak{g}_{2}$ and $\\mathfrak{g}_{5}$,\n\\item[] $A_3$ yields $\\mathfrak{g}_{3}$ and $\\mathfrak{g}_{6}$. \\qedhere \n\\end{itemize}\n\\end{proof}\n\n\\begin{theorem} \\label{LCB_class}\nLet $\\mathfrak{g}$ be a six-dimensional almost abelian Lie algebra which does not admit balanced or LCK structures. If $\\mathfrak{g}$ is nilpotent, then it admits an LCB structure $(J,g)$ if and only if it is isomorphic to one of the following:\n\\begin{itemize}\n\\setlength{\\itemindent}{-1em}\n\\item[] $(0,0,0,0,0,f^{12})$,\n\\item[] $(0,0,0,f^{12},f^{13},f^{14})$.\n\\end{itemize}\nIf $\\mathfrak{g}$ is non-nilpotent, then it admits an LCB structure $(J,g)$ if and only if it is isomorphic to one of the following:\n\\begin{itemize}\n\\setlength{\\itemindent}{-1em}\n\\item[] $\\mathfrak{l}_{1} \\hspace{1.35mm} =(f^{16},pf^{26},pf^{36},qf^{46},qf^{56},0)$, \\, $pr \\neq 0$, $p \\neq \\pm q$, \\smallskip\n\\item[] $\\mathfrak{l}_{2} \\hspace{1.35mm} =(f^{16},pf^{26}+f^{36},pf^{36},pf^{46}+f^{56},pf^{56},0)$, \\, $p \\neq 0$, \\smallskip\n\\item[] $\\mathfrak{l}_{3} \\hspace{1.35mm} =(pf^{16},qf^{26},qf^{36},rf^{46}+f^{56},-f^{46}+rf^{56},0)$, \\, $pq \\neq 0$, $q \\neq \\pm r$, \\smallskip\n\\item[] $\\mathfrak{l}_{4} \\hspace{1.35mm} =(pf^{16},qf^{26}+f^{36},-f^{26}+qf^{36},rf^{46}+sf^{56},-sf^{46}+rf^{56},0)$, \\, $pqs \\neq 0$, $q\\neq \\pm r$,\\smallskip\n\\item[] $\\mathfrak{l}_{5} \\hspace{1.35mm} =(pf^{16},qf^{26}+f^{36}-f^{46},-f^{26}+qf^{36}-f^{56},qf^{46}+f^{56},-f^{46}+qf^{56},0)$, \\, $pq \\neq 0$, \\smallskip\n\\item[] $\\mathfrak{l}_{6} \\hspace{1.35mm} =(f^{16},f^{26},0,0,0,0)$, \\smallskip\n\\item[] $\\mathfrak{l}_{7} \\hspace{1.35mm} = (f^{16},f^{26}+f^{36},f^{36},0,0,0)$, \\smallskip\n\\item[] $\\mathfrak{l}_{8} \\hspace{1.35mm} =(pf^{16}+f^{26},-f^{16}+pf^{26},0,0,0,0)$, \\, $p \\neq 0$, \\smallskip\n\\item[] $\\mathfrak{l}_{9} \\hspace{1.35mm} =(f^{16},pf^{26},pf^{36},0,0,0)$, \\, $p \\neq 0$,\\smallskip\n\\item[] $\\mathfrak{l}_{10} =(pf^{16},qf^{26}+f^{36},-f^{26}+qf^{36},0,0,0)$, \\, $pq \\neq 0$, \\smallskip\n\\item[] $\\mathfrak{l}_{11} =(f^{16},f^{26},pf^{36},pf^{46},0,0)$, \\, $p \\neq 0,\\pm1$,\\smallskip\n\\item[] $\\mathfrak{l}_{12} =(f^{16},f^{26},f^{46},0,0,0)$,\\smallskip\n\\item[] $\\mathfrak{l}_{13} =(f^{16},f^{26},qf^{36}+rf^{46},-rf^{36}+qf^{46},0,0)$, \\, $q \\neq \\pm 1$, $r \\neq 0$, \\smallskip\n\\item[] $\\mathfrak{l}_{14} =(pf^{16}+f^{26},-f^{16}+pf^{26},f^{46},0,0,0)$, \\smallskip\n\\item[] $\\mathfrak{l}_{15} =(f^{16}+f^{26},f^{26},f^{36}+f^{46},f^{46},0,0)$, \\smallskip\n\\item[] $\\mathfrak{l}_{16} =(pf^{16}+f^{26},-f^{16}+pf^{26},qf^{36}+rf^{46},-rf^{36}+qf^{46},0,0)$, \\, $r \\neq 0$, $p^2+q^2 \\neq 0$, $p \\neq \\pm q$, \\smallskip\n\\item[] $\\mathfrak{l}_{17} =(pf^{16}+f^{26}-f^{36},-f^{16}+pf^{26}-f^{46},pf^{36}+f^{46},-f^{36}+pf^{46},0,0)$, \\, $p \\neq 0$.\n\\end{itemize}\nAmong these, only $\\mathfrak{l}_1^{q=-\\frac{1}{2}-p}$, $\\mathfrak{l}_{2}^{p=-\\frac{1}{4}}$, $\\mathfrak{l}_{3}^{r=-\\frac{p}{2}-q}$, $\\mathfrak{l}_{4}^{r=-\\frac{p}{2}-q}$, $\\mathfrak{l}_{5}^{q=-\\frac{p}{4}}$, $\\mathfrak{l}_{9}^{p=-\\frac{1}{2}}$, $\\mathfrak{l}_{10}^{q=-\\frac{p}{2}}$ and $\\mathfrak{l}_{14}^{p=0}$ are unimodular. \\end{theorem}\n\\begin{proof}\nLet $(J,g)$ be an LCB structure on $\\mathfrak{g}$. As in Theorem \\ref{LCK_class}, we need to examine each matrix $A_i$ in \\eqref{A} to see whether they can satisfy the LCB condition $A_i^tv=0$ for some suitable metric and vector $v$. Of course $v=0$ is a sufficient condition and, in this case, after discarding the algebras admitting balanced or LCK structures (including the nilpotent $(0,0,0,0,f^{12},f^{13})$, which admits balanced structures, by \\cite{Uga}), we have that\n\\begin{itemize}\n\\item[] $A_1$ yields $\\mathfrak{l}_{1}$, $\\mathfrak{l}_{6}$, $\\mathfrak{l}_{9}$ and $\\mathfrak{l}_{11}$,\n\\item[] $A_2$ yields $\\mathfrak{l}_{3}$, $\\mathfrak{l}_{8}$, $\\mathfrak{l}_{10}$ and $\\mathfrak{l}_{13}$, \\item[] $A_3$ yields $\\mathfrak{l}_{4}$ and $\\mathfrak{l}_{16}$,\n\\item[] $A_4$ yields $\\mathfrak{l}_{2}$ and $\\mathfrak{l}_{15}$,\n\\item[] $A_5$ yields $\\mathfrak{l}_{5}$ and $\\mathfrak{l}_{17}$.\n\\end{itemize}\nTo complete the classification, we now assume $v \\neq 0$. Then $A^tv=0$ forces $A$ to be degenerate: we are then left with $A_1^{q=0}$, $A_2^{q=0}$ (both with $p$ possibly vanishing) and $A_4^{p=0}$. If $a=g([e_6,e_1],e_1)$ is not an eigenvalue of $A_i$, then $\\operatorname{im}(A-a\\,\\text{Id}_{\\mathfrak{n}_1})=\\mathfrak{n}_1$, so that $v=AX-aX$ for some $X \\in \\mathfrak{n}_1$ and the matrix $B$ corresponding to $\\text{ad}_{e_6}\\rvert_{\\mathfrak{n}}$ can be brought into the form\n\\[\nB=\\begin{pmatrix} a & 0 \\\\ 0 & A_i \\end{pmatrix},\n\\]\nsimply by replacing $e_1$ with $e_1^\\prime=e_1-X$, so that eventually we get some of the previously found Lie algebras. Otherwise, if $a$ is an eigenvalue of $A_i$, the algebraic multiplicity of $a$ as an eigenvalue of $B$ (namely, its multiplicity as a root of the characteristic polynomial) might exceed its algebraic multiplicity for $A_i$ by one: this happens exactly when $v \\notin \\operatorname{im}(A-a\\,\\text{Id}_{\\mathfrak{n}_1})$ and, in this case, $B$ is similar to a $5 \\times 5$ matrix obtained by taking $A_i$ and raising the rank of a Jordan block relative to the eigenvalue $a$ by one: this can occur for $A_1^{q=0}$, when $a=p$ or $a=0$, for $A_2^{q=0}$ when $a=0$ and for $A_4^{p=0}$, $a=0$.\n\nFor the cases $A_{i}^{q=0}$, $i=1,2$, with $a=0$, one can simply assume that the basis $\\{e_2,\\ldots,e_5\\}$ with respect to which $A_i$ is in the form \\eqref{A} is orthonormal, with $Je_2=e_3$, $Je_4=e_5$, and take $v=e_4$, for instance. \n\nFor $A_4^{p=0}$, assume again that $\\{e_2,e_3,e_4,e_5\\}$ is orthonormal, this time satisfying $Je_2=e_4$, $Je_3=e_5$ and take $v=e_3$, for example.\n\nFor the remaining case $A_1^{q=0}$, $a=p \\neq 0$, one can consider for example the Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ determined by the data\n\\[\na=p,\\quad v= \\left( \\begin{smallmatrix} 0\\\\0\\\\1\\\\0 \\end{smallmatrix} \\right), \\quad A=\\left( \\begin{smallmatrix} p&1&0&0\\\\0&0&0&0 \\\\ 0&0&0&0 \\\\ 0&0&1&p \\end{smallmatrix} \\right),\\,p \\neq 0,\n\\]\nwith respect to an adapted unitary basis $\\{e_1,\\ldots,e_6\\}$, $Je_i=e_{7-i}$, $i=1,2,3$. Then, it is easy to check that $A$ is similar to $A_1^{q=0}$ and that $A^tv=0$, $v \\notin \\operatorname{im}(A-p\\,\\text{Id}_{\\mathfrak{n}_1})$, so that the structure is LCB and the whole matrix $B$ is similar to\n\\[\n\\left(\\begin{smallmatrix} p&1&0&0&0 \\\\ 0&p&0&0&0 \\\\ 0&0&p&0&0 \\\\ 0&0&0&0&0 \\\\ 0&0&0&0&0 \\end{smallmatrix} \\right)\n\\]\nas desired.\n\nThese new cases with $v \\neq 0$ yield Lie algebras isomorphic to $\\mathfrak{l}_{7}$, $\\mathfrak{l}_{12}$, $\\mathfrak{l}_{14}$ or one of the two nilpotent Lie algebras of the statement, concluding the proof.\n\\end{proof}\n\n\n\\begin{remark}\nIt can be shown that, among the unimodular Lie groups whose Lie algebra appears in Theorem \\ref{LCB_class}, the ones with Lie algebra\n$\\mathfrak{l}_1^{q=-\\frac{1}{2}-p}$, $\\mathfrak{l}_{2}^{p=-\\frac{1}{4}}$, $\\mathfrak{l}_{3}^{r=-\\frac{p}{2}-q}$ and $\\mathfrak{l}_{9}^{p=-\\frac{1}{2}}$\ndo not admit any compact quotients by lattices.\n\nWe prove this only for $\\mathfrak{l}_1^{q=-\\frac{1}{2}-p}$, since the discussion for the other two Lie algebras is analogous.\nFollowing \\cite{Bock}, a co-compact lattice exists on such Lie groups if and only if there exists a non-zero $t_0 \\in \\mathbb{R}$ and a basis of $\\mathfrak{n}$ such that the matrix associated with $\\text{exp}(t_0\\text{ad}_{f_6})\\rvert_{\\mathfrak{n}}$ has integer entries. In the basis $\\{f_1,\\ldots,f_5\\}$ one easily computes \n\\begin{equation} \\label{exptad}\n\\text{exp}(t\\,\\text{ad}_{f_6})\\rvert_{\\mathfrak{n}}=\\text{diag}\\big(e^t,e^{pt},e^{pt},e^{-pt-\\frac{1}{2}t},e^{-pt-\\frac{1}{2}t}\\big).\n\\end{equation}\nIts minimal polynomial, namely the monic polynomial $P_t$ of least degree such that $P_t(\\text{exp}(t\\,\\text{ad}_{f_6})\\rvert_{\\mathfrak{n}})=0$, is of the form $P_t(x)=\\sum_{i=0}^3 a_i(t,p) x^i$, with coefficients\n\\[\na_0=-e^{\\frac{t}{2}},\\quad a_1=e^{t(1+p)}+e^{-\\frac{t}{2}}+e^{t\\left(\\frac{1}{2}-p\\right)},\\quad a_2=-e^{pt}-e^t-e^{-t\\left(\\frac{1}{2}+p\\right)},\\quad a_3=1.\n\\]\nIf \\eqref{exptad} is conjugate to an integer matrix for some $t_0$, then necessarily $P_{t_0}(x)$ is an integer polynomial, so that $a_0(t_0,p) \\in \\mathbb{Z}$ forces $t_0=2\\log k$, for some $k \\in \\mathbb{Z}_{>0}$. \nAssuming $a_2(t_0,p) \\in \\mathbb{Z}$, one computes\n\\[\nk^2\\left(k^2+a_2(t_0,p)\\right)+a_1(t_0,p)=\\tfrac{1}{k},\n\\]\nwhich is integer if and only if $k=1$, that is, $t_0=0$, a contradiction.\n\nInstead, for some choices of the parameters, the Lie groups with Lie algebra $\\mathfrak{l}_{4}^{r=-\\frac{p}{2}-q}$, $\\mathfrak{l}_{10}^{q=-\\frac{p}{2}}$ and $\\mathfrak{l}_{14}^{p=0}$ admit co-compact lattices (see \\cite{FP} and the references therein). Some results are known for the remaining Lie groups, namely the ones corresponding to $\\mathfrak{l}_{5}^{q=-\\frac{p}{4}}$ \\cm{$\\mathfrak{k}_{12}^{p,-\\frac{p}{4}}$} (see \\cite{CM}), but the existence of lattices on them is still an open problem.\n\\end{remark}\n\n\n\\section{Compatibility results between Hermitian metrics} \\label{sec_comp}\n\nIn this section, we ask whether a (unimodular) almost abelian Lie algebra endowed with a fixed complex structure may admit two different kinds of special Hermitian metrics.\n\nIn order to carry over the results to almost abelian solvmanifolds, we exploit the well-known ``symmetrization'' process. We summarize the results we need in the next lemma.\nRecall that a solvable Lie group is called \\emph{completely solvable} if all the eigenvalues of $\\text{ad}_X$ are real, for every $X$ in its Lie algebra.\n\\begin{lemma} {\\normalfont (\\cite{Bel, FG, Uga, Saw, AU})} \\label{symm}\nLet $\\Gamma \\backslash G$ be a compact solvmanifold endowed with a left-invariant complex structure $(J,g)$. Then, the existence of a balanced (resp.\\ SKT) metric implies the existence of a left-invariant balanced (resp.\\ SKT) metric.\nIf $G$ is completely solvable, the analogous results hold for LCK and LCB metrics.\n\\end{lemma}\n\n\\subsection{SKT and LCB} The SKT condition and the balanced condition are two ``transversal'' generalizations of the K\\\"ahler condition. Indeed, by \\cite{AI} a Hermitian metric which is both SKT and balanced is K\\\"ahler and it has been conjectured in \\cite{FV} that a compact complex manifold admitting an SKT metric and a balanced metric necessarily admits a K\\\"ahler metric as well. For almost abelian solvmanifolds, the conjecture was proven in \\cite{FP1}. \n\nThe same transversality no longer holds when considering the weaker LCB condition instead of the balanced condition, and the same Hermitian metric can even be SKT and LCB at the same time: in \\cite{FT1}, it was proven that\nevery non-K\\\"ahler compact homogeneous complex surface admits a compact torus bundle carrying an SKT and LCB metric; moreover, an example of compact nilmanifold in any even dimension admitting a left-invariant metric which is both SKT and LCB with respect to a fixed left-invariant complex structure was exhibited in \\cite{OOS}.\n\nIn addition, recalling that LCK metrics are particular instances of LCB metrics, it is easy to see that a non-K\\\"ahler LCK almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ is also SKT if and only if it satisfies \\eqref{LCK} with $a \\neq 0$ and $\\lambda=-\\frac{a}{2}$ or if $n=2$, $\\mathfrak{g} \\cong \\mathfrak{h}_3 \\oplus \\mathbb{R}$ or $\\mathfrak{g} \\cong \\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$ (cf. also \\cite{FP1}).\n\n\\begin{proposition}\nLet $\\mathfrak{g}$ be an almost abelian Lie algebra endowed with a complex structure $J$. If $(\\mathfrak{g},J)$ admits an SKT metric, then it admits an LCB metric as well.\n\\end{proposition} \n\\begin{proof} Let $g$ denote the SKT metric. By \\cite{AL}, with respect a unitary basis $\\{e_1,\\ldots,e_{2n}\\}$ of $\\mathfrak{g}$ adapted to the splitting $\\mathfrak{g}=J\\mathfrak{k} \\oplus \\mathfrak{n}_1 \\oplus \\mathfrak{k}$, the matrix $B$ associated with $\\text{ad}_{e_{2n}}\\rvert_{\\mathfrak{n}}$ is of the form \\eqref{B}, with $[A,A^t]=0$ and the eigenvalues of $A$ having real part equal to $-\\frac{a}{2}$ or $0$.\n\nDecompose $v \\in \\mathfrak{n}_1=\\operatorname{im}(A-a\\,\\text{Id}_{\\mathfrak{n}_1}) \\oplus \\left(\\operatorname{im}(A-a\\,\\text{Id}_{\\mathfrak{n}_1})\\right)^{\\perp_g}$ as $v=AX-aX+v^\\prime$ for some $X \\in \\mathfrak{n}_1$ and $v^\\prime \\in \\left(\\operatorname{im}(A-a\\,\\text{Id}_{\\mathfrak{n}_1})\\right)^{\\perp_g}$, that is, $(A-a\\,\\text{Id}_{\\mathfrak{n}_1})^tv^\\prime=0$.\n\nConsider the new $J$-Hermitian metric $g^\\prime=g\\rvert_{\\mathfrak{n}_1} + ({e^1}^\\prime)^2 + ({e^{2n}}^\\prime)^2$, with $e_1^\\prime=e_1-X$, $e_{2n}^\\prime=Je_1^\\prime$. Then, the matrix $B^\\prime$ associated with $\\text{ad}_{e_{2n}^\\prime}\\rvert_{\\mathfrak{n}}$ with respect to the new adapted unitary basis $\\{e_1^\\prime,e_2,\\ldots,e_{2n-1}\\}$ for $\\mathfrak{n}$ is of the form\n\\[\nB^\\prime=\\begin{pmatrix} a & 0 \\\\ v^\\prime & A \\end{pmatrix},\n\\]\nwith $A$ as above and $(A-a\\,\\text{Id}_{\\mathfrak{n}_1})^tv^\\prime=0$. If $a \\neq 0$, $a$ is not an eigenvalue of $A$, so that $v^\\prime=0$. Instead, if $a=0$, we have $A^tv^\\prime=0$. In either case, the metric $g^\\prime$ is LCB.\n\\end{proof}\n\nUsing Lemma \\ref{symm}, we get\n\\begin{corollary}\nLet $\\Gamma \\backslash G$ be a compact almost abelian solvmanifold endowed with a left-invariant complex structure $J$. If $(\\Gamma \\backslash G,J)$ admits an SKT metric, then it admits an LCB metric as well.\n\\end{corollary}\n\n\\begin{example} \\label{exampleSKTLCB}\nWe now exhibit an example of compact almost abelian solvmanifold in any even dimension admitting a (left-invariant) Hermitian structure which is at the same time SKT and LCB. For any $n \\geq 2$, consider the $2n$-dimensional simply connected unimodular almost abelian Lie group $S_{2n}$ having indecomposable Lie algebra $\\mathfrak{s}_{2n}$ endowed with a fixed coframe $\\{e^1,\\ldots,e^{2n}\\}$ satisfying the structure equations\n\\begin{gather*}\nde^1=a\\,e^1 \\wedge e^{2n},\\quad de^2=-\\tfrac{a}{2} \\, e^2 \\wedge e^{2n} + e^3 \\wedge e^{2n},\\quad de^3=- e^2 \\wedge e^{2n} - \\tfrac{a}{2} \\, e^3 \\wedge e^{2n}, \\quad de^{2n}=0, \\\\\nde^{2i}=c \\, e^{2i+1} \\wedge e^{2n},\\quad de^{2i+1}=-c \\, e^{2i} \\wedge e^{2n},\\quad i=2,\\ldots,n-1,\n\\end{gather*}\nfor some $a,c \\in \\mathbb{R} - \\{0\\}$, with $c$ depending on $a$ in a way which we shall explain. \nNow, it is easy to check that the left-invariant Hermitian structure $(J,g)$ on $S_{2n}$ defined by\n\\[\nJe_1=e_{2n}, \\quad Je_{2i}=e_{2i+1},\\,i=1,\\ldots,n-1, \\quad g=\\sum_{i=1}^{2n} (e^i)^2,\n\\]\nis both SKT and LCB, satisfying in particular $v=0$.\n\t\nAs we shall now show, $S_{2n}$ admits compact quotients by lattices, for all $n$, for some values of $a$ and $c$: by \\cite{Bock}, this is equivalent to proving that there exists $t_0 \\in \\mathbb{R}-\\{0\\}$ such that $\\text{exp}(t_0B_{2n})$ is similar to an integer matrix, $B_{2n}$ being the $(2n-1) \\times (2n-1)$ matrix representing $\\text{ad}_{e_{2n}}\\rvert_{\\mathfrak{n}}$ in the fixed basis. The claim is true for $n=2$ for countably many values of $a \\in \\mathbb{R}-\\{0\\}$, with compact quotients of $S_4$ biholomorphic to Inoue surfaces (see \\cite[Section 3.2.2]{AO} for a detailed discussion). Fixing $a \\in \\mathbb{R}$ such that $S_4$ admits co-compact lattices, let $t_0\\in \\mathbb{R}-\\{0\\}$ be such that $\\text{exp}(t_0B_{4})$ is similar to an integer matrix and set $c \\coloneqq \\tfrac{2\\pi}{t_0}$, so that\n\\[\n\\text{exp}\\left(t_0 \\left( \\begin{smallmatrix} 0 & c \\\\ -c & 0 \\end{smallmatrix} \\right)\\right)=\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}\n\\]\nis an integer matrix. The claim then easily follows in any dimension by induction.\n\\end{example}\n\n\\subsection{Balanced and LCK}\nBy \\cite{AO}, almost abelian Lie groups which admit left-invariant LCK structures and compact quotients by lattices only exist in real dimension four. The resulting solvmanifolds are biholomorphic to primary Kodaira surfaces, Inoue surfaces, hyperelliptic surfaces or complex tori: out of these, the only ones admitting K\\\"ahler metrics (recall that K\\\"ahler is equivalent to balanced, in real dimension four) are\ncomplex tori or hyperellyptic surfaces, which, by \\cite{HK}, cannot admit non-K\\\"ahler LCK metrics. Thus, we phrase the next result only in terms of structures on Lie algebras and not on compact almost abelian solvmanifolds, where the situation is already completely understood.\n\nIt was proven in \\cite{OOS} that a nilpotent Lie algebra cannot admit a balanced metric and a non-K\\\"ahler LCK metric both compatible with the same complex structure. In the almost abelian setting, the situation is analogous, apart from one exception in the non-unimodular case.\n\\begin{proposition} \\label{prop_balLCK}\n\tLet $\\mathfrak{g}$ be an almost abelian Lie algebra endowed with a complex structure $J$. If $(\\mathfrak{g},J)$ admits a balanced metric, then it does not admit any non-K\\\"ahler LCK metrics, unless $\\mathfrak{g} \\cong \\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$.\n\\end{proposition}\n\\begin{proof}\n\tAs we have recalled in Proposition \\ref{Herm-alm-ab}, an LCK almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ can either satisfy\n\t\\eqref{LCK} or $n=2$, $A=0$, which corresponds to $\\mathfrak{g} \\cong \\mathfrak{h}_3 \\oplus \\mathbb{R}$ (if $a=0$, $v \\neq 0$), $\\mathfrak{g} \\cong \\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$ (if $a \\neq 0$) or to $4\\mathbb{R}$ (if $a=0$, $v=0$).\n\t\n\tLet $g$ denote an LCK metric and assume \\eqref{LCK}. The result readily follows by observing that, in order to admit a balanced metric, $(\\mathfrak{g},J)$ must satisfy $\\operatorname{tr} A=\\operatorname{tr} \\text{ad}_X \\rvert_{\\mathfrak{n}_1}=0$ for all $X \\in \\mathfrak{g}$, so that $A \\in \\mathfrak{u}(\\mathfrak{n}_1,J_1,g)$. This implies that $g$ is K\\\"ahler.\n\t\n\tIf $n=2$ (in which case balanced implies K\\\"ahler) and we have $A=0$, note that $\\mathfrak{h}_3 \\oplus \\mathbb{R}$ does not admit K\\\"ahler structures, while all Hermitian structures on $4\\mathbb{R}$ are K\\\"ahler. On $\\mathfrak{aff}_2 \\oplus 2\\mathbb{R} = (f^{12},0,0,0)$, consider the complex structure defined by $Jf_1=f_2$, $Jf_3=f_4$. The Hermitian metric $g=\\sum_{i=1}^4 (f^i)^2$ is K\\\"ahler, while, denoting $f^i \\odot f^j=\\frac{1}{2}(f^i \\otimes f^j +f^j \\otimes f^i)$,\n\t\\[\n\tg^\\prime=2(f^1)^2+2(f^2)^2+(f^3)^2+(f^4)^2+2\\,f^1 \\odot f^3+2\\,f^2 \\odot f^4\n\t\\]\n\tis non-K\\\"ahler and LCK, with fundamental form $\\omega^\\prime=2f^{12}+e^{14}-e^{23}+e^{34}$ satisfying $d\\omega^\\prime=f^{124}=(f^2+f^4) \\wedge \\omega^\\prime$, $d(f^2+f^4)=0$.\n\\end{proof}\n\n\\subsection{Balanced and LCB}\nBalanced metrics are trivially LCB. One could ask whether there exist non-K\\\"ahler compact solvmanifolds endowed with a left-invariant complex structure admitting both balanced metrics and non-balanced LCB metrics. \n\nFor nilmanifolds, the answer is affirmative, as shown in \\cite{OOS}.\nAs a corollary of the next proposition, the analogous result is not true for completely solvable almost abelian solvmanifolds.\n\\begin{proposition} \\label{prop_balLCB}\nLet $\\mathfrak{g}$ be a unimodular almost abelian Lie algebra endowed with a complex structure $J$. If $(\\mathfrak{g},J)$ carries a balanced metric, then it cannot admit any non-balanced LCB metrics.\n\\end{proposition}\n\\begin{proof}\nBy the characterization of unimodular balanced almost abelian Lie algebras, we know that $[\\mathfrak{g},\\mathfrak{g}] \\subset \\mathfrak{n}_1$, since there exists an adapted unitary basis with respect to the balanced metric satisfying $a=0$, $v=0$. In particular, $\\text{rank}(\\text{ad}_X)=\\text{rank}(A)$ for all $X \\in \\mathfrak{g}$ transverse to $\\mathfrak{n}$. Assume a non-balanced LCB metric $g$ exists. Then, any adapted unitary basis for $(\\mathfrak{g},J,g)$ satisfies $a=0$, $\\operatorname{tr} A=0$, $A^tv=0$, with $v \\neq 0$ to ensure the metric is non-balanced. Now, this implies $\\text{rank}(\\text{ad}_X)=\\text{rank}(A)+1$ for all $X \\in \\mathfrak{g}$ transverse to $\\mathfrak{n}$, since $v \\notin \\operatorname{im} A$, a contradiction.\n\\end{proof}\n\nRecalling Lemma \\ref{symm}, we obtain\n\\begin{corollary}\nLet $\\Gamma \\backslash G$ be a completely solvable almost abelian solvmanifold endowed with a left-invariant complex structure $J$. If $(\\Gamma \\backslash G,J)$ carries a balanced metric, then it cannot admit any non-balanced LCB metrics.\n\\end{corollary}\n\n\\begin{remark}\nWe note that Proposition \\ref{prop_balLCB} is no longer true if one drops the hypothesis of unimodularity: this is clear from the example on the four-dimensional Lie algebra $\\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$ in the proof of Proposition \\ref{prop_balLCK}, recalling that LCK implies LCB. However, one can easily find other examples of complex structures of higher-dimensional almost abelian Lie algebras admitting both balanced and non-balanced LCB metrics: consider the six-dimensional almost abelian Lie algebra (see \\cite{FP1})\n\\[\n\\mathfrak{b}_2=(f^{16},f^{36},0,f^{56},0,0),\n\\]\nendowed with the complex structure defined by $Jf_1=f_6$, $Jf_2=f_4$, $Jf_3=f_5$. On it, one has the balanced metric $g=\\sum_{i=1}^6 (f^i)^2$ and the non-balanced and non-LCK LCB metric\n\\[\ng^\\prime=3(f^1)^2+(f^2)^2+(f^3)^2+(f^4)^2+(f^5)^2+3(f^6)^2+2(f^1 \\odot f^2 + f^1 \\odot f^3 + f^4 \\odot f^6 + f^5 \\odot f^6),\n\\]\nwhose associated Lee form is the closed $1$-form $\\theta^\\prime=f^5+f^6$, as shown by a direct computation.\n\\end{remark}\n\n\n\\section{Locally conformally hyperk\\\"ahler metrics} \\label{LCHK}\nWe now turn our attention to the study of (locally conformally) hyperk\\\"ahler metrics on almost abelian Lie algebras.\n\nIn the nilpotent setting, these structures were studied in \\cite{OOS}, where it was proven that compact nilmanifolds never admit left-invariant LCHK structures, unless they are tori.\n\nIn the next theorem, we classify almost abelian Lie algebras admitting LCHK structures. Recall that the spectrum of a matrix (or an endomorphism) $D$, denoted by $\\operatorname{Spec}(D)$, is the set of its eigenvalues. Given $z \\in \\mathbb{C}$, we denote by $m_D(z)$ its algebraic multiplicity for $D$, namely its multiplicity as a root of the characteristic polynomial of $D$. When $D$ is complex-diagonalizable, $m_D(z)$ is also equal to the (complex) dimension of the corresponding eigenspace.\n\n\\begin{theorem}\nA $4m$-dimensional almost abelian Lie algebra $\\mathfrak{g}=\\mathbb{R}^{4m-1} \\rtimes_D \\mathbb{R}$ admits an LCHK structure if and only if $D \\in \\mathfrak{gl}_{4m-1}$ is complex-diagonalizable and\n\\begin{itemize}\n\t\\item[\\normalfont (i)] $\\operatorname{Spec}(D) \\subset a + \\mathbb{R} i$, for some $a \\in \\mathbb{R}$,\n\t\\item[\\normalfont (ii)] $m_D(a) \\geq 3$,\n\t\\item[\\normalfont (iii)] $m_D(a+ib) \\in 2 \\mathbb{Z}$, for every $b \\in \\mathbb{R}-\\{0\\}$.\n\\end{itemize}\nThe Lie algebra $\\mathfrak{g}$ admits a hyperk\\\"ahler structure if and only if the above holds, with $a=0$, in which case $\\mathfrak{g}$ is unimodular and decomposable ($\\mathfrak{g}=\\mathfrak{g}^\\prime \\oplus (m_D(0))\\mathbb{R}$, with $\\mathfrak{g}^\\prime$ indecomposable) and every LCHK structure on $\\mathfrak{g}$ is hyperk\\\"ahler. In particular, there do not exist unimodular almost abelian Lie algebras admitting non-hyperk\\\"ahler LCHK structures.\n\\end{theorem}\n\\begin{proof}\nAssume $\\mathfrak{g}$ admits an LCHK structure $(I_1,I_2,I_3,g)$. In particular, $(I_1,g)$ is an LCK structure, so that there exists an adapted $(I_1,g)$-unitary basis $\\{e_1,\\ldots,e_{4m}\\}$ of $\\mathfrak{g}$ such that the matrix $B$ associated with $\\text{ad}_{e_{4m}}\\rvert_{\\mathfrak{n}}$ is of the form \\eqref{B} with the conditions \\eqref{LCK} or $m=1$, $A=0$, by \\cite{AO}. Note that, up to conjugation, we obtain the same $B$ when considering a basis adapted to $(I_2,g)$ or $(I_3,g)$.\n\nAssume \\eqref{LCK} holds. It follows that $B$ is complex-diagonalizable, hence $D$ is. Moreover, $\\operatorname{Spec}(B) \\subset \\{a\\} \\cup \\lambda + \\mathbb{R} i$. We claim that $\\lambda=a$: if this were not the case, in order for $(I_2,g)$ to be LCK, one should have that $I_1(e_{4m})=\\pm I_2(e_{4m})$, implying $I_3(e_{4m})=I_1I_2(e_{4m})= \\pm e_{4m}$, contradicting $I_3^2=-\\text{Id}_{\\mathfrak{g}}$. For the same reason, it follows that $m_{B}(a) \\geq 3$, to accommodate for the fact that $I_1(e_{4m})$, $I_2(e_{4m})$ and $I_3(e_{4m})$ should be eigenvectors for $B$ with real eigenvalue, hence equal to $a$.\n\nNow, denote by $V_z \\subset \\mathfrak{n} \\otimes \\mathbb{C}$ the eigenspace for $B$ corresponding to the eigenvalue $z \\in \\mathbb{C}$, and define \n\\[\n\\mathfrak{m} \\coloneqq \\mathfrak{n} \\cap I_1\\mathfrak{n} \\cap I_2 \\mathfrak{n} \\cap I_3 \\mathfrak{n}=\\text{span}\\left^{\\perp_g}.\n\\]\nWe note that $I_1$, $I_2$, $I_3$ must preserve $W_z \\coloneqq (V_{z}+\\overline{V_{z}}) \\cap \\mathfrak{m}$, for all $z \\in \\text{Spec}(B)$, since $\\text{ad}_{e_{4m}}\\rvert_{\\mathfrak{m}}$ commutes with the restriction of each of the three complex structures on $\\mathfrak{m}$. Note that, when $z$ is not real, we have that $W_z$ is the set of real elements of $V_{z}\\oplus \\overline{V_{z}}$. It follows that $W_z$ inherits a hyperhermitian structure, so that its real dimension is a multiple of four, which implies that the complex dimension of $V_z$ is a multiple of $2$, when $z$ is not real. Up to rescaling $B$ to recover $D$, points (i), (ii) and (iii) of the statement follow. \n\nAssume now that $B$ satisfies $A=0$, with $m=1$. In particular we have that $\\mathfrak{g}$ is four-dimensional, with $\\dim [\\mathfrak{g},\\mathfrak{g}]=1$, so, by \\cite[Proposition 3.2]{Bar}, $\\mathfrak{g}$ does not admit hypercomplex structures.\n\nConversely, assume $\\mathfrak{g}=\\mathbb{R}^{4m-1} \\rtimes_D \\mathbb{R}$ satisfies $D$ being complex-diagonalizable and requirements (i), (ii) and (iii). It follows that, up to a change of basis $\\{e_1,\\ldots,e_{4m-1}\\}$ of $\\mathbb{R}^{4m-1}$, $D$ is of the form\n\\begin{equation} \\label{hyperD}\nD=\\text{diag}\\left(C_{1},C_{2},\\ldots,C_{m-1},a,a,a\\right),\n\\end{equation}\nwhere, for $i=1,\\ldots,m-1$, $C_i$ is a $4 \\times 4$ matrix of the form\n\\[\nC_i= \\left(\\begin{smallmatrix} \na & b_i & 0 & 0 \\\\\n-b_i & a & 0 & 0 \\\\\n0 & 0 & a & -b_i \\\\\n0 & 0 & b_i & a\n\\end{smallmatrix}\\right),\n\\]\nfor some (possibly vanishing) $b_i \\in \\mathbb{R}$. Denoting by $e_{4m}$ the generator of the extra $\\mathbb{R}$, an explicit LCHK structure on $\\mathfrak{g}$ is given by $(I_1,I_2,I_3,g)$, with $g=\\sum_{i=1}^{4m} (e^i)^2$ and, with respect to the fixed basis, $I_i=\\text{diag}(K_i,\\ldots,K_i)$, $i=1,2,3$, with\n\\[\nK_1=\\left( \\begin{smallmatrix}\n\t0&-1&0&0\\\\\n\t1&0&0&0\\\\\n\t0&0&0&-1 \\\\\n\t0&0&1&0\n\\end{smallmatrix} \\right), \\quad\nK_2=\\left( \\begin{smallmatrix}\n\t0&0&-1&0\\\\\n\t0&0&0&1\\\\\n\t1&0&0&0 \\\\\n\t0&-1&0&0\n\\end{smallmatrix} \\right), \\quad\nK_3=\\left( \\begin{smallmatrix}\n\t0&0&0&-1\\\\\n\t0&0&-1&0\\\\\n\t0&1&0&0 \\\\\n\t1&0&0&0\n\\end{smallmatrix} \\right).\n\\]\nThe three induced Lee forms are all equal to the closed $1$-form $\\theta=-(4m-2)ae^{4m}$.\n\nThe part of the claim regarding hyperk\\\"ahler structures easily follows from the fact that the K\\\"ahler condition on almost abelian Lie algebras corresponds to \\eqref{LCK}, with $\\lambda=0$. In particular, in this case, we note that, if $D$ is of the form \\eqref{hyperD}, with $m_D(0)=3+4h$, we can assume $b_i=0$, $i=m-h,\\ldots,m-1$, so that $\\text{span}\\left$ is an abelian subalgebra of dimension $m_D(0)=3+4h$, while its complement, $\\text{span}\\left$, is an indecomposable almost abelian Lie algebra.\n\\end{proof}\n\nThe previous theorem can be used to get a more precise list of almost abelian Lie algebras admitting hyperk\\\"ahler or LCHK structures: in the next proposition, we cover dimensions $4$, $8$ and $12$.\n\n\\begin{proposition}\nLet $\\mathfrak{g}$ be a $4m$-dimensional almost abelian Lie algebra.\n\\begin{itemize}[leftmargin=3em, itemindent=-1em]\n\\item\nIf $m=1$, $\\mathfrak{g}$ admits a hyperk\\\"ahler structure if and only if $\\mathfrak{g}=4\\mathbb{R}$, while it admits a non-hyperk\\\"ahler LCHK structure if and only if it is isomorphic to $(f^{14},f^{24},f^{34},0)$. \\smallskip\n\\item\nIf $m=2$, $\\mathfrak{g}$ admits a hyperk\\\"ahler structure if and only if it is isomorphic to one among \\smallskip\n\\begin{itemize}\n\t\\setlength{\\itemindent}{-1em}\n\t\\item[] $8\\mathbb{R}=(0,0,0,0,0,0,0,0)$,\\smallskip\n\t\\item[] $(f^{28},-f^{18},f^{48},-f^{38},0,0,0,0)$, \\smallskip\n\\end{itemize}\nwhile it admits a non-hyperk\\\"ahler LCHK structure if and only if it is isomorphic to \\smallskip\n\\begin{itemize}\n\t\\setlength{\\itemindent}{-1em}\n\t\\item[] $(f^{18},f^{28},f^{38},f^{48},f^{58},f^{68},f^{78},0)$,\\smallskip\n\t\\item[] $(f^{18},f^{28},f^{38},f^{48}+pf^{58},-pf^{48}+f^{58},f^{68}+pf^{78},-pf^{68}+f^{78},0)$, \\,$p \\neq 0$.\n\\end{itemize}\\smallskip\n\\item\nIf $m=3$, $\\mathfrak{g}$ admits a hyperk\\\"ahler structure if and only if it is isomorphic to one among \\smallskip\n\\begin{itemize}\n\t\\setlength{\\itemindent}{-1em}\n\t\\item[] $12\\mathbb{R}=(0,0,0,0,0,0,0,0,0,0,0,0)$,\\smallskip\n\t\\item[] $(f^{2,12},-f^{1,12},f^{4,12},-f^{3,12},0,0,0,0,0,0,0,0)$,\\smallskip\n\t\\item[] $(f^{2,12},-f^{1,12},f^{4,12},-f^{3,12},pf^{6,12},-pf^{5,12},pf^{8,12},-pf^{7,12},0,0,0,0)$, \\,$p \\neq 0$, \\smallskip\n\\end{itemize}\nwhile it admits a non-hyperk\\\"ahler LCHK structure if and only if it is isomorphic to \\smallskip\n\\begin{itemize}\n\\setlength{\\itemindent}{-1em}\n\t\\item[] $(f^{1,12},f^{2,12},f^{3,12},f^{4,12},f^{5,12},f^{6,12},f^{7,12},f^{8,12},f^{9,12},f^{10,12},f^{11,12},0)$,\\smallskip\n \\item[] $(f^{1,12},f^{2,12},f^{3,12},f^{4,12},f^{5,12},f^{6,12},f^{7,12},f^{8,12}+pf^{9,12},-pf^{8,12}+f^{9,12},f^{10,12}+pf^{11,12},\\\\-pf^{10,12}+f^{11,12},0)$, \\,$p \\neq 0$,\\smallskip\n \\item[] $(f^{1,12},f^{2,12},f^{3,12},f^{4,12}+pf^{5,12},-pf^{4,12}+f^{5,12},f^{6,12}+pf^{7,12},-pf^{6,12}+f^{7,12},f^{8,12}+qf^{9,12},\\\\-qf^{8,12}+f^{9,12},f^{10,12}+qf^{11,12},-qf^{10,12}+f^{11,12},0)$, \\,$pq \\neq 0$.\n\\end{itemize}\n\\end{itemize}\n\\end{proposition}\n\n\\medskip \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nCryptocurrencies like Bitcoin \\cite{nakamoto2008bitcoin} have gained popularity as an alternative method of payment. Blockchain, a cryptographically secure, tamper proof ledger, forms the backbone of such decentralized network, guaranteeing pseudonymity of participant. The records stored in this distributed ledger can be verified by anyone in the network. Consensus algorithms like Proof-of-Work \\cite{nakamoto2008bitcoin}, \\cite{o2014bitcoin}, \\cite{bano2017consensus}, Proof-of-Stake \\cite{king2012ppcoin}, \\cite{li2017securing}) are used for reaching agreement on state change in the ledger across the network participants. However, computation time taken by such consensus algorithm is the major bottleneck in scalability of blockchain based transaction \\cite{croman2016scaling}, \\cite{poon2016bitcoin}. To be at par with traditional methods of payment like Visa, PayPal, scaling blockchain transactions is an important concern which needs to be addressed, without compromising on the privacy.\n\nSeveral solutions like sharding \\cite{luu2016secure}, \\cite{gencer2016service}, alternate consensus architecture \\cite{kiayias2017ouroboros}, \\cite{miller2014permacoin}, \\cite{buterin2017casper}, \\cite{park2018spacemint}, \\cite{eyal2016bitcoin}, \\cite{pass2017hybrid}, side-chains \\cite{back2014enabling} have been proposed in Layer-one. But this requires revamping the trust assumptions of the base layer and changing the codebase. A more modular approach is exploring scalability in Layer-two \\cite{gudgeon2019sok}. It massively cuts down data processing on the blockchain by running computations off-chain. The amount of data storage on Layer-one is minimized. Taking transactions off the base layer, while still anchored to it, would free up processing resources to do other things. Also Layer-two relies on Layer-one for security. Several solutions like \\cite{decker2018eltoo}, \\cite{decker2015fast}, \\cite{luu2016secure} have been proposed. \\textit{Payment Channel} \\cite{decker2015fast} stands out as a practically deployable answer to the scalability issue. \n\nAny two users, with mutual consent, can open a payment channel by locking their funds in a deposit. Users can perform several off-chain payments between each other without recording the same on blockchain. This is done by locally agreeing on the new deposit balance, enforced cryptographically by smart contracts \\cite{poon2016bitcoin}, key based locking \\cite{malavoltamulti} etc. Whenever one of the party wants to close the payment channel, it broadcasts the transaction on blockchain with the final balance. None of the parties can afford to cheat by claiming payment for an older transaction. Opening of new payment channel between parties which are not connected directly has its overhead where funds get locked for a substantial amount of time. This can be avoided by leveraging on the set of existing payment channels for executing a transaction, proving beneficial in terms of resource utilization. These set of payment channels form the \\textit{Payment Channel Network (PCN)}. Several problem such as routing, security and interoperability needs to be addressed in such a network.\n\n\n\n\n\n \nThe major challenge in designing any protocol for PCN is to ensure privacy of payer and payee and hiding the payment value transferred. No party, other than the payer and payee, should get any information about the transaction. Thus any routing algorithm designed for such a network must be decentralized, where individual nodes take decision based on the information received from its neighbourhood. Several distributed routing algorithms exists but they suffer from various disadvantages - Elias et al. \\cite{elias} requires a single node to maintain list of active vertices for executing push relabel algorithm on single source-sink pair, Flare \\cite{prihodko2016flare} requires intermediate users to reveal the current capacity of their payment channels to the sender for computation of the maximum possible value to be routed through a payment path, Canal \\cite{viswanath2012canal} entrusts a single node for computing maximum flow in a graph. Landmark-based routing algorithms \\cite{silentwhispers}, \\cite{speedymurmur} decide the number of landmarks by trial and error. If the total number of landmarks is $k$, then the payment value is split into $k$ microtransactions randomly without considering the nature of the graph. Such a myopic approach for routing each microtransaction may result in failure as it does not allow optimal utilization of the available capacities present across multiple paths.\n\nIt was first mentioned in Elias et al. \\cite{elias} that push relabel fits better as a routing algorithm for PCN because it proceeds locally, taking into account the residual capacity of each payment channel. However, the push relabel algorithm used for single source-sink pair \\cite{elias} is not decentralized in nature. A distributed version of the same was implemented in their paper for multiple source-sink pair but it is not well defined. It is not clear how many payment transfer can be allowed at a time through a channel. Further, it was assumed that each payment value for a source-sink pair is unsplittable. This assumption does not work in real life since the payment value might be higher than the bottleneck capacity of a single path. Deciding feasible routes even for a single payment transfer is an involved process in a distributed network. This motivated us to design a new routing algorithm for PCN which is privacy-preserving, efficient as well as scalable.\n\n\n\n\n\n\\vspace{-0.2cm}\n\\subsection{\\textbf{Our Contributions}}\nThe following contributions have been made in this paper :\n\\begin{itemize}[leftmargin=*]\n\n\\vspace{-0.1cm}\n\\item We have proposed a privacy preserving distributed routing algorithm, \\textit{HushRelay}, in payment channel network. \n\n\\item We have implemented the scheme and its performance has been compared with SpeedyMurmur \\cite{speedymurmur} in terms of \\textit{success ratio} and \\textit{time taken to route (TTR)} a payment. Testing was done on real instances of Ripple Network and Lightning Network \\footnote{In the absence of widespread PCN, we use the statistics of such real instances to create the network} and it is observed that \\emph{HushRelay} attains a success ratio of 1 in both the cases. However \\emph{SpeedyMurmur} attained a maximum success ratio of 0.9815 and 0.907 respectively, when number of landmarks is 6. The time taken to execute the routing algorithm in Ripple like Network and Lightning Network are 2.4s and 0.15189s for \\emph{HushRelay} but it takes 4.736s and 1.937s for \\emph{SpeedyMurmur}. These statistics justify our claim of the algorithm being efficient and scalable. The code is given in \\cite{Code}.\n\n\\item The proposed routing algorithm is modular and it can be combined with any other privacy preserving payment protocol.\n\\end{itemize}\n \n\\subsection{\\textbf{Organization}}\nSection \\ref{2} discusses the state-of-the-art in PCN. Section \\ref{3} gives a brief overview of the preliminaries. Section \\ref{4} defines the problem statement and Section \\ref{5} provides discusses \\emph{HushRelay} with Section \\ref{route} dealing with Generic Construction and Section \\ref{correct} providing the proof of correctness. Performance analysis of each subprotocol of \\textit{HushRelay} has been stated in Section \\ref{6} and Section \\ref{7} concludes the paper.\n\\vspace{-0.2cm}\n\\section{Related Work}\n\\label{2}\nA payment channel network is a peer-to-peer, path based transaction (PBT) network, where each party operates independent of other parties. Several P2P path-based transaction networks such as such as the Lightning Network for Bitcoin \\cite{poon2016bitcoin}, the Raiden Network for Ethereum \\cite{raiden}, SilentWhispers \\cite{silentwhispers}, InterLedger \\cite{thomas2015protocol}, Atomic-swap \\cite{atomic}, TeeChain \\cite{lind2017teechain} etc. have been developed over the years. Perun \\cite{dziembowski2017perun} proposes a more efficient network structure which is built around payment hubs. An extension of payment channel, State Channel Network \\cite{dziembowski2018general}, not only supports off chain payment but allows execution of complex smart contract. Spider network \\cite{sivaraman2018routing} adheres to a packet-switched architecture for payment channel network. Payment is split into several transaction units and it is transmitted over time across different paths. However the split does not take into account the bottleneck capacity of each path which might lead to failure of payment. BlAnC \\cite{panwar2019blanc}, a fully decentralized blockchain-based network, has been proposed which transfers credit between a sender and receiver on demand. \n\nTill date, the routing algorithms proposed for payment channel network are as follows : Canal \\cite{viswanath2012canal} - uses a centralized server for computing the path, Flare \\cite{prihodko2016flare} - requires intermediate nodes to inform source node about their residual capacity, SilentWhispers \\cite{silentwhispers} - a distributed PBT network without using any public ledger, SpeedyMurmur \\cite{speedymurmur} - a privacy preserving embedded based routing, extending Voute \\cite{roos2016voute}, depending on presence of landmark nodes. SpeedyMurmur is the most relevant privacy preserving distributed routing algorithm. However, it makes use of repeated trials to figure out a suitable split of the total transaction value across multiple paths. Elias et al. \\cite{elias} proposed an extended push relabel for finding payment flow in the payment network. They are the first to point out the flaw in assumption of considering transaction \\textit{unsplittable} for existing routing techniques. In real life, splitting of fund across multiple path is inevitable since the bottleneck capacity of a single path may be lower than the total value of fund transfer. Later, a distributed approach for PCN routing, CoinExpress \\cite{yu2018coinexpress}, was proposed for finding routes that fulfill payment with higher success ratio. A routing algorithm based on swarm intelligence, ant colony optimization \\cite{grunspan2018ant} has been explored. Hoenisch et al. \\cite{hoenisch2018aodv} proposed an adaptation of an Ad-hoc On-demand Distance Vector (AODV)-based routing algorithm which supports different cryptocurrencies allowing transactions across multiple blockchains. We observe that none of the past works provide an efficient and secure routing algorithm. It is either susceptible to leaking of sensitive information or there exist a central entity controlling the routing algorithm.\n\n\n\n\n\n\\section{Background}\n\\label{3}\nIn this section, we provide the required background on payment channel network. \nThe terms source\/payer means the sender node. Similarly, sink\/payee\/destination means the receiver node and transaction means payment transfer.\n\n\\subsection{\\textbf{Payment Channel Network}}\n\\begin{definition}\n\\label{basic}\nA Payment Channel Network (PCN) \\cite{malavolta} is defined as a bidirected graph $G:=(V,E)$, where $V$ is the set of accounts dealing with cryptocurrency and $E$ is the set of payment channels opened between a pair of accounts. A PCN is defined with respect to a blockchain. Only opening and closing of payment channel gets recorded on blockchain apart from disputed transactions where settlement is done by broadcasting the transaction on blockchain.\n\\end{definition}\n\nBasic operations of PCN \\cite{malavolta}- \n\\begin{itemize}[leftmargin=*]\n\\item \\texttt{openPaymentChannel$(v_1,v_2,\\alpha,t,m)$} : For a given pair of accounts $v_1,v_2 \\in V$, channel capacity $\\alpha$ (initial balance escrowed), timeout value of $t$ and processing fee charged $m$, \\texttt{openPaymentChannel} creates a new payment channel $(id_{(v_1,v_2)},\\alpha,t,m) \\in E$, where $id_{(v_1,v_2)}$ is the channel identifier, provided both $v_1$ and $v_2$ has authorized to do so and the funds contributed by each of them sum up to value $\\alpha$. \n\n\\item \\texttt{closePaymentChannel$(id_{(v_1,v_2)},\\tilde{\\alpha})$} : Given a channel identifier $id_{(v_1,v_2)}$ with balance $\\tilde{\\alpha}$, \\texttt{closePaymentChannel} removes the channel from $G$ provided it is authorized to do so by both $v_1,v_2 \\in V$. The balance $\\tilde{\\alpha}$ gets written on blockchain and this amount is distributed between $v_1$ and $v_2$ as per the net balance recorded.\n\n\\item \\texttt{payVal$(p(s,r),val)$} : $p(s,r)$ denotes a path between sender $s$ and receiver $r$. It is defined by a set of identifiers $id_{(s,v_1)},id_{(v_1,v_2)},\\ldots,id_{(v_n,r)}$, $s,v_1,v_2,\\ldots,v_n,r \\in V$, having enough credit to allow transfer of $val$ from $s$ to $r$, if for each payment channel denoted by $id_{(v_i,v_{i+1})}$ has capacity of at least $\\beta \\geq val_i', val_i'=val+\\Sigma_{j=i+1}^{n} fee(v_j), 0 \\leq i \\leq n, v_0=s \\ and \\ v_{n+1}=r$, where $fee(v_j)$ is the processing fee charged by each intermediate node $v_j$ in $p(s,r)$. A successful \\texttt{payVal} operation leads to a decrease of capacity of each payment channel $id_{(v_i,v_{i+1})}$ by $val_i'$. Else the capacity of the channel remains unaltered. \n\\end{itemize} \n\n\n\\subsection{\\textbf{Payment Flow problem}}\n\\label{flowpay}\nConsider a directed graph $G:=(V,E) : \\ n=|V|, m=|E|, m \\geq n-1$, having two distinguished vertices, source $s \\in V$, sink $r \\in V, s \\neq r$, as a \\textit{flow network.} For a pair of vertices $v,w$, distance from $v$ to $w$ in graph $G$ is defined by $d_G(v,w)$, the minimum number of edges on the path from $v$ to $w$; if there is no path from $v$ to $w$, $d_G(v,w)=\\infty$. A positive real-valued capacity $c(v,w)$, defined by $c : E \\rightarrow \\mathbb{R}$, is the amount of funds that can be\ntransferred between two nodes sharing an edge. For every edge $(v,w) \\in E$ ; if $(v,w) \\not \\in E$, then $c(v,w)=0$. A flow $f$ on $G$ is a real-valued function on vertex pairs satisfying the constraints \\cite{tarjan}, \\cite{thuy2005distributed} :\n\\begin{equation}\n\\label{eq1}\n\\begin{matrix}\nf(v,w) \\leq c(v,w), \\ \\forall (v,w) \\in V \\times V \\ \\textrm{(capacity)}, \\\\\nf(v,w)=-f(w,v), \\ \\forall (v,w) \\in V \\times V \\ \\textrm{(antisymmetry)}, \\\\\n\\Sigma_{u \\in V} f(u,v)=0 \\ \\forall v \\in V-\\{s,r\\} \\ \\textrm{(flow-conservation)}, \\\\\n\\end{matrix}\n\\end{equation}\nThe net flow into the sink is given by $f$, where:\n\\begin{equation}\nf=\\Sigma_{v \\in V} f(v,r)\n\\end{equation}\nA payment channel network can be mapped to flow network with channels forming the edges and funds locked on each channel can be considered as the edge capacity. Finding the maximum flow value from source to sink for a flow network is termed as the \\textit{Maximum Flow problem}. In the context of PCN, given a payment value $val$, one has to find a feasible flow from payer to payee, which is termed here as \\textit{Payment Flow problem}. Any max-flow algorithm with subtle modifications can be applied here, taking into account the preflow $f$ of each vertex (except the source and sink) on the network. A preflow is a real-valued function on a vertex pair which satisfies the first two constraints of Eq. \\ref{eq1} and a weaker form of the third constraint :\n\\begin{equation}\n\\Sigma_{u \\in V} f(u,v)\\geq 0, \\ \\forall v \\in V-\\{s,r\\} \\ \\textrm{(non-negativity constraint)},\n\\end{equation}\n\nA residual capacity of an edge $(v,w) \\in E$ is the amount of capacity remaining after the preflow $f$, i.e. $c(v,w)-f(v,w)$ and it is denoted by $r_f(v,w)$. A residual graph $G_f=(V,E_f)$ for a preflow $f$ is the graph whose vertex set is $V$ and edge set $E_f$ is the set of residual edges $(v,w) \\in E \\ : \\ r_f(v,w)>0$. \nThe \\textit{flow excess e(v)} of a vertex $v$ is the net balance of funds in node $v$ denoted by $\\Sigma_{u \\in V} f(u,v)$. The algorithm ends with all vertices except $s$ and $r$ having zero excess flow. If sink is unreachable or if the network does not have adequate capacity for transferring the amount $val$, then the excess value is pushed back to source $s$. \n\n\n\n \n\n\n\n\n\\section{Problem Statement}\n\\label{4}\nIt is not always possible to route the transaction across a single path as the value may be quite high compared to minimum capacity of the designated path. Hence it is better to find set of paths such that the total amount to be transferred is split across each such path. We define the problem as follows -\n\\begin{problem}\n\\textit{Given a payment channel network $G(V,E)$, a transaction request ($s,r,val$) for a source-sink pair $(s,r)$, the objective is to find a set of paths $p_1,p_2,\\ldots,p_m$ for transferring the fund from $s$ to $r$ such that $p_1$ transfers $val_1$, $p_2$ transfers $val_2,\\ldots,$ $p_m$ transfers $val_m : val=\\Sigma_{i=1}^m val_i$ without violating \\textit{transaction level privacy} i.e. neither the sender nor the receiver of a particular transaction must be identified as well as hiding the actual transaction value from intermediate parties.}\n\\end{problem}\n\n\n\\section{Our Proposed Construction}\n\\label{5}\nIn this section we provide a detailed overview of the routing algorithm, \\textit{HushRelay}. The payment network comprises set of payment channels denoted by channel identifier $id_{(i,j)}, (i,j)\\in E$.\nWe describe state the model and the assumptions made. \n\n\\subsection*{\\textbf{Network Model and its Assumptions}}\n\\begin{itemize}[leftmargin=*]\n \\item The network is static i.e. no opening of new payment channel or closing of existing payment channel is considered.\n \\item The topology of the network is known by any node in the network since any opening or closing of channel is recorded on the blockchain.\n \\item Atmost one timelock contract is allowed to be established on a payment channel at a time.\n \\item Sender of a payment chooses set of paths to the receiver according to her own criteria.\n \\item The current value on each payment channel is not published but instead kept locally by the users sharing a payment channel. \n \\item Pairs of users sharing a payment channel communicate through secure and authenticated channels\n(such as TLS).\n\n \n\\end{itemize} \n\n\\subsection{Generic Algorithm}\n\\label{route}\nSince we consider PCN as a flow network, for solving the \\textit{payment flow problem} in the given network for executing a transaction request $(s,r,val)$, we propose a routing algorithm inspired from distributed push relabel algorithm stated in \\cite{tarjan}, \\cite{thuy2005distributed}. The algorithm proceeds locally by exchange of messages between neighbouring nodes. No single entity controls the flow in the network.\n\nBefore discussing the algorithm, we briefly describe the \\textit{Push Relabel} algorithm for a single source-sink pair (as stated in \\cite{tarjan}):\n\\begin{itemize}[leftmargin=*]\n\\item The instruction \\textit{push} redirects the excess flow of a vertex to the sink via its neighbouring vertices. The amount of excess flow that can be \\textit{pushed} from a vertex $v$ to one of its neighbouring vertex $w$ is $\\delta=min(e(v),r_{f(v,w)})$, where $r_{f(v,w)}$ is the residual capacity of edge $(v,w)$. The value $\\delta$ is added to the preflow value $f(v,w)$ (subtracted from $f(w,v)$) and subtracted from $e(v)$. Any push which results in zero residual capacity of the edge is said to be \\textit{saturating}. \n\n\\item A valid labeling function $d : V \\rightarrow I^{+}\\cup\\{0,\\infty\\}$ is used for estimating the distance of a vertex $v$ from sink $r$. $d(s)=n, d(r)=0$ and $d(v)\\leq d(w)+1$ for every residual edge $(v,w)$. The label $d(v) < n$ forms the lower bound on the actual distance from $v$ to $r$ in the residual graph $G_f$ and if $d(v)\\geq n$, then $d(v)-n$ is a lower bound on the actual distance from $r$ in the residual graph. \n\\item A relabeling operation is initiated when a vertex with excess flow has a label less than or equal to that of the neighbouring vertex. Once relabeling is done, it can initiate a push operation. So one can think labels to denote the potential level, where flow can occur from a region of higher potential to a region of lower potential.\n\n\\item A vertex $v$ is defined as active $v \\in V - \\{s,r\\}, d(v) < \\infty$, and $e(v)>0$. The maximum-flow algorithm is initialized with preflow value $f$, which is summation of the edge capacities of all edges incident from the source vertex $s$ and rest all other edges have zero flow. \n\\end{itemize}\n\nFor our distributed algorithm, \\textit{HushRelay}, the basic operations is \\textit{Push} and \\textit{Relabel}, with all the nodes acting as individual processing unit in parallel. The network model considered for payment channel network is asynchronous. \nSynchronization across all the nodes is achieved via use of \\textit{acknowledgements} \\cite{tarjan}. A vertex $v$ tries to push excess flow to one of its neighbouring vertex $w$ if and only if, as per the information maintained by $v$, label $d(v)=d(w)+1$. It first sends a request message with the information $(v,\\delta,d(v),e(v))$. Vertex $w$ can either accept the push by sending an acknowledgement or it may reject it by sending a negative acknowledgement $(NAK)$. If $d(v)=d(w)+1$, then $w$ sends to $v$ a message of the form $(accept,w,\\delta,d(w))$ and $v$ initiates the push. Otherwise, if $d(w)\\geq d(v)$ or $d(v) < d(w)+1$, then it sends a message $(reject,w,\\delta,d(w))$ where $d(w)$ is the updated distance label of $w$. A reject message will cause $v$ to update the value of $d(w)$. When a distance label of the vertex increases, it sends the information of new label to all its neighbouring nodes. \n\nAs seen in \\textit{Push Relabel} algorithm \\cite{tarjan}, the label initially set for source and sink node reveals the identity of payer and payee. To obfuscate their identity from other intermediate nodes in the network, we use a dummy source vertex $s'$ for node $s$ and a dummy sink vertex $r'$ for node $r$. Note that the existence of dummy node is known only by the source and sink. In the initialization phase of \\textit{HushRelay}, a directed virtual edge from $s'$ to $s$ and from $r$ to $r'$ is established. Since $s',r'$ are virtual entities, introduction of edge $(s',s)$ and $(r,r')$ is not recorded in the blockchain. The capacity is initialized to $c(s',s)=val, c(r,r')=val$ and the label is set as $d(s')=n+2, d(s)=0, d(r)=0$ and $d(r')=0$. The flow $f(s',s)$ is set to $val$, $f(r,r')=0$, excess flow $e(s)=val$, $e(r)=e(r')=0$. For all vertices $v \\in V-\\{s,r\\}$, $d(v)=0,e(v)=0$, $f(w,v)=0, (w,v) \\in E, w,v \\in V-\\{s,r\\}$. We mention the procedure of \\textit{Push}, \\textit{Push-request} and \\textit{Relabel} for a vertex in Procedure \\ref{algo:v4}, \\ref{algo:vpr} and \\ref{algo:v5} respectively.\n\\begin{proc}[!ht]\n \\SetKwInOut{Input}{Input}\n \\SetKwInOut{Output}{Output}\n\n \\Input{ Active vertex $v \\in V, e(v)>0$, vertices $w: (v,w) \\in E$ }\n \n \\caption{Push(v,w,d)}\n \\label{algo:v4}\n \\If{ $v\\neq r'$ and $v \\neq s'$}\n {\n Set $find\\_neighbour$=0 \\\\\n \\While{neighbour $w$ of $v$ : $e(v)>0, r_f(v,w)>0 \\ and \\ d(w) < d(v)$}\n {\n \t$v$ generates a push of the value $\\delta=min(e(v),r_f(v,w)$ \\\\\n \t$f(v,w)=f(v,w)+\\delta$\\\\\n \t$e(v)=e(v)-\\delta$\\\\\n \t$r_f(v,w)=c(v,w)-f(v,w)$\\\\\n \n \tSend \\textit{Push-request(w,v,$f(v,w),\\delta$)} to node $w$ \\\\\n \t\\If{\\textit{NAK} received}\n \t{\n \tUpdate information $d(w)$ \\\\\n \t$f(v,w)=f(v,w)-\\delta$\\\\\n \t$e(v)=e(v)+\\delta$\\\\\n \t$r_f(v,w)=c(v,w)+f(v,w)$\\\\\n \t}\n \t\\Else\n \t{\n \t Set $find\\_neighbour=1$\\\\\n \t \n \t}\n \n }\n \\If{$find\\_neighbour=0$}\n {\n Call \\textit{Relabel} function.\n }\n }\n\\end{proc}\n\n\n\n\n\\begin{proc}[!ht]\n \\SetKwInOut{Input}{Input}\n \\SetKwInOut{Output}{Output}\n\n \\Input{ Active vertex $w \\in V, e(w)>0$, vertex $v: (w,v) \\in E$ }\n \n \\caption{Push-request(v,w,$f(w,v)$,$\\delta$)}\n \\label{algo:vpr}\n \\If{ $d(v)0$, vertices $w: (v,w) \\in E, r_f(v,w)>0, d(v)\\leq d(w)$ }\n \n \\caption{Relabel(v,w,d)}\n \\label{algo:v5}\n \\begin{enumerate}\n \\item Update $d(v)=min(d(w) ,(v,w) \\in E) +1$\n \\item Inform all the neighbours of vertex $v$ about the updated label $d(v)$.\n \\end{enumerate}\n\\end{proc} \n\n \n\n\n\nThe algorithm terminates when there are no active vertex left (except the dummy source and dummy sink) in the graph. The number of messages exchanged (\\textit{for push request, push accepted\/NAK, height updation}) is also bounded. The communication complexity, termination condition of the algorithm and an upper bound on the $d$ value of a node in the given graph is stated in \\cite{thuy2005distributed}. $\\mathcal{O}(n^2m)$ messages are exchanged in the asynchronous implementation with $\\mathcal{O}(n^2)$ runtime. The overhead lies in the interprocessor communication between a vertex and its neighbours.\n\n\\begin{example}\nConsider a network given in Fig. \\ref{fig:example1}. Sender $S$ intends to make a payment of 15 units to receiver R. Dummy vertices $S'$ and $R'$ is added to the network with edges $(S',S)$ and $(R,R')$. The edge capacities are as follows : $c(S',S)=15, c(S,A)=10, c(S,B)=10, c(A,C)=10, c(B,C)=15, c(C,R)=20$ and $c(R,R')=15$. \\textit{HushRelay} is implemented on the network to obtain a feasible flow of value 15 from source node S to sink node R. The initial state is given in Fig. \\ref{fig:example1} (a). \n\\begin{figure}[!ht]%\n \\centering\n \\label{img1}\n\n \\subfloat[Initial state]{{\\includegraphics[width=6.3cm]{fig1} }}%\n \\end{figure}\n\nIn the initialization phase, each nodes is assigned a label of 0 except dummy vertex $S'$ where d(S') is the count of the number of nodes (except S') in the network. As given in Fig. \\ref{fig:example1} (b), S' sends a push request of 15 units to S. \n\n\\begin{figure}[!ht]\\ContinuedFloat\n\n \\centering \n \\subfloat[state]{{\\includegraphics[width=6.3cm]{fig02} }}%\n \\end{figure}\n In Fig. \\ref{fig:example1} (c), S accepts the push request as $d(S)0$ is an integral value, since our transaction failed. If there exists no more augmenting paths, then $f_m = \\tilde{d} - \\gamma$, which implies $f_m < \\tilde{d}$. This contradicts the fact that $f_m > \\tilde{d}$. Hence our assumption was wrong. \n\n\\begin{lemma}\n\\label{lm2}\nFor $ v \\in V\\setminus \\{s',t'\\}, e(v)=0$ on termination.\n\\end{lemma}\n\n\n\\textbf{Proof of Lemma \\ref{lm2}}. Assume that there exist one vertex $\\hat{v} \\in V\\setminus \\{s',t'\\} : e(\\hat{v})>0$ after termination. But since termination condition has been reached, it means there vertex $s$ is not reachable from this vertex $\\hat{v}$. Let set of vertices not reachable from $\\hat{v}$ be denoted by $V'$ and those reachable from $\\hat{v}$ be $V-V'$.\n\\begin{equation}\n\\begin{matrix}\n\n\ne(v)=\\Sigma_{k \\in V, (v,k) \\in E} f(k,v) \\\\\n = \\Sigma_{k \\in V',(v,k) \\in E} f(k,v) + \\Sigma_{k \\in V-V',(v,k) \\in E} f(k,v) \\\\\n = \\Sigma_{k \\in V',(v,k) \\in E} f(k,v) \\\\ (\\because \\Sigma_{k \\in V-V',(v,k) \\in E} f(k,v)=0 , \\textrm{flow conservation constraint})\n\\end{matrix}\n\\end{equation} \nBut since $e(v)>0$, then $\\Sigma_{k \\in V',(v,k) \\in E} f(k,v)>0$, which means there still exists some augmenting path from $s$ to $v$ ($s \\in V'$). Hence it contradicts the assumption of termination. \n\n\\begin{lemma}\n\\label{lm3}\nFor all edges $(v,w) \\in E, v,w \\in V$, $f(v,w)\\leq c(v,w)$.\n\\end{lemma}\n\n\\textbf{Proof of Lemma \\ref{lm3}}. In the algorithm \\textit{Push}, the flow value $\\delta$ for a given edge $(v,w) \\in E$, the flow value $f(v,w)$ from vertex $v$ to vertex $w$ is decided by $\\min(e(v),r_f(v,w))$. Since $r_f(v,w)=c(v,w)-f(v,w)$, $f(v,w)\\leq c(v,w)$ and $e(v) \\leq \\tilde{d}$. Flow value will be bounded by $\\tilde{d}$, if $\\tilde{d} < c(v,w)$ or $c(v,w)$ otherwise. \n\n\n\n\\subsubsection{\\textbf{Propagating the flow information to source node}}\n\\label{hide}\nEach edge $e \\in E$ involved in transfer of payment from source to sink will generate temporary key $k_e$ for encrypting the flow message to be propagated back to the source node. The sink node, on termination, generates a key $k_{sink}$ as well some random message $rm$, equivalent to size of the packet or its multiple. Each such packet contains flow information which is shared with a predecessor node. It constructs a message $m'$ containing the information of identity of preceding vertex $w$, the non negative flow $f_{wv}>0$ along with key $k_{wv}$. It encrypts the packet with $k_{sink}$, $E'=Enc_{k_{sink}}(w,f_{wv},k_{wv})$, and concatenates the randomly generated message $rm$ with the encrypted packet to construct message $E'||rm$. It shares this information with $w$. If $w$ is honest, it will construct a similar message $m'$, for its neighbour say $u$ containing the identity of $u$, flow $f_{uw}$, key $k_{uw}$. It is encrypted with $k_{wv}$ to get $E''$. The encrypted message is concatenated with the message received from its successor i.e. $E''||E'||rm$. This continues till all the packets reach the source vertex $s$. The sink vertex shares $k_{sink}$ and set of randomly generated message $rm$ with source vertex $s$ via secure communication channel. $s$ discards $rm$ from the received message and starts decrypting, beginning with the message encrypted by sink. On decryption, it retrieves the flow information, identity of vertex and key with which it will decrypt the next encrypted packet. All duplicate information on flow is discarded and the remaining one is used for reconstructing the flow across the network. This is the routing information denoted by $\\mathcal{P}$.\n\n\n\n \\begin{table*}[!ht]\n\\begin{center}\n\n\n \\caption{SpeedyMurmur vs HushRelay - Performance Analysis on Real Instances }\n \\label{tab:1}\n \\scalebox{0.9}[0.9]{\n\n\n \\begin{tabular}{|p{3cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|} \n \\hline\n\\multirow{4}{*}{Network\/Algorithm} &\n \\multicolumn{8}{c|}{SpeedyMurmur} &\n \\multicolumn{2}{c|}{HushRelay} \\\\\n \\cline{2-11}\n \n & \\multicolumn{4}{c|}{Success Ratio} &\\multicolumn{4}{c|}{Time taken}\n & Success Ratio &Time taken\\\\\n\n &\n \\multicolumn{4}{c|}{Number of Landmarks} &\n \\multicolumn{4}{c|}{Number of Landmarks} & & \\\\\n \\cline{2-9}\n &1 &2 &4 &6 &1 &2&4 &6 & &\\\\\n\\hline\nRipple Network &0.38 &0.7 &0.92 &0.98 &1.66s &2.2s &3.23s &4.74s &1 &2.4s\\\\\n\\hline\nLightning Network &0.42 &0.65 &0.83 &0.91 &0.61s &0.69s &0.83s &1.94s &0.99 &0.15s\\\\\n\\hline\n \\end{tabular}\n\n}\n\\end{center}\n\\end{table*}\n\n\n\\section{Performance Analysis of \\textit{HushRelay}}\n\\label{6}\n\\subsubsection*{Experimental Setup}\nIn this section, we define the experimental setup. The code for \\textit{HushRelay} is available in \\cite{Code}. System configuration used is : \\texttt{Intel Core i5-8250U CPU, Kabylake GT2 octa core processor, frequency 1.60 GHz}, OS : \\textit{Ubuntu-18.04.1 LTS} (64 bit). The programming language used is C, compiler - gcc version 5.4.0 20160609. The library \\textit{igraph} was used for generating random graphs of size ranging from 50 to 25000, based on Bar\\'{a}basi-Albert model \\cite{albert2002statistical}, \\cite{barabasi2003scale}. Payment Channel Network follows the scale free network where certain nodes function as hub (like central banks), having higher degree compared to other nodes \\cite{javarone2018bitcoin}. For implementing the cryptographic primitives, we use the library \\textit{Libgcrypt}, version-1.8.4 \\cite{libgcrypt}, which is based on code from GnuPG. \n\\begin{figure}[h]\n\\centering\n\\subfloat[Success Ratio vs Number of Nodes]{{\\includegraphics[width=8cm]{success1}}}%\n\\qquad\n\\subfloat[Time To Route vs Number of Nodes]{{\\includegraphics[width=8cm]{time1} }\n\\qquad\n\\caption{Analysis of \\textit{HushRelay} and \\textit{SpeedyMurmur}}%\n\\label{fig:example2}%\n\\end{figure}\n\\subsubsection*{Evaluation}\nFollowing metrics are used to compare the performance of the routing algorithm, \\textit{HushRelay} with \\textit{SpeedyMurmur} \\cite{speedymurmur} \n\\begin{itemize}[leftmargin=*]\n\\item Success Ratio : It is the ratio of number of successful payment to the total number of payment transfer request submitted in an epoch. \n\\item TTR \\textit{(Time Taken to Route)} : Given a payment transfer request, it is the time taken from the start of routing protocol till its completion (returning the set of feasible paths).\n\\end{itemize}\nWe allow just one trial (i.e. $a=1$) of \\textit{SpeedyMurmur} since \\textit{HushRelay} executes just once. The number of landmarks is varied as 1,2,4 and 6.\n\n\\begin{itemize}[leftmargin=*]\n\\item Real Instances - \\emph{HushRelay} and \\emph{SpeedyMurmur} has been executed on real instances - Ripple Network \\cite{malavolta}, Lightning Network \\cite{seres2019topological}. The results are tabulated in Table \\ref{tab:1}.\n\\item Simulated Instances - The capacity of each payment channel is set between 20 to 100 and each transaction value ranges from 10 to 80. For each synthetic graph, we have executed a set of 2000 transactions, with original state of the graph being restored after a transaction gets successfully executed. The source code for SpeedyMurmur is available in \\cite{crysp}. It is written in Java and makes use of the graph analysis tool GTNA\\footnote{https:\/\/github.com\/BenjaminSchiller\/GTNA}. From the graphs plotted in Fig. \\ref{fig:example2} a) and b), it is seen that as the number of landmarks increases, SpeedyMurmur gives better success ratio but at the cost of delayed routing. On the other hand, our routing algorithm, which is independent of any landmark, achieves a better success ratio in less time. \n\\end{itemize}\nFrom the results, we can infer that random splitting of capacity without any knowledge of residual graph may lead to failure in spite of presence of routes with the required capacity. \n\n\\section{Conclusion}\n\\label{7}\nIn this paper, we have proposed a novel privacy preserving routing algorithm for payment channel network, \\textit{HushRelay} suitable for simultaneous payment across multiple paths. From the results, it was inferred that our proposed routing algorithm outperforms landmark based routing algorithms in terms of success ratio and the time taken to route. Currently all our implementations assume that the network is static. In future, we would like to extend our work for handling dynamic networks as well. Our algorithms have been defined with respect to a transaction between a single payer and payee but it can extended to handle multiple transaction by enforcing blocking protocol or non blocking protocol to resolve deadlocks in concurrent payments. \\cite{malavolta}.\n\n\n\n\n\\bibliographystyle{IEEEtranS}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nRecently, Ancona proved the numerical Hodge standard conjecture for abelian fourfolds \\cite{Ancona}. \nIn fact, he proved a general theorem for certain rank 2 pure motives in mixed characteristic \\cite{Ancona}*{8.1}, and showed that this general theorem is applicable to abelian fourfolds over finite fields. In this paper, we point out some other cases where the general theorem can be applied. (See \\cite{Ancona}*{A.9} for another example.) \n\nIn Ancona's work, the main cases are\n\\begin{enumerate}\n \\item an absolutely simple abelian fourfold, and\n \\item the product of a simple abelian threefold and an elliptic curve. \n\\end{enumerate}\n(See \\cite{Ancona}*{A.8} for some discussion.)\nIn this paper, we generalize the second case as follows: \n\n\\begin{thm}\\label{main:example}\nLet $A$ be a simple abelian variety over a field $k$. Assume either\n\\begin{itemize}\n\\item $\\dim A$ is \\emph{prime}, or\n\\item some specialization of $A$ to a finite field is absolutely simple and \\emph{almost ordinary}\\footnote{This means its Newton polygon is the same as the one of the product of a supersingular elliptic curve and $\\dim A-1$ ordinary elliptic curves. Such a simple abelian variety exists \\cite{LO}.}. \n\\end{itemize}\nLet $E$ be an elliptic curve. The numerical Hodge standard conjecture holds for $A\\times A$ and $A\\times E$. \n\\end{thm}\n\nAs in \\cite{Ancona}*{1.6}, this combined with \\cite{Clozel} implies\n\n\\begin{cor}\nThe numerical equivalence on $A\\times A, A\\times E$ coincides with the $\\ell$-adic homological equivalence on $A\\times A, A\\times E$ for infinitely many $\\ell$. \n\\end{cor}\n\n\\begin{rem}\nAssume that $k$ is a finite field. \nIn both cases, the Tate conjecture for $A$ is known, and, all algebraic classes come from the intersection of divisors \\cites{Tate, Tankeev:prime, LZ}. Therefore, the numerical Hodge standard conjecture holds for $A$ itself and the numerical equivalence on $A$ coincides with the $\\ell$-adic homological equivalence on $A$ for every $\\ell$ \\cite{Milne:polarization}*{3.7}, \\cite{Ancona}*{Section 5}. \nHowever, if $\\dim A\\geq 3$, $A^2$ and $A \\times E$ may have an exotic Tate class in the middle degree, i.e., a class that cannot be written using Tate classes of degree 2, and the Tate conjecture is not known except the case of the product of a simple threefold and an ordinary elliptic curve \\cite{Milne2022}\\footnote{This relies on the work of Markman \\cite{Markman} on Weil classes.}. \n\\end{rem}\n\nWe prove a slightly more general statement. Let $A$ be an absolutely simple abelian variety of dimension $g$ over a finite field $\\bF_q$. Let $\\alpha_1, \\dots, \\alpha_{2g}$ be the Frobenius eigenvalues of the first cohomology so that $\\overline{\\alpha}_i=\\alpha_{i+g}$. \nSet $\\beta_{i}\\coloneqq q\/ \\alpha_i^2, 1\\leq i\\leq g$. Let $\\Gamma'$ denote the multiplicative group generated by $\\beta_i, 1\\leq i \\leq g$ inside $\\bQ(\\alpha_1, \\dots, \\alpha_{2g})$. The rank of $\\Gamma'$ has been studied, e.g., \\cites{Zarhin:nonsimple, Zarhin:eigenvalues}. Following \\cite{DKZB}, we call it the \\emph{angle rank} of $A$. The angle rank is always less than or equal to $g$. If the angle rank is $g$ or $A$ is a supersingular elliptic curve\\footnote{The angle rank is $0=g-1$ in this case.}, all the Tate classes on $A^n$ for a positive integer $n$ can be written using Tate classes of degree 2, and the Tate conjecture holds for $A^n$. \n(This is the case for all abelian surfaces and elliptic curves.)\nThe converse is also true. \nRecall that such a Tate class is called \\emph{Lefschetz} and a Tate class is \\emph{exotic} if it is not Lefschetz. \nWe are interested in the easiest case with possible exotic Tate classes:\n\n\\begin{thm}\\label{main:general}\nIf the angle rank of $A$ is $g-1$ or $g$ and $\\dim A >1$ is odd, then the numerical Hodge standard conjecture holds for $A\\times A$ and $A\\times E$, where $E$ is an elliptic curve. \n\\end{thm}\n\n\\begin{rem}\nTankeev \\cite{Tankeev:prime}*{p.332} showed that the angle rank is $g-1$ or $g$ if $g=\\dim A$ is an odd prime\\footnote{Tankeev excludes the case $g=3$, but the same argument actually works.}. \nLenstra and Zarhin \\cite{LZ} showed that that if $A$ is almost ordinary, the angle rank is $g-1$ when $g$ is odd and $g$ when $g$ is even; see \\cite{LZ}*{6.7} (and \\cite{DKZB}*{1.5}) for a slightly more general case. \n\\end{rem}\n\n\\begin{rem}\nIf the angle rank of $A$ is $g-1$ and $E$ is ordinary, then $A\\times E$ has no exotic Tate classes; see Corollary \\ref{ordinary}. The same holds trivially if the angle rank of $A$ is $g$ and $E$ is supersingular. \n\\end{rem}\n\n\\begin{rem}\\label{even}\nIf we assume instead that $g$ is even and the angle rank is $g-1$ (or $g$), then we can show that the numerical Hodge standard conjecture holds for $A$ itself. This partly generalizes the case of absolutely simple abelian fourfolds in \\cite{Ancona} because one can show that the angle rank is $\\geq 3$ for an absolutely simple abelian fourfold if its Frobenius generates a CM field of degree $8$. \n\\end{rem}\n\nNow, Theorem \\ref{main:general} clearly implies Theorem \\ref{main:example}, so we will focus on Theorem \\ref{main:general}. \nWe shall show that $A\\times A$ may have an exotic Tate class only in the middle degree, and they form a 2-dimensional space so that we can apply \\cite{Ancona}*{8.1}. The case of $A\\times E$ is similar. \n\nFinally, let us mention that we study the Tate conjecture and the Hodge standard conjecture for self-products of K3 surfaces in \\cite{IIK}. \n\n\\section{A lemma on the Hodge standard conjecture}\nLet $A$ be an abelian variety of dimension $g$ over a field with a polarization $L$. \nLet $\\cZ^{n}_{\\num}(A)_{\\bQ}$ denote the space of algebraic cycles of codimension $n$ modulo numerical equivalence. \nRecall that the Lefschetz standard conjecture holds for $A$ and $L$, and we define the primitive part $\\cZ^{n, \\prim}_{\\num}(A)_{\\bQ}$ of $\\cZ^{n}_{\\num}(A)_{\\bQ}$. \n\n\\begin{conj}[The numerical Hodge standard conjecture]\nFor a nonnegative integer $n\\leq g\/2$, \nThe pairing\n\\[\n\\langle -, -\\rangle_n \\colon \\cZ^{n, \\prim}_{\\num}(A)_{\\bQ} \\times \\cZ^{n, \\prim}_{\\num}(A)_{\\bQ} \\to \\bQ; (\\alpha, \\beta) \\mapsto\n(-1)^{n} \\alpha \\cdot \\beta \\cdot L^{g-2n}\n\\]\nis positive definite. \n\\end{conj}\n\nLet us say that a class in $\\cZ^n_{\\num}(A)_{\\bQ}$ is \\emph{exotic} if it cannot be written as the intersection of divisors. \n\n\\begin{lem}\\label{independence}\nIf $A$ has exotic classes only in the middle degree, then the numerical Hodge standard conjecture is independent of $L$. \n\\end{lem}\n\n\\begin{proof}\nLet $\\cL^{n}_{\\num}(A)_{\\bQ}$ denote the subspace of $\\cZ^n_{\\num}(A)_{\\bQ}$ spanned by the intersections of divisors. \nThe numerical Hodge standard conjecture is known for $\\cL^{n}_{\\num}(A)_{\\bQ}$ by specializing to a finite field \\cite{Milne:polarization}*{3.7}, \\cite{Ancona}*{Section 5}. \nIn particular, only the middle degree is a problem. There is an orthogonal decomposition with respect to $\\langle -,- \\rangle_n$\n\\[\n\\cZ^{g\/2}_{\\num}(A)_{\\bQ}=\\cL^{g\/2}_{\\num}(A)_{\\bQ}\\oplus \\cE^{g\/2}_{\\num}(A)_{\\bQ}, \n\\]\nwhere $\\cE^{g\/2}_{\\num}(A)_{\\bQ}$ is the space of exotic classes. This decomposition is independent of $L$. \nThe numerical Hodge conjecture holds for $A$ and $L$ if and only if $\\langle -, - \\rangle_{g\/2}$ is positive definite on $\\cE^{g\/2}_{\\num}(A)_{\\bQ}$, and the latter statement is independent of $L$. \n\\end{proof}\n\n\\section{Exotic Tate classes}\nWe assume that $A$ is an absolutely simple abelian variety of dimension $g >2$ defined over a finite field $\\bF_q$ of characteristic $p$. \nWe use the notation $\\alpha_i, \\beta_i$ as in the introduction. \nSuppose first that the angle rank of $A$ is $g-1$. This implies that $\\End (A)\\otimes \\bQ$ is a number filed of degree $2g$ generated by Frobenius. \n\n\\begin{lem}\nAssume that the angle rank is $g-1$. \nAfter replacing $\\alpha_i$ by $\\alpha_{i+g}$ if necessary, the only relation among $\\beta_1, \\dots \\beta_g$ has the form of\n\\[\n(\\beta_1 \\cdots \\beta_g)^N = 1\n\\]\nfor some $N$. \n\\end{lem}\n\n\\begin{proof}\nLet $\\beta_1^{\\bZ} \\cdots \\beta_g^{\\bZ}$ denote the free abelian group of rank $g$ with the basis $\\beta_1, \\dots, \\beta_g$, and let $\\Gamma_1$ be the kernel of the natural map\n\\[\n\\beta_1^{\\bZ} \\cdots \\beta_g^{\\bZ} \\to \\bQ(\\alpha_1, \\dots, \\alpha_{2g})\\setminus\\{0\\}. \n\\]\nBy assumption, $\\Gamma_1$ is a free abelian group of rank 1. \nSo, the Galois group of $\\bQ(\\alpha_1, \\dots, \\alpha_{2g})$ acts naturally on $\\Gamma_1$ via $\\{\\pm 1\\}\\subset \\Aut (\\Gamma_1)$. Note that the Galois group acts on $\\{\\{\\beta_1^{\\pm 1}\\}, \\dots, \\{\\beta_g^{\\pm 1}\\}\\}$ by permutation and the action is transitive, and the Galois group contains the complex conjugation so that $\\overline{\\beta}_i=\\beta_i^{-1}$. \nThis implies that a generator of $\\Gamma_1$ has the form of\n\\[\n\\beta_1^{\\pm N} \\beta_2^{\\pm N} \\cdots \\beta_g^{\\pm N}\n\\]\nfor some $N$. \n\\end{proof}\n\n\\begin{cor}\nLet $\\ell$ be a prime different from $p$. \nIf $g$ is odd (resp. even), any exotic $\\ell$-adic Tate class of $A\\times A$, $A\\times E$ (resp. $A$) is in the middle degree. If an exotic Tate class exists, then the space of exotic Tate classes is two-dimensional for $A\\times A$ (resp. $A$) and four-dimensional for $A\\times E$. \n\\end{cor}\n\n\\begin{cor}\\label{ordinary}\nIf $g$ is odd and $E$ is ordinary, then $A\\times E$ has no exotic Tate classes. \n\\end{cor}\n\nA similar argument shows the following:\n\n\\begin{lem}\nSuppose the angle rank of $A$ equals $g$ and $g$ is odd, then any exotic $\\ell$-adic Tate class of $A\\times E$ is in the middle degree. If an exotic Tate class exists, then $E$ is ordinary and the space of exotic Tate classes is two-dimensional. \n\\end{lem}\n\nNext, we construct a motivic counterpart of possible exotic Tate classes using complex multiplication. \nLet us first recall some facts about the motive of $A$ \\cite{Ancona}*{Section 4, Section 6}. \nSet $B\\coloneqq \\End (A)\\otimes\\bQ$ and write $L\\subset \\overline{\\bQ}$ for the Galois closure of $B$ with $\\Sigma\\coloneqq \\Hom (B, L)$. As in \\cite{Ancona}*{6.6}, there is the following decomposition in the category of Chow motives with coefficients in $L$:\n\\[\n[H^1 (A)]=\\bigoplus_{\\sigma \\in \\Sigma} M_{\\sigma}\n\\]\nthat induces \\cite{Ancona}*{6.7 (1)}\n\\[\n[H^g (A)]=\\bigoplus_{I\\subset \\Sigma, \\# I =g} M_I, \n\\]\nwhere $M_I=\\otimes_{i \\in I} M_i$. \nThis further induces the following decomposition \\cite{Ancona}*{6.7 (2)}, in the category of Chow motives with coefficients in $\\bQ$, \n\\[\n[H^g (A)]=\\bigoplus M_{[I]}, \n\\]\nwhere $[I]$ denotes the Galois orbit of $I$ and $M_{[I]}$ is the direct sum of $M_?$ over the Galois orbit. \nThis decomposition is orthogonal as numerical motives with respect to $\\langle -, -\\rangle^{\\otimes g}_{1, \\textnormal{mot}} $ defined in \\cite{Ancona}*{3.6}. \nSimilarly, $[H^{2g}(A\\times A)]$ has such a decomposition and we have summands like\n\\[\nM_{I^2}\\coloneqq M_I\\otimes M_{I}, \\quad M_{[I^2]}. \n\\]\n\n\\begin{prop}\nAssume that the angle rank is $g-1$. \n\\begin{enumerate}\n \\item If $g$ is even, there exists at most one $[I]$ such that the $\\ell$-adic realization of $M_{[I]}$ is exotic. The numerical algebraic classes in $M_{[I]}$ is zero or two-dimensional. \n \\item If $g$ is odd and $E$ is supersingular, there exists at most one $[I]$ such that the $\\ell$-adic realization of $M_{[I]}\\otimes H^1 (E)$ is exotic. The numerical algebraic classes in $M_{[I]}$ is zero or four-dimensional. \n \\item If $g$ is odd, there exists at most one $[I^2]$ such that the $\\ell$-adic realization of $M_{[I^2]}$ is exotic. The numerical algebraic classes in $M_{[I^2]}$ is zero or two-dimensional. \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nThis follows from the description of exotic Tate classes and \\cite{Ancona}*{6.8}. The key claim here is that the relevant Galois orbit only has two elements, and it controls the dimension of numerical algebraic classes. \n\\end{proof}\n\nWe call $M_{[I]}, M_{[I]}\\otimes H^1(E), M_{[I^2]}$ \\emph{exotic} if it has a nonzero numerical algebraic class. \nIf it is the case, their $\\ell$-adic realizations are the only exotic Tate classes. \nBy \\cite{Ancona}*{5.3} and Lemma \\ref{independence}, the numerical Hodge standard conjecture for $A, A\\times A, A\\times E$ reduces to the corresponding problem on $M_{I}, M_{[I]}\\otimes H^1(E), M_{[I^2]}$ respectively, with respect to $\\langle -, -\\rangle^{\\otimes g}_{1, \\textnormal{mot}}, \\langle -, -\\rangle^{\\otimes g+1}_{1, \\textnormal{mot}}. \\langle -, -\\rangle^{\\otimes 2g}_{1, \\textnormal{mot}}$ for some polarization. \n\nA similar construction makes sense for $A \\times E$ if $g$ is odd, the angle rank is $g$, and $E$ is ordinary. \n\nFinally, when $g$ is odd and $E$ is supersingular, an exotic $M_{[I]}\\otimes H^1(E)$ has a decomposition into rank 2 motives\n\\[\nM_{[I]}\\otimes H^1(E) =M_1 \\oplus M_2\n\\]\northogonal with respect to $\\langle -, -\\rangle^{\\otimes g+1}_{1, \\textnormal{mot}}$. \nMore precisely, the Galois action on $[I]$ gives rise to an imaginary quadratic field $F$ inside $B$ and there is an embedding $F\\hookrightarrow \\End (E_{\\overline{\\bF}_q})\\otimes \\bQ$ by exactly the same argument as in the proof of \\cite{Ancona}*{7.16}. The actions of $F$ on $M_{[I]}$ and $H^1(E)$ induce the above decomposition. \n\n\\section{Ancona's theorem for rank 2 motives}\nTo conclude the proof of Theorem \\ref{main:general}, we recall Ancona's theorem and then use CM liftings to apply it. \n\nLet $K$ be a $p$-adic field with the ring of integers $O_K$ with residue field $k$. \nFix an embedding $\\sigma \\colon K\\hookrightarrow \\bC$. \nWe shall use the language of relative Chow motives over $O_K$, equipped with base changes to $\\bC$ via $\\sigma$ and to $k$ via the specialization. \nFor a relative Chow motive $M$ over $O_K$, we write $V_B$ for the Betti realization of $M_{\\bC}$. \nLet $V_Z$ denote the space of numerical algebraic cycles in $M_k$, i.e., homomorphisms from $\\mathbbm{1}$ modulo numerical equivalences. Both $V_B$ and $V_Z$ are $\\bQ$-vector spaces. \nIf $M$ has a quadratic form\n\\[\nq\\colon \\Sym^2 (M) \\to \\mathbbm{1}, \n\\]\nthen it induces ($\\bQ$-valued) quadratic forms $q_B, q_Z$ on $V_B, V_Z$ respectively. \n\n\\begin{thm}[Ancona \\cite{Ancona}*{8.1}]\\label{rank 2}\nLet $M$ be a relative Chow motive over $O_K$ with a quadratic form $q$. \nAssume that\n\\begin{itemize}\n \\item $\\dim_{\\bQ} V_B=\\dim_{\\bQ} V_Z=2$, and \n \\item $q_B\\colon V_B\\times V_B \\to \\bQ$ is a polarization of Hodge structures. \n\\end{itemize}\nThen, $q_Z$ is positive definite. \n\\end{thm}\n\n\\begin{proof}[Proof of Theorem \\ref{main:general}]\nFollowing the proof of \\cite{Ancona}*{3.18}, we use CM liftings to prove Theorem \\ref{main:general}. \nLet $A$ be as in Theorem \\ref{main:general}. \nSet $B\\coloneqq \\End (A_{\\bF_q})\\otimes \\bQ$. \nAfter enlarging $\\bF_q$, we can find a finite extension $O_K$ of $W(\\bF_q)$ and an abelian scheme $\\cA$ over $O_K$ with $B\\to \\End (\\cA)$ such that the reduction $\\cA_{\\bF_q}$ is $B$-isogenous to $A$. \nWe may replace $A$ by $\\cA_{\\bF_q}$ and assume that a polarization on $\\cA_{\\bF_q}$ lifts to a polarization on $\\cA$. \n\nIf $A^2$ has no exotic classes, there is nothing to prove. So, assume some $M_{[I^2]}$ is exotic. \nBy \\cite{Ancona}*{5.3}, it suffices to show the the paring $\\langle -, - \\rangle^{\\otimes 2g}_{1, \\textnormal{mot}}$ is positive definite on the exotic $M_{[I^2]}$. By the construction of $M_{[I^2]}$ and the paring, it lifts to a relative Chow motive with a quadratic form over $O_K$ (cf. \\cite{Ancona}*{4.1, 4.2} and references therein, and the proof of \\cite{Ancona}*{3.18}). \nBy the definition of the exotic $M_{[I^2]}$, this lift satisfies the assumption of Theorem \\ref{rank 2}. \nSo, $\\langle -, - \\rangle^{\\otimes 2g}_{1, \\textnormal{mot}}$ is positive definite on $M_{[I^2]}$. \n\nThe case of $A\\times E$ is similar as in \\cite{Ancona}. Let us consider the case $E$ is supersingular. The decomposition\n\\[\nM_{[I]}\\otimes H^1(E) =M_1 \\oplus M_2\n\\]\nis constructed using the action of the imaginary quadratic field $F\\subset B$ on $E$, and it may also lifts by taking a lift of $E$ with the action of $F$. \n\\end{proof}\n\n\\begin{rem}\nConsider the case $A\\times A$. \nThe Hodge type of the Betti realization of the lifts of the exotic classes have the form of $(2a, 2b), (2b. 2a)$ with $a+b=g$. In particular, it is never $(g,g)$ and any exotic class cannot be lifted to an algebraic class of $\\cA_{\\bC} \\times \\cA_{\\bC}$. Therefore, Ancona's theorem is essential. \n\\end{rem}\n\nRemark \\ref{even} can be proved in the same way. \n\n\\begin{bibdiv}\n\\begin{biblist}\n\\bib{Ancona}{article}{\n author={Ancona, Giuseppe},\n title={Standard conjectures for abelian fourfolds},\n journal={Invent. Math.},\n volume={223},\n date={2021},\n number={1},\n pages={149--212},\n issn={0020-9910},\n review={\\MR{4199442}},\n doi={10.1007\/s00222-020-00990-7},\n}\n\\bib{Clozel}{article}{\n author={Clozel, L.},\n title={Equivalence num\\'{e}rique et \\'{e}quivalence cohomologique pour les\n vari\\'{e}t\\'{e}s ab\\'{e}liennes sur les corps finis},\n language={French},\n journal={Ann. of Math. (2)},\n volume={150},\n date={1999},\n number={1},\n pages={151--163},\n issn={0003-486X},\n review={\\MR{1715322}},\n doi={10.2307\/121099},\n}\n\\bib{DKZB}{article}{\n author={Dupuy, Taylor},\n author={Kedlaya, Kiran S.}, \n author={Zureick-Brown, David},\n title={Angle ranks of abelian varieties},\n eprint={https:\/\/arxiv.org\/abs\/2112.02455}, \n}\n\\bib{IIK}{misc}{\n author={Ito, Kazuhiro},\n author={Ito, Tetsushi},\n author={Koshikawa, Teruhisa},\n note={in preparation},\n}\n\\bib{LO}{article}{\n author={Lenstra, Hendrik W., Jr.},\n author={Oort, Frans},\n title={Simple abelian varieties having a prescribed formal isogeny type},\n journal={J. Pure Appl. Algebra},\n volume={4},\n date={1974},\n pages={47--53},\n issn={0022-4049},\n review={\\MR{354686}},\n doi={10.1016\/0022-4049(74)90029-2},\n}\n\\bib{LZ}{article}{\n author={Lenstra, Hendrik W., Jr.},\n author={Zarhin, Yuri G.},\n title={The Tate conjecture for almost ordinary abelian varieties over\n finite fields},\n conference={\n title={Advances in number theory},\n address={Kingston, ON},\n date={1991},\n },\n book={\n series={Oxford Sci. Publ.},\n publisher={Oxford Univ. Press, New York},\n },\n date={1993},\n pages={179--194},\n review={\\MR{1368419}},\n}\n\\bib{Markman}{article}{\n author={Markman, Eyal}, \n title={The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians}, \n eprint={https:\/\/arxiv.org\/abs\/1805.11574}, \n}\n\\bib{Milne:polarization}{article}{\n author={Milne, J. S.},\n title={Polarizations and Grothendieck's standard conjectures},\n journal={Ann. of Math. (2)},\n volume={155},\n date={2002},\n number={2},\n pages={599--610},\n issn={0003-486X},\n review={\\MR{1906596}},\n doi={10.2307\/3062126},\n}\n\\bib{Milne2022}{article}{\n author={Milne, J. S.}, \n title={The Tate and standard conjectures for certain abelian varieties}, \n eprint={https:\/\/arxiv.org\/abs\/2112.12815},\n}\n\\bib{Tankeev:prime}{article}{\n author={Tankeev, S. G.},\n title={Cycles of abelian varieties of prime dimension over finite and\n number fields},\n language={Russian},\n journal={Izv. Akad. Nauk SSSR Ser. Mat.},\n volume={47},\n date={1983},\n number={2},\n pages={356--365},\n issn={0373-2436},\n review={\\MR{697300}},\n}\n\\bib{Tate}{article}{\n author={Tate, John},\n title={Endomorphisms of abelian varieties over finite fields},\n journal={Invent. Math.},\n volume={2},\n date={1966},\n pages={134--144},\n issn={0020-9910},\n review={\\MR{206004}},\n doi={10.1007\/BF01404549},\n}\n\\bib{Zarhin:nonsimple}{article}{\n author={Zarhin, Yu. G.},\n title={The Tate conjecture for nonsimple abelian varieties over finite\n fields},\n conference={\n title={Algebra and number theory},\n address={Essen},\n date={1992},\n },\n book={\n publisher={de Gruyter, Berlin},\n },\n date={1994},\n pages={267--296},\n review={\\MR{1285371}},\n}\n\\bib{Zarhin:eigenvalues}{article}{\n author={Zarhin, Yuri G.},\n title={Eigenvalues of Frobenius endomorphisms of abelian varieties of low\n dimension},\n journal={J. Pure Appl. Algebra},\n volume={219},\n date={2015},\n number={6},\n pages={2076--2098},\n issn={0022-4049},\n review={\\MR{3299720}},\n doi={10.1016\/j.jpaa.2014.07.024},\n}\n\\end{biblist}\n\\end{bibdiv}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:1}\n\nThe substantial homogeneity of Type~Ia supernovae (SNe~Ia), together with the fact\nthat the peak luminosity of individual objects can be observationally determined\nthrough comparisons with well-calibrated samples of nearby SNe~Ia, has propelled \nthem to ``gold standard'' status among extragalactic distance indicators. Precise \ndistance measurements out to $z \\approx 1.7$ have been made \\citep{Riess01,Riess04},\nrevealing the surprising cosmological result that the expansion rate of the\nuniverse is currently accelerating \\citep{Riess98,Perlmutter99}; see\n\\citet{Filippenko04a,Filippenko05} for extensive reviews. Although the\nprogenitor systems have not yet been conclusively identified, the general \nconsensus is that SNe~Ia arise from carbon-oxygen white dwarfs (CO WDs) that \naccrete matter through some mechanism until they achieve a density of $\\sim 3 \n\\times 10^9 {\\rm\\ g\\ cm}^{-3}$ in their centers, leading to a runaway thermonuclear \nreaction that incinerates the star \\citep{Woosley86}. This occurs, coincidentally, \nwhen a CO WD's mass is nearly the Chandrasekhar limit of $\\sim$1.4~$M_\\odot$.\nSimulations of exploding CO WDs are able to reproduce the main spectral and\nphotometric characteristics of SNe~Ia (e.g., \\citealt{Leibundgut00}, and\nreferences therein), lending theoretical support to this scenario.\n\nHowever, many questions remain concerning the SN~Ia progenitors and \nexplosion mechanism. What is the nature of the ``donor'' that is\nresponsible for providing the material that the WD accretes? Is there more\nthan one channel by which the accretion can take place? Where does the\nthermonuclear runaway begin inside the WD, and how does it propagate throughout\nthe star? As with so many things, the ``devil is in the details,'' and there\nis a growing sentiment that the answers to these fundamental questions may come\nfrom careful study of the {\\it differences} seen among SNe~Ia, rather than from\nthe similarities alone.\n\nSpectropolarimetry offers the only direct probe of early-time SN geometry, and\nthus is an important diagnostic tool for discriminating among SN~Ia progenitor\nsystems and theories of the explosion physics. The essential idea is this: A\nhot, young SN atmosphere is dominated by electron scattering, which by its\nnature is highly polarizing. For an unresolved source that has a spherical\ndistribution of scattering electrons, the directional components of the\nelectric vectors of the scattered photons cancel exactly, yielding zero net\nlinear polarization. Any asymmetry in the distribution of the scattering\nelectrons, or of absorbing material overlying the electron-scattering\natmosphere, results in incomplete cancellation, and produces a net polarization\n\\citep[see, e.g.,][]{Leonard11}.\n\nInitial broad-band polarimetry studies found SNe~Ia to possess zero or, at\nmost, very weak ($< 0.2\\%$) intrinsic polarization \\citep{Wang96}, suggesting a\nhigh degree of symmetry for their scattering atmospheres. More recent\nspectropolarimetric studies capable of resolving individual line features,\nhowever, are revealing a complex picture, with both continuum and line\npolarization now convincingly established for at least a subset of the SN~Ia\npopulation.\n\nTo date, three SNe~Ia have been examined in detail with spectropolarimetry at\nearly times: SN~1999by \\citep{Howell01}, SN~2001el \\citep{Kasen03,Wang03}, and,\nmost recently, SN~2004dt \\citep{Wang05}. In this paper we present single-epoch\nspectropolarimetry of four SNe~Ia: SN~1997dt, SN~2002bf, SN~2003du, and\nSN~2004dt, obtained about $21$, 3, 18, and 4 days (respectively) after maximum\nlight. \n\nA particular motivation for this multi-object investigation is to attempt to\nlink spectral and photometric peculiarities of individual SNe~Ia with their\nspectropolarimetric characteristics. Accordingly, the four events span a range\nof properties: SN~1997dt is likely somewhat subluminous, SN~2002bf and\nSN~2004dt exhibit unusually high-velocity absorption lines (in the case of\nSN~2002bf, the highest ever seen in an SN~Ia for the epochs considered), and\nSN~2003du is slightly overluminous.\n\nThis paper is organized as follows. We briefly review the present photometric,\nspectroscopic, and spectropolarimetric state of knowledge of SNe~Ia in\n\\S~\\ref{sec:2}, focusing particular attention on objects sharing\ncharacteristics with the SNe~Ia included in our spectropolarimetric survey. We\ndescribe and present the spectropolarimetry in \\S~\\ref{sec:3}; to assist in the\nclassification, we also include optical photometry and additional spectroscopy\nfor two of the events, SN~2002bf and SN~2003du. We analyze the data in\n\\S~\\ref{sec:4}, and present our conclusions in \\S~\\ref{sec:5}. Note that\npreliminary discussions of the spectropolarimetry of SN~1997dt and SN~2002bf\nhave been given by \\citet{Leonard1} and \\citet{Filippenko04}, respectively.\n\n\\section{Background}\n\\label{sec:2}\n\\subsection{Photometric Properties of SNe~Ia}\n\\label{sec:2.1}\n\nType Ia SNe typically rise to peak $B$-band brightness in about 20 days,\ndecline by about $3$ mag over the next $35$ days, and then settle into a nearly\nconstant descent of $\\sim$1.55 mag (100\\ day)${^{-1}}$ for the next year.\nHowever, it is now confirmed beyond doubt that deviations exist from the\ncentral trend; see, for example, \\citet{Phillips99}. Some events rise and fall\nmore slowly, producing ``broader'' light curves (e.g., SN~1991T), whereas\nothers rise and fall more quickly, yielding ``narrower'' light curves (e.g.,\nSN~1991bg). The width of the light curves near peak correlates strongly with\nluminosity, in the sense that broader light curves generally indicate\nintrinsically brighter objects (but see \\citealt*{Jha05}, for exceptions), with\na total spread of about a factor of sixteen in absolute peak $B$-band\nbrightness among the population \\citep{Altavilla04}.\n\nThe ability to individually determine the luminosity of SNe~Ia through\nexamination of their light-curve shapes has been the primary driver for\ncosmological applications. There are several photometric calibration\ntechniques in common use that correlate the observed light-curve shape, or\nsupernova color, with luminosity (see \\citealt{Wang05a}, and references\ntherein). The simplest calibration technique involves measuring the decline in\n$B$ during the first 15 days after maximum $B$-band brightness, $\\Delta\nm_{15}(B)$. A typical value is $\\Delta m_{15}(B) \\approx 1.1$ mag, which\ncorresponds to $M_B = -19.26 \\pm 0.05$ mag according to the calibrations of\n\\citet{Hamuy96}. Underluminous, SN 1991bg-like objects yield values up to\n$\\Delta m_{15}(B) = 1.94$ mag, while overluminous, SN 1991T-like objects \ngo as low as $\\Delta m_{15}(B) = 0.81$ mag \\citep{Altavilla04}. There is a\ncontinuum of values in between the two extremes.\n\nFor the two SNe with optical photometry presented in this paper, we shall use\nthe ``multicolor light-curve shape'' method \\citep[MLCS; e.g.,][]{Riess96},\nwhich has been revised (and hereafter referred to as MLCS2k2) by \\citet{Jha02}\nand S. Jha et al. (in preparation) to include $U$-band light curves from\n\\citet{Jha05}, a more self-consistent treatment of extinction, and an improved\ndetermination of the unreddened SN~Ia color. This method is based on\ndetermining $\\Delta$ (a dimensionless number related to magnitude that \nparameterizes the light-curve shape), $t_0$ (the time of $B$-band maximum light), \n$\\mu_0$ (the distance modulus for an assumed value of $H_0$, here taken to be \n$65\\ {\\rm km\\ s\\ }^{-1}{\\rm\\ Mpc}^{-1}$), and $A_V^0$ (the visual extinction at $t_0$).\n\nThe mechanism that produces the dispersion in SN~Ia luminosity is not known;\npresent speculations range from different progenitor systems to differences in\nthe explosion mechanism and flame propagation (e.g., deflagration, detonation,\ndelayed-detonation, or, most recently, gravitationally confined detonation; see,\nrespectively, \\citealt*{Nomoto84,Arnett69,Khokhlov91,Plewa04}).\nSome have further speculated that global asymmetries in the expanding ejecta\nmay result in viewing-angle dependent luminosity \\citep{Wang03}. One\nobservational fact that all theories must confront is that the most luminous\nSNe~Ia have thus far only been seen in late-type galaxies; the faintest\nobjects tend to prefer elliptical galaxies, but have also been found in spirals\n\\citep[e.g., ][and references therein]{Howell01a,Benetti05}.\n\n\\subsection{Spectroscopic Properties of SNe~Ia}\n\\label{sec:2.2}\n\nEarly-time spectra of SNe~Ia (see \\citealt{Filippenko97} for a review)\ntypically exhibit lines of intermediate-mass elements (IMEs), such as\nmagnesium, silicon, sulfur, and calcium, with some contribution from iron-peak\nelements, especially at near-ultraviolet wavelengths. As time progresses, and\nthe photosphere recedes deeper into the ejecta, lines of Fe come to dominate\nthe spectrum. This spectral evolution suggests a burning front that\nincinerates some of the progenitor's carbon and oxygen all the way to the\niron peak deep inside the ejecta, but then leaves the outer layers only\npartially burned. It is interesting to note that overluminous SNe~Ia show\nenhanced Fe features at early times, whereas subluminous ones show weak\nearly-time Fe features.\n\nAt early times (e.g., $-20 {\\rm\\ d} < t < 20 {\\rm\\ d}$ from the date of maximum\n$B$ brightness, $B_{\\rm max}$), the spectra of typical SNe~Ia evolve rapidly\nand with such uniformity that it is possible to determine the age of an event\nrelative to the date of $B_{\\rm max}$ to within $\\sim$2 days from a single\nspectrum alone \\citep{Riess97,Foley05}. The spectroscopic peculiarities of subluminous\nand overluminous events currently preclude their ``spectral feature ages'' from\nbeing derived accurately through comparison with average SN~Ia spectra. For\ninstance, pre-maximum spectra of overluminous events lack the strong\n\\ion{Si}{2} $\\lambda 6355$ absorption that is so prominent in normal and\nunderluminous SN~Ia spectra \\citep{Filippenko92a}.\n\nAnother point of distinction among SN~Ia spectra comes from the blueshifts of\nthe spectral lines. Increasing attention is being paid to a small but growing\ngroup of ``high-velocity'' (HV) SNe~Ia, whose spectra around maximum light are\ncharacterized by unusually broad and highly blueshifted absorption troughs in\nmany line features, indicating optically thick ejecta moving about\n4000--5000~km s$^{-1}$\\ faster than is typically seen for SNe~Ia\n\\citep{Branch87,Benetti05}. Well-studied examples of HV~SNe~Ia include\nSN~1983G \\citep[][and references therein]{Branch93}, SN~1984A\n\\citep{Branch87,Barbon89}, SN~1997bp \\citep{Anupama97}, SN~1997bq \\citep[][and\nreferences therein]{Lentz01}, SN~2002bo \\citep{Benetti04}, SN~2002dj\n\\citep{Benetti04}, and SN~2004dt \\citep{Wang05}. \\citet{Wang05} note that in a\nspectrum of one HV~SN~Ia, SN~2004dt, some line features do not possess\nabnormally high velocity, such as those identified with \\ion{S}{2}. Since two\nof our objects, SN~2002bf and SN~2004dt, are HV~SNe~Ia, we briefly review the\nsalient features that are known about this class of objects.\n\nThe most thorough and recent study of the diversity of SN~Ia expansion\nvelocities is that of \\citet{Benetti05}, who apply a statistical treatment to\ndata from 26 SNe~Ia and find that HV SNe~Ia indeed make up a kinematically\ndistinct group. Its members have normal peak luminosity (based on their\n$\\Delta m_{15}(B)$ values: $1.09 \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} \\Delta m_{15}(B) \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 1.37$ mag),\nreside in all types of host galaxies (e.g., both ellipticals and spirals, but\nwith a preference for later types), and have very strong \\ion{Si}{2} features.\nThey are further distinguished by a large temporal velocity gradient,\n$\\dot{v}$\\ $ > 70\\ {\\rm km\\ s}^{-1} {\\rm\\ day}^{-1}$, where $\\dot{v}$ is defined to be the\naverage daily rate of decrease of the expansion velocity between maximum light\nand the time the \\ion{Si}{2} $\\lambda 6355$ feature disappears. HV~SNe~Ia\nappear to have similar photometric characteristics to ordinary Type~Ia events,\nalthough there is some indication of subtle differences in their color\nevolution \\citep{Benetti04}.\n\nWhether HV SNe~Ia represent the extreme end of a continuum of more typical\nSNe~Ia, or require a different explosion mechanism, progenitor system, or\nexplosion physics, remains controversial. What is certain is that any model\nmust have significant optical depth in the IMEs at high velocities at early\ntimes. \\citet{Branch93} originally proposed that HV~SNe~Ia may simply result\nfrom more energetic explosions. However, \\citet{Benetti04} show that the Fe\nnebular lines have velocities comparable to those of normal SNe~Ia, which tends\nto weaken the argument for higher overall kinetic energy. There is also\nevidence from early-time spectra of SN~2002bo that HV~SNe~Ia possess very\nlittle unburned carbon in their outer layers, in contrast with more typical\nSNe~Ia like SN~1994D \\citep{Benetti04}. The high velocities of the IMEs\ncoupled with the lack of primordial carbon may indicate that burning to Si\npenetrates to much higher layers in HV~SNe~Ia than it does in more normal\nevents. This is a feature of some delayed-detonation models, such as those\nstudied by \\citet{Lentz01}. However, \\citet{Wang05} argue that a strong\ndetonation wave is unlikely to generate the clumpy and asymmetrically\ndistributed silicon layer that is inferred from pre-maximum spectropolarimetric\nobservations of SN~2004dt. As another alternative, \\citet{Benetti04} propose\nthat IMEs produced at deeper layers may simply be more efficiently mixed\noutward in HV~SNe~Ia than is typical. A prediction of this mechanism is that\nwhile the IME products of explosive carbon burning will be mixed in with the\nprimordial C and O in the outer layers, the C and O should also be mixed inward\nwithin these events, and exist at lower velocities than are normally seen.\n\nIt is appropriate at this point to mention the recently proposed\ngravitationally confined detonation (GCD) model of \\citet{Plewa04}, which\nstands as an intriguing alternative to the standard deflagration or\ndelayed-detonation scenarios. The mechanism involves the slightly off-center\nignition of a deflagration that produces a buoyancy-driven bubble of material\nthat reaches the stellar surface at supersonic speeds, where it laterally\naccelerates the outer stellar layers. This material, gravitationally confined\nto the white dwarf, races around the star and, in $\\sim$2~s, converges at a\npoint opposite to the location of the bubble's breakout, creating conditions\ncapable of igniting the nuclear fuel and triggering a detonation that can\nincinerate the white dwarf and result in an energetic explosion. \n\nOf interest to the present study is the recent work of \\citet{Kasen05}, who\ninvestigate the spectral, and spectropolarimetric, consequences of the GCD\nmodel. They focus their investigation on the interaction of the expanding\nejecta with an ellipsoidal, metal-rich extended atmosphere formed from the\nbubble of deflagration products (taken to be 57\\% Si, 27\\% S, 7.1\\% Fe, and\n2.7\\% Ca, plus smaller amounts of other metals; see \\citealt{Khokhlov93}), and\nfind that a dense, optically thick pancake of metal-rich material is formed at\npotentially large velocity on the side of the ejecta where the bubble emerged.\nFor low atmosphere masses (e.g., resulting from a bubble of mass $0.008\nM_\\odot$), the pancake of material spans the velocity range 17,000--28,000\nkm s$^{-1}$\\ and is geometrically detached from the bulk of the SN ejecta. This might\nexplain the detached, high-velocity \\ion{Ca}{2} near-infrared (IR) triplet\nabsorption seen in pre-maximum spectra of some SNe~Ia (e.g., SN~2001el, see\n\\S~\\ref{sec:2.3.3}). For larger bubble and, hence, atmosphere masses ($m_{\\rm\natm} \\gtrsim 0.016 M_\\odot$), the absorbing pancake moves at lower velocities\n(e.g., 10,000--21,000 km s$^{-1}$\\ for $m_{\\rm atm} = 0.08 M_\\odot$) and could blend\nwith the region of IMEs in the SN ejecta and potentially increase the blueshift\nof several of the spectral features. This could provide an\norientation-dependent explanation for the origin of HV~SNe~Ia, whereby sight\nlines in which the pancake more completely blocks the photosphere produce the\nanomalously large velocities. The spectropolarimetric consequences of this\nmodel are discussed in \\S~\\ref{sec:2.3.1}.\n\n\\subsection{Spectropolarimetric Properties of SN~Ia}\n\\label{sec:2.3}\n\\subsubsection{Supernova Polarization Mechanisms}\n\\label{sec:2.3.1}\n\nThe first definitive proof that some SNe are polarized came from\nobservations of the Type II SN~1987A \\citep{Cropper88}, which exhibited a\nmodest temporal increase in continuum polarization during the photospheric\nphase, as well as sharp polarization modulations across strong P-Cygni flux\nlines \\citep{Jeffery91a}. While intrinsic polarization has now been\nestablished in over a dozen SNe (for recent reviews, see \\citealt{Wheeler00};\n\\citealt{Filippenko04}; \\citealt{Leonard11}), including at least three SNe~Ia, \nthe exact origin of both continuum and line polarization remains controversial.\n\nThere are essentially two mechanisms by which supernova continuum polarization\nis thought to be produced: (1) a globally aspherical photosphere and\nelectron-scattering atmosphere\n\\citep[e.g.,][]{Shapiro82,Hoflich91,Jeffery91,Leonard2,Wang01}, and (2)\nionization asymmetry produced by the decay of asymmetrically distributed\nradioactive $^{56}{\\rm Ni}$, perhaps flung out into \\citep{Chugai92,Hoflich01}\nor beyond \\citep{Kawabata02,Leonard8} the expanding ejecta in clumps. The\nsimplest, and most well-studied, globally aspherical geometry is that of an\nellipsoid, and we shall make frequent reference to the ``ellipsoidal model'' in\nthe following discussion. In the second model, in which an ionization asymmetry\nis present, continuum polarization is generated by light from the (either\nspherical or aspherical) photosphere scattering off of asymmetrically\ndistributed free electrons that exist in clouds surrounding clumps of\nradioactive $^{56}{\\rm Ni}$. Variations on both of these polarization\nmechanisms are also possible. For instance, an aspherical distribution of\n$^{56}{\\rm Ni}$ could also result in SN polarization by providing an asymmetry\nin a source of optical photons (produced by the thermalization of\n$\\gamma$-rays) relative to the scattering medium.\n\nTo explain polarization modulations seen across spectral lines, it is important\nto differentiate between the emission peaks and blueshifted absorption troughs\nthat are characteristic of P-Cygni profiles in total-flux spectra of SNe. Line\npeaks have usually been assumed to consist of intrinsically unpolarized\nphotons. This is because although resonance scattering by a line is an\ninherently polarizing process \\citep{Jeffery91}, directional information for\nscattered photons is lost in an SN atmosphere since the timescale for\nrandomizing collisional redistribution of the relative level populations within\nthe fine structure of the atomic levels of a line transition is much shorter\nthan the characteristic timescale for absorption and reemission in a line\n\\citep{Hoflich96}. The assumption of intrinsically unpolarized emission lines\nis also commonly used to derive the interstellar polarization (ISP;\n\\S~\\ref{sec:2.3.2}). Note, though, that if ionization asymmetry exists above\nthe photosphere (from, e.g., clumps of radioactive $^{56}{\\rm Ni}$), then even\nintrinsically unpolarized emission-line photons may become polarized by\nelectron scattering within the SN atmosphere.\n\nFor SNe~Ia, line-blanketing, due largely to Fe, is particularly severe at\nwavelengths below $\\sim$5000~\\AA, where theoretical models \\citep{Howell01}\nsuggest that nearly complete depolarization of any ``continuum'' light may be\nassumed. Conversely, the broad spectral region $6800 \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} \\lambda\n\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 7800$~\\AA\\ is largely devoid of line opacity \\citep{Kasen04,Howell01},\nat least near maximum light, and thus may give a good indication of true\ncontinuum polarization level.\n\nThe explanation of polarization changes in absorption troughs is controversial.\nOne well-studied hypothesis, which follows logically from the ellipsoidal\nmodel, is that selective blocking of more forward-scattered and, hence, less\npolarized, light in P-Cygni absorption troughs results in trough polarization\nincreases \\citep{Jeffery91,Leonard1,Leonard3}, potentially producing\n``inverted'' P-Cygni polarization profiles when coupled with emission peak\ndepolarization. Some SN modelers have claimed that this ``geometrical\ndilution'' mechanism, however, is rather poor at polarizing SN~Ia light, and\nthat polarization {\\it decreases} should actually be seen in absorption\ntroughs, since much of the light reaching the observer in those spectral\nregions has been absorbed and reemitted by the line \\citep{Howell01}. \n\nA key point is that, in either case, to produce line trough polarization changes\nunder the ellipsoidal model, some continuum polarization must exist. In fact,\n\\citet{Leonard4} show that the strength of a line trough polarization feature\ncan be used to place a lower bound on the true intrinsic continuum polarization\nlevel under the ellipsoidal model according to\n\n\\begin{equation}\np_{cont} \\geq \\frac{\\Delta p_{tot}} {(I_{cont}\/I_{trough}) - 1},\n\\label{eqn:1}\n\\end{equation}\n\n\\noindent \nwhere $I_{cont}$ is the interpolated value of the continuum flux\nat the location of the line trough, $I_{trough}$ is the total flux at the\nline's flux minimum, and $\\Delta p_{tot}$ is the total polarization change\nobserved in the line trough. Thus, a critical test of the ellipsoidal model is\nto see whether the continuum polarization observed in the spectral region\n6800--7800~\\AA\\ is sufficient to explain observed line trough polarization\nchanges. Lack of significant continuum polarization in an object with strong\nline features argues against the ellipsoidal model. An additional prediction\nis that {\\it no} rotation of the polarization angle (PA) should be seen across\nthe line, since the continuum and line-forming regions share the same geometry.\n\nA final way to produce a polarization change in a line trough is through\nasymmetry in the distribution of elements in the ejecta material located above\nthe photosphere along the line-of-sight (l-o-s), hereafter referred to as the\n``clumpy ejecta'' model. Asymmetry in the distribution of material with\nsignificant optical depth may unevenly block the underlying photospheric light,\nthereby producing a net polarization change and\/or PA rotation in a line\ntrough, even when the photosphere is spherical. The GCD model discussed in\n\\S~\\ref{sec:2.2} provides a natural mechanism by which to generate line-trough\npolarization through this model, since it predicts an optically thick pancake\nof high-velocity, metal-rich material overlying the photosphere on the side in\nwhich the bubble emerged. In particular, if the GCD model provides the correct\nmechanism by which to produce HV~SNe~Ia, then polarization changes in the\nabsorption troughs of the strong, HV metal lines in these objects should be\nparticularly pronounced since the l-o-s necessarily intercepts a substantial\nfraction of the pancake.\n\nFor SNe~Ia, an interesting alternative to the simple ellipsoidal model for\nproducing {\\it both} line and continuum polarization is presented by\n\\citet{Kasen04}, who explore the polarization consequences of a conical hole in\nthe ejecta due to the interaction with a companion star, which we shall refer\nto as the ``ejecta-hole'' model. By considering various viewing angles and\nhole sizes, \\citet{Kasen04} demonstrate that both continuum and line\npolarization and PA changes can be generated. In general, viewing angles\nalmost directly down the hole yield low continuum polarization with\npolarization increases in strong line troughs, whereas sight lines more nearly\nperpendicular to the hole result in larger continuum polarization (prominently\nseen in the spectral region 6800--7800~\\AA) and ``inverted P-Cygni'' line\npolarization profiles; this latter case is nearly indistinguishable from the\npredictions of the simple ellipsoidal model.\n\nIonization asymmetry is also capable of generating both continuum polarization\nas well as PA and polarization level changes through line features, since both\ncontinuum and line photons will scatter off of concentrations of free electrons\nin (or beyond) the ejecta. An impressive example of this mechanism potentially\nbeing at work is given by \\citet{Chugai92} for the case of SN~1987A, in which\nthe spectropolarimetry data, as well as asymmetries in the flux line profiles,\nare convincingly reproduced by the effects of two clumps of $^{56}{\\rm Ni}$ in\nthe far (receding) hemisphere of the ejecta.\n\n\\subsubsection{Removing Interstellar Polarization}\n\\label{sec:2.3.2}\n\nA problem that plagues interpretation of all SN polarization measurements is\nproper removal of the ISP. Since directional extinction resulting from\naspherical interstellar dust grains aligned by some mechanism along the l-o-s\nto an SN can contribute a large polarization to the observed signal, an attempt\nmust be made to remove it prior to analyzing SN spectropolarimetry data. This\nis notoriously difficult, although a number of different techniques have been\nadvanced over the years, which we here summarize.\n\nAn excellent way to derive Galactic ISP is through observations of distant,\nintrinsically unpolarized, ``probe stars'' close to (within $\\sim$0.5$\\ensuremath{^{\\circ}}$)\nthe l-o-s to the SN \\citep[e.g.,][]{Leonard8,Leonard7}. Deriving the total\nISP, which includes the contribution from dust in the host galaxy, however, is\nmore difficult.\n\nThe most basic technique is to {\\it place limits on the ISP from reddening\nconsiderations.} Since the same dust that polarizes starlight should redden it\nas well, it seems logical to expect a correlation between reddening and ISP.\nBecause the alignment of dust grains is not total (or has multiple preferred\norientations due to non-uniformity of the magnetic field along the l-o-s), and\ngrains are probably only moderately elongated particles, it is not surprising\nthat, through the analysis of thousands of reddened Galactic stars, only an\nupper bound on the polarization efficiency of Galactic dust can be derived\n\\citep{Serkowski75}: ${\\rm ISP} \/ E_{B-V} < 9.0\\%\\ {\\rm mag}^{-1}$. However,\nthe polarization efficiency of the dust in external galaxies is not well\nstudied, and in one of the few investigations carried out to date,\n\\citet{Leonard7} find compelling evidence for polarization efficiency well in\nexcess of the empirical Galactic limit for dust in NGC~3184 along the l-o-s to\nthe Type II-P SN~1999gi. It is not clear at this point whether meaningful\nconstraints can thus be placed on the ISP of extragalactic SNe that are\nsignificantly reddened by host-galaxy dust. Nonetheless, total reddening\narguments are still often used to set ``reasonable'' limits on the ISP.\n\nAnother method to get a handle on the ISP is to assume axisymmetry for any SN\nasphericity, a situation that reveals itself through a straight-line\ndistribution of points when the spectropolarimetry is plotted in the $q$--$u$\nplane. If axisymmetry exists, the ISP is constrained to lie along the axis\ndefined by the line \\citep{Howell01}. Its absolute value, however, is\nuncertain without additional input.\n\nA more precise technique relies on the theoretical expectation that {\\it\nunblended emission lines consist of unpolarized light}, and that any\npolarization observed in emission-line photons comes from the ISP\n\\citep{Jeffery91a,Tran97,Wang96,Leonard3}. This assumption is thought to be\nmost valid at early times, when any $^{56}{\\rm Ni}$ concentrations are likely\nto be below the photosphere. Note, though, that to use this method, care must\nbe taken to isolate the emission-line photons from the underlying continuum\nlight \\citep[e.g.,][]{Tran97}, which may be intrinsically polarized.\n\nA related method is to assume that a particular spectral region is\nintrinsically {\\it completely} unpolarized, with all of the observed\npolarization therefore coming from ISP. Some empirical support exists that, in\nsome objects, this may be the case for very strong emission lines\n\\citep{Kawabata02,Leonard8,Wang04a}. For SNe~Ia in particular, it has\nsometimes been assumed that the far blue spectral region (e.g., typically $<\n5000$~\\AA) satisfies this criterion due to the effect of the heavy\nline-blanketing and multiple, depolarizing, line scatters, largely due to\niron-group elements. This qualitative expectation is demonstrated\nquantitatively by \\citet{Howell01}, who present model polarization spectra\nresulting from delayed-detonation models in realistic SN Ia atmospheres. More\nrecently, \\citet{Wang05} update this approach by choosing only specific\nspectral regions at blue wavelengths ($\\lambda < 5000$~\\AA) that are not\ndominated by obvious individual line features in either flux or polarization\nfor the ISP determination, rather than the arbitrary blue edge of the spectrum, as \nadopted by \\citet{Howell01}. The central idea here is that while strong individual line \nfeatures might impart their own polarization signature (e.g., through geometrical \ndilution in ellipsoidal models, or through blocking of the parts of the photosphere \nin clumpy-ejecta models), the spectral regions in between specific features\nprobably decrease any effective continuum polarization by the numerous\noverlapping spectral lines at blue wavelengths.\n\nFinally, if polarimetry is obtained after the electron-scattering\noptical depth of the atmosphere has dropped well below unity, then it\nmay be adequate to assume the observed polarization to be due entirely\nto ISP across the whole spectrum \\citep[e.g.,][]{Wang03}.\n\nImproperly removed, ISP can increase or decrease the derived intrinsic\npolarization, and it can change ``valleys'' into ``peaks'' (or vice\nversa) in the polarization spectrum. Since it is so difficult to be\ncertain of accurate removal of ISP, it is generally safest to focus on\n(a) temporal changes in the polarization with multiple-epoch data, (b)\ndistinct line features in spectropolarimetry having high\nsignal-to-noise ratio (S\/N), and (c) continuum polarization unlike the\ncharacteristic ``Serkowski-law'' wavelength dependence imparted by\ndust \\citep[e.g.,][]{Whittet92}.\n\n\\subsubsection{Previous SN~Ia Polarimetry Studies}\n\\label{sec:2.3.3}\n\nEvidence for intrinsic polarization in SNe~Ia was initially difficult to find,\nas the first investigations detected only marginally significant polarization\namong normal-brightness events observed near maximum, $p < 0.2$\\%\n\\citep[e.g.,][]{Wang96,Wang97}. Broad-band polarimetry of one overluminous\nevent observed over a month after maximum also found no intrinsic polarization\ndown to a level of $p \\approx 0.3\\%$ \\citep{Wang96}. Significant advances have\nbeen made in the last few years. \\citet{Leonard1} reported the first\nconvincing, albeit weak, features in the polarization of an SN~Ia, SN~1997dt,\nwhich was likely a subluminous event; these data are presented and analyzed in\nmore detail in the present study.\n\nThe first thorough spectropolarimetric study of an SN~Ia is that by\n\\citet{Howell01}. In spectropolarimetry obtained at maximum light, the\nsubluminous SN~1999by exhibits a polarization change of $\\sim$0.8\\% from\n4800~\\AA\\ to 7100~\\AA, and a sharp polarization modulation of $\\sim$0.4\\%\nacross the strong \\ion{Si}{2} $\\lambda6355$ absorption. These features are\nexplained within the context of an ellipsoidal model with a global asphericity\nof $\\sim$20\\%, observed equator-on. This physical picture was achieved by\nassuming the ISP to be the observed polarization at the blue edge of the\nspectrum, which was about $0.2\\% {\\rm\\ at\\ } 4800$~\\AA. With this choice of\nISP, the inferred intrinsic polarization rises from $0\\%$ to about $0.8\\%$ from\nblue to red, with a sharp depolarization from $0.4\\%$ to near $0\\%$ across the\n\\ion{Si}{2} $\\lambda 6355$ feature. An argument supporting both the\nellipsoidal model as well as the ISP choice is that after ISP removal the PA\nbecomes nearly independent of wavelength. \\citet{Howell01} note, on the other hand, \nthat a wavelength-independent PA also results if one assumes the ISP to be the\nobserved polarization of the far {\\it red} edge of the spectrum; in this case,\nthe polarization modulation across the \\ion{Si}{2} $\\lambda 6355$ becomes a\npolarization increase. Such a scenario could result from selective blocking of\nforward-scattered light, as described in \\S~\\ref{sec:2.3.1}. However,\n\\citet{Howell01} find the theoretical arguments supporting the former ISP to be\nmore compelling. With this ISP, the redward rise in intrinsic polarization is\nattributed to the decreasing importance of line opacities, and the increased\ninfluence of continuum electron scattering at longer wavelengths. The\ndepolarization across the \\ion{Si}{2} $\\lambda6355$ absorption is also\nattributed to the depolarizing effect of line scattering. Despite the\ndetection of intrinsic polarization for SN~1997dt and SN~1999by, their unusual\n(subluminous) nature nevertheless left some doubt about intrinsic polarization in {\\it\nnormal} SNe~Ia.\n\n That doubt has recently been put to rest with the work of \\citet{Kasen03} on\nSN~2001el. For this normal-luminosity event, the percent polarization changed\nfrom blue to red by $\\sim$0.4\\% in spectropolarimetry obtained 1 week before\nmaximum brightness. However, the extraordinary feature here is the existence\nof distinct high-velocity \\ion{Ca}{2} near-infrared (IR) triplet absorption\n($v$ = 18,000--25,000 km s$^{-1}$) in addition to the usual, lower-velocity\n\\ion{Ca}{2} feature. A similar, but much weaker, high-velocity feature had\nbeen previously observed in SN~1994D, and perhaps in other SNe~Ia as well; the\nnumber of pre-maximum spectra covering the near-IR spectral range is small.\nThe polarization is seen to change dramatically in this feature, by $\\sim$0.4\\%. \nThe \\ion{Ca}{2} near-IR feature is examined by \\citet{Kasen03} in an\nelegant study, which concludes that it is likely due to photospheric\nobscuration by a clumped shell of high-velocity material. Using multi-epoch\ndata, \\citet{Wang03} demonstrate that the nature of the polarization changes\nover the course of two weeks following this early epoch, becoming nearly\nundetectable a week after maximum brightness, further solidifying the case for\nintrinsic polarization at early times. Although not commented on by either\nstudy, it appears that the \\ion{Si}{2} $\\lambda 6355$ line also shows a\npolarization modulation in the earliest epoch, and with a different PA from the\n\\ion{Ca}{2} near-IR feature. Although it does not affect the main results, a\ncautionary note on the difficulty of ISP determination is set by the fact that\nthe two studies arrive at quite different values: \\citet{Kasen03} derive an ISP\nof very nearly $0\\%$ by assuming the blue edge of the spectrum to be\nunpolarized, whereas \\citet{Wang03} obtain an ISP of $\\sim$0.6\\% by\nattributing the observed polarization $38$ days after maximum light, when\nSN~2001el is argued to be in the nebular phase, entirely to ISP.\n\nMost recently, \\citet{Wang05} analyze a single epoch of pre-maximum\nspectropolarimetry of SN~2004dt, an HV~SN~Ia for which an ISP of $q_{\\rm ISP} =\n0.2 \\pm 0.1\\%$ and $u_{\\rm ISP} = -0.2 \\pm 0.1\\%$ is derived from the observed\npolarization of a handful of narrow, blue spectral regions in which no single\nspectral feature dominates in the total-flux spectrum. This results in rather\nlow continuum polarization, $p \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 0.4\\%$, but very strong modulations\nacross spectral lines. The polarization spikes reach $2\\%$ in the deep troughs\ndue to \\ion{Si}{2} $\\lambda 4130$ and $\\lambda 6355$; lesser peaks are observed\nin features identified with \\ion{Mg}{2} $\\lambda 4471$, a blend of \\ion{Si}{2}\n$\\lambda\\lambda 5041, 5056$ and \\ion{Fe}{2} $\\lambda\\lambda 4913, 5018, 5169$,\nand the \\ion{Ca}{2} near-IR triplet. All line polarization has similar directional\nbehavior in the $q$--$u$ plane, suggesting a common origin. Interestingly,\nwhereas other strong line features in the total-flux spectrum are characterized\nby strong polarization modulations, \\ion{O}{1} $\\lambda7774$ shows no\npolarization signature. These features are explained in terms of optically\nthick bubbles of IMEs, the result of partial burning, that are asymmetrically\ndistributed within an essentially spherical oxygen substrate that remains from\nthe progenitor. Note that \\citet{Wang97a} find polarization\nvariation at a level of $> 0.5\\%$ across the strong \\ion{Si}{2}\n$\\lambda 6355$ feature of another HV~SN~Ia, SN~1997bp, in unpublished\nspectropolarimetry obtained near maximum light.\n\nFrom this small sample, the observations thus far suggest that normal and,\nperhaps, overluminous events are weakly polarized ($p \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 0.4\\%$), with\nsubluminous ones possessing somewhat larger values ($p \\approx 0.8\\%$). HV~SNe~Ia\nhave modest continuum polarization, but possess the highest line polarization\nof all, achieving polarizations of up to $2\\%$ in the strongest features. We\nassess the robustness of these tentative trends with the present study of\nfour SNe~Ia.\n\n\\subsection{The Type Ia Supernovae 1997dt, 2002bf, 2003du, and 2004dt}\n\\label{sec:2.4}\n\nSN~1997dt was discovered \\citep{Qiao97} by the Beijing Astronomical Observatory\nSupernova Survey \\citep{Li96} on 1997 November 22.44 (UT dates are used\nthroughout this paper) at an unfiltered magnitude of $\\sim$15.3 in the Sbc\ngalaxy NGC~7448. Images of the same field taken eight days earlier show no star\nat the position of the SN to a limiting unfiltered magnitude of about 18.5. An\noptical spectrum obtained immediately (0.06 days) after discovery showed it to\nbe a Type~Ia event \\citep{Qiao97}; a subsequent examination by \\citet{Li01a}\nestimates the age at discovery to be $-7 \\pm 5$ days relative to maximum light,\nwhich places the date of maximum at very roughly 1997 November 29.\n\\citet{Tonry03} report that SN~1997dt suffers from a host extinction of $A_V =\n0.46 {\\rm\\ mag}$, the median of the values derived from MLCS and ``Bayesian\nAdapted Template Match Method'' \\citep[J. L. Tonry et al. 2005, in\npreparation;][]{Tonry03} analyses of its unpublished light curves. An MLCS2k2\nfit (S. Jha et al., in preparation) suggests that it is subluminous, with\n$\\Delta = 0.94$ (corresponding to a $\\Delta m_{15}(B) \\approx 1.8$ mag; see\n\\S~\\ref{sec:2.1}), although there is a long shoulder of probability to lower\n$\\Delta$ values, indicating that there is a wide range of light-curve\nshapes that can fit the sparse number of photometric points (2 in $B$, 3 in\n$V$, and 2 in $I$). From a pre-maximum spectrum posted at the Center for\nAstrophysics' Recent Supernovae\nPage,\\footnote{\\url{http:\/\/cfa-www.harvard.edu\/cfa\/oir\/Research\/supernova\/RecentSN.html}}\nwe estimate $\\cal R$(\\ion{Si}{2}) $\\approx 0.3$ (see \\citealt{Nugent95}), which\nimplies $\\Delta m_{15}(B) \\approx 1.33$ mag from the correlation derived by\n\\citet{Benetti04}, also consistent with a somewhat subluminous classification.\n\nSN~2002bf was discovered \\citep{Martin02} by the Lick Observatory and Tenagra\nObservatory Supernova Searches \\citep[LOTOSS;][]{Schwartz00} on 2002 February\n22.30 at an unfiltered magnitude of $\\sim$17, very close to the nucleus of the\nSb galaxy PGC~029953. An image of the same field taken twenty days earlier\nshowed nothing at the position of SN~2002bf to a limiting unfiltered magnitude\nof $\\sim$19. Optical spectra obtained on 2002 March 6.21 by \\citet{Matheson02}\nand on 2002 March 7.41 by \\citet{Filippenko02} identified it as a Type~Ia\nevent. Both groups noted that the expansion velocity derived from the\nabsorption minimum of the \\ion{Si}{2} $\\lambda 6355$~\\AA\\ line was\nsignificantly greater than normal for an SN~Ia near maximum light, indicating\nthat it may be an HV~SN~Ia. A ``spectral feature age'' \\citep{Riess97} of $0\n\\pm 2$ days was derived from the March 6 spectrum \\citep{Matheson02}. The\nMLCS2k2 analysis of the light curves presented in \\S~\\ref{sec:4.2.1} yields\n2002 March $4.37 \\pm 0.50$ as the date of maximum $B$ light, consistent with\nthe age on March 6 derived by \\citet{Matheson02}.\n\nSN~2003du was discovered \\citep{Schwartz03} by LOTOSS on 2003 April 22.4 at an\nunfiltered magnitude of $\\sim$15.9 in the SBd galaxy UGC~9391. An image of the\nsame field taken fifteen days before discovery showed nothing at the position\nof SN~2003du to a limiting unfiltered magnitude of $\\sim$19. An optical\nspectrum obtained shortly thereafter, on 2003 April 24.06, identified it as a\nType~Ia event roughly two weeks before maximum light \\citep{Kotak03}. The\nMLCS2k2 analysis of the light curves presented in \\S~\\ref{sec:4.2.1} yields\n2003 May $6.12 \\pm 0.50$ as the date of maximum $B$ light, consistent with the\nearlier spectral age estimate. It also agrees with the epochs of maximum\ndetermined by the recent photometric studies of SN~2003du by \\citet{Anupama05}\nand \\citet{Gerardy04}.\n\nSN~2004dt was discovered \\citep{Moore04} by the Lick Observatory Supernova\nSearch \\citep{Filippenko03a} at an unfiltered magnitude of $\\sim$16.1 in the\nSBa galaxy NGC 799 on 2004 August 11.48. An image of the same field taken ten\ndays earlier showed nothing at the position of SN~2004dt to a limiting\nunfiltered magnitude of $\\sim$18. Spectra taken within 2 days of discovery by\n\\citet{Galyam04}, \\citet{Patat04}, and \\citet{Salvo04} confirmed it to be an\nSN~Ia before maximum light. \\citet{Patat04} noted that several absorption\nlines showed high expansion velocities, a result confirmed by the recent study\nby \\citet{Wang05}, suggesting that, like SN~2002bf, SN~2004dt is an HV~SN~Ia.\nA preliminary analysis of the light curves of SN~2004dt shows that maximum\nlight occurred near 2004 August 20 (W. Li, personal communication). A series\nof {\\it Hubble Space Telescope (HST)} UV spectral observations was obtained as\npart of program GO-10182 (P.I. Filippenko), and will be analyzed in a future\npaper.\n\n\n\\section{Observations and Reductions}\n\\label{sec:3}\n\nWe obtained single-epoch spectropolarimetry of SN~1997dt, SN~2002bf, SN~2003du,\nand SN~2004dt on days 21, 3, 18, and 4 (respectively) after maximum light. We\nalso obtained additional optical spectroscopy and \\bvri\\ photometry of\nSN~2002bf and SN~2003du. For SN~2002bf, our photometry samples $-10$ to $57$\ndays from the time of maximum light, with one additional flux spectrum taken on\nday 9 after maximum. For SN~2003du, our ground-based photometry covers $-5$ to\n$113$ days from maximum, with one additional epoch on day 436 taken using the\nHigh Resolution Channel (HRC) of the Advanced Camera for Surveys (ACS) on board\n{\\it HST}. Five additional spectral epochs sample its development from days\n$24$ to $82$ after maximum.\n\n\n\\subsection{Photometry}\n\\label{sec:3.1}\n\n\\subsubsection{Ground-Based Photometry of SN 2002bf and SN 2003du}\n\\label{sec:3.1.1}\n\nAll ground-based photometric data were obtained using either the 0.76-m Katzman\nAutomatic Imaging Telescope \\citep[KAIT; ][]{Filippenko01,Li03} or the Nickel 1\nm reflector \\citep{Li01}, both located at Lick Observatory.\nFigures~\\ref{fig:1} and \\ref{fig:2} show KAIT images of PGC~029953 and\nUGC~9391, the host galaxies of SN~2002bf and SN~2003du, respectively. Also\nlabeled in the KAIT images are the ``local standards'' in both fields that were\nused to measure the relative SN brightness on non-photometric nights. We\nobtained 12 epochs of Johnson-Cousins \\bvri\\ photometry (\\citealt[][for\n$BV$]{Johnson66}; \\citealt[][for $RI$]{Cousins81}) for SN 2002bf, all taken\nwith KAIT, and 38 epochs of \\bvri\\ photometry for SN~2003du, 33 of them taken\nwith KAIT and five with the Nickel telescope. We also obtained three\npre-maximum unfiltered observations of SN~2002bf with KAIT (approximating the\n$R$ band, see \\citealt{Li03}), and one additional epoch of $RI$ photometry with\nKAIT of SN~2003du. \n\nFor the photometry we employed the usual techniques of galaxy ``template''\nsubtraction \\citep[][and references therein]{Li00}, point-spread function\nfitting \\citep[][and references therein]{Stetson91}, and using ``local\nstandards'' to determine the \\bvri\\ brightnesses of the SNe on non-photometric\nnights; in general, we closely followed the technique detailed by\n\\citet{Leonard6}. We note that the galaxy subtraction procedure for SN~2002bf\nwas particularly challenging since it is only $4\\farcs1$ from its host galaxy's\ncenter.\n\nThe absolute calibration of the SN~2002bf field was accomplished on the\nphotometric nights of 2002 May 14 and 2004 March 17 with the Nickel telescope,\nand 2003 February 3 and 2004 March 18 with KAIT, by observing several fields of\nLandolt (1992) standards over a range of airmasses in addition to the SN~2002bf\nfield. The absolute calibration of the SN~2003du field was similarly derived\nfrom data taken on the photometric nights of 2003 May 31, June 1, June 26, and\nAugust 27 with the Nickel telescope, and of 2003 May 22 and 2004 March 18 with\nKAIT. The color terms used to transform the filtered instrumental magnitudes\nto the standard Johnson-Cousins system are those of \\citet{Foley03}. We list\nthe measured \\bvri\\ magnitudes and the $1\\sigma$ uncertainties, taken as the\nquadrature sum of a typical photometric error and the $1\\sigma$ scatter of the\nphotometric measurements from all of the photometric nights, of the local\nstandard stars in Tables~\\ref{tab:1} and \\ref{tab:2}.\n\nAfter deriving the \\bvri\\ magnitudes of the SNe based on a comparison with each\nof the local standards, we took the weighted mean of the individual estimates\nas the final standard magnitude of the SNe at each epoch in each filter. The\nresults of our ground-based photometric observations are given in\nTables~\\ref{tab:3} and \\ref{tab:4} and shown in\nFigures~\\ref{fig:3} and \\ref{fig:4}. The reported uncertainties come from the\nquadratic sum of the photometric errors (reported by DAOPHOT) and the\ntransformation errors. For SN~2002bf, the uncertainty produced by the\ndifficult galaxy-subtraction process contributed the majority of the error on\nnights with low S\/N.\n\n\\subsubsection{{\\it Hubble Space Telescope} Photometry of SN 2003du}\n\\label{sec:3.1.2}\n\nWe obtained {\\it HST}\\ images during the course of two orbits of a\n$29^{\\prime\\prime} \\times 26^{\\prime\\prime}$ field of view centered on\nSN~2003du on 2004 July 15, 434 days after $B_{\\rm max}$, with the ACS\/HRC\ndetector through filters F435W, F555W, F625W, and F814W (hereafter referred to\nas $B, V, R, {\\rm\\ and\\ } I$, respectively), as part of our Snapshot survey\nprogram (GO-10272; P.I. Li) to investigate the late-time photometric behavior and\nenvironment of nearby SNe. SN~2003du was detected in all images. Total\nexposure times in \\bvri\\ were, respectively, $1680 {\\rm\\ s}$ (data archive\ndesignation j8z441011\/3011), $960 {\\rm\\ s}$ (j8z442011\/4011), $720 {\\rm\\ s}$\n(j8z441021\/3021), and $1440 {\\rm\\ s}$ (j8z442021\/4021). \n\nTo derive the {\\it HST} photometry, we followed as closely as possible the\nprocedure detailed by \\citet{Sirianni05}, including correction for the effects\nof SN light contaminating the background region, aperture corrections, the\n``red-halo'' effect (for the $I$-band), and CTE degradation \\citep{Riess03}.\nWe translated the resulting instrumental magnitudes to the standard\nJohnson-Cousins \\bvri\\ system by using the coefficients and color corrections\ntabulated by \\citet{Sirianni05}. The final results of our {\\it HST} photometry\nare included in Table~\\ref{tab:4} and displayed in Figure~\\ref{fig:4}.\n\n\\subsection{Spectropolarimetry and Spectroscopy}\n\\label{sec:3.2}\n\nWe obtained single epochs of spectropolarimetry for SN~2002bf and SN~2003du on\n2002 March 7 and 2003 May 24, respectively, with the Low-Resolution Imaging\nSpectrometer \\citep{Oke95} in polarimetry mode (LRISp; Cohen\n1996)\\footnote{Instrument manual available at\n\\url{http:\/\/www2.keck.hawaii.edu\/inst\/lris\/pol\\_quickref.html}.} at the\nCassegrain focus of the Keck-I 10-m telescope. We observed SN~1997dt on 1997\nDecember 20 with LRISp using the Keck II 10-m telescope, and SN~2004dt on 2004\nAugust 24 with the Kast double spectrograph \\citep{Miller93} with polarimeter\nat the Cassegrain focus of the Shane 3-m telescope at Lick Observatory. We\nreduced the polarimetry data according to the methods outlined by\n\\citet{Miller88} and detailed by \\citet{Leonard3} and \\citet{Leonard4}. \n\nThe polarization angle offset between the half-wave plate and the sky\ncoordinate system was determined by observing the following polarized standard\nstars from the list of \\citet{Schmidt92a} and setting the observed $V$-band\npolarization position angle (i.e., $\\theta_V$, the debiased, flux-weighted\naverage of the polarization angle over the wavelength range 5050--5950~\\AA; see\n\\citealt{Leonard3}) equal to the cataloged value: BD $+64^\\circ106$ (1997\nDecember 20), BD $+59^\\circ389$ (2002 March 7), and HD 161056 (2003 May 24).\nOn the night of 2004 August 24, we averaged the polarization angle offsets\nderived from observations of three polarized standards from the\n\\citet{Schmidt92a} list, HD 204827, BD $+59^\\circ389$, and HD 19820; the\nindividual offsets were internally consistent to within $1^\\circ$. To check\nfor instrumental polarization, the following null standards taken from the\nlists of \\citet{Turnshek90}, \\citet{Mathewson70}, \\citet{Schmidt92a}, and\n\\citet{Berdyugin95}, were also observed: HD 94851 (1997 December 20), HD 57702\n(2002 March 7), HD 109055 and BD $+32^\\circ3739$ (2003 May 24), and HD 212311\n(2004 August 24). All stars were measured to be null to within $0.1\\%$, which\nis also our estimate of the systematic uncertainty of a continuum polarization\nmeasurement made with either the Keck or Lick spectropolarimeters\n\\citep[e.g.,][]{Leonard3}.\n\nAdditional specifics of the observations of SN~2002bf taken on 2002 March 7,\nincluding an investigation of the potential impact of second-order light\ncontamination and instrumental polarization (both shown to be minimal) in the\nsetup used on this night, are given by \\citet{Leonard8}. Second-order light\ncontamination is not a concern for our observation of SN~1997dt due to its\nlimited spectral range. For SN~2003du, the use of a dichroic (D560) to split\nthe beam near 5600~\\AA\\ eliminates second-order light contamination on the\nred side. Our spectropolarimetric observation of SN~2004dt was taken in a\nsetting that included the use of an order-blocking filter (GG455) to prevent\ncontamination by second-order light at red wavelengths. Beyond $\\sim$9000~\\AA, \nhowever, second-order contamination may exist, but for reasons similar to\nthose discussed by \\citet{Leonard8} we believe it to have minimal\nimpact for this particular object.\n\nTo derive the total-flux spectra, we extracted all one-dimensional\nsky-subtracted spectra optimally \\citep{Horne86} in the usual manner. Each\nspectrum was then wavelength and flux calibrated, and was corrected for\ncontinuum atmospheric extinction and telluric absorption bands\n\\citep{Wade88,Bessell99,Matheson00}. With the exception of the\nspectropolarimetric observations of SN~1997dt and SN~2002bf, all spectra were\ntaken near the parallactic angle \\citep{Filippenko82}, so the spectral shape\nshould be quite accurate. Table~\\ref{tab:5} lists the spectropolarimetric and\nspectral observations for all four SNe. Figures~\\ref{fig:5}--\\ref{fig:8}\nshow the observed spectropolarimetry data of the four objects, and\nFigures~\\ref{fig:9} and \\ref{fig:10} show the complete series of spectra\nobtained for SN~2002bf and SN~2003du, respectively.\n\n\\section{Analysis}\n\\label{sec:4}\n\n\\subsection{Spectroscopy}\n\\label{sec:4.1}\n\nOur spectrum of SN~1997dt, taken $\\sim$21 days after maximum, shows typical\nfeatures for an SN~Ia at this phase (Fig.~\\ref{fig:5}a). Similarly, our\nspectral sequence of SN~2003du (Fig.~\\ref{fig:10}) closely follows the\nevolution of the normal-luminosity SN~Ia~1994D \\citep{Patat96,Filippenko97} in\nterms of the strengths and blueshifts of line features. This is consistent\nwith the analyses of \\citet{Anupama05} and \\citet{Gerardy04}, in which spectra\ntaken before and shortly after maximum light were also examined. This\nconvincingly establishes SN~2003du as a spectroscopically ``typical'' SN~Ia\n\\citep{Branch93a}.\n\nAs discussed in \\S~\\ref{sec:2.4}, the spectra of both SN~2002bf and SN~2004dt\nare peculiar in one regard: the blueshifts of many of the spectral lines, most\nnoticeably \\ion{Si}{2} $\\lambda 6355$, occur at significantly higher velocity\nthan is typical for an SN~Ia at this phase, and indicate that these are both\nHV~SNe~Ia.\n\nFor comparison, we have measured the velocities of the \\ion{Si}{2}\n$\\lambda6355$ line in a number of other HV~SNe~Ia from our database, along with\nthe spectroscopically normal SN~1994D, the subluminous SN~1991bg\n\\citep[e.g.,][]{Filippenko92b}, and the overluminous SN~1991T\n\\citep[e.g.,][]{Filippenko92a}. We present the results in Table~\\ref{tab:6}\nand Figure~\\ref{fig:11}, from which it is clear that both SN~2002bf and\nSN~2004dt belong to the class of HV SNe~Ia; indeed, SN~2002bf is the most\nextreme HV~SN~Ia yet observed for its epochs. The figure also suggests that\nline velocity is not strongly correlated with luminosity, although the\nexpansion velocity of the subluminous SN~1991bg is somewhat lower than typical\nvalues.\n\nFigure~\\ref{fig:12} presents a spectral comparison near maximum light of three\nHV~SNe~Ia (SN~2002bf, SN~2002bo, and SN~2004dt) with the spectroscopically\nnormal SN~1994D. The excessive blueshift of the \\ion{Si}{2} $\\lambda 6355$\ntrough is obvious for the three HV~SN~Ia compared with SN~1994D. While it must\nbe cautioned that the spectra span a range of ages of about five days near\nmaximum light, a time when significant spectral development occurs, the\nblueshift differences are much greater than can be explained by age differences\nalone. In addition, the \\ion{Si}{2} line is significantly stronger in the\nHV~SNe~Ia compared with SN~1994D, with equivalent widths of $\\gtrsim 140$~\\AA\\\ncompared with $\\sim$100~\\AA\\ for SN~1994D. Conversely, the \\ion{O}{1}\n$\\lambda 7774$ absorption is relatively weaker in the HV~SN~Ia events, with\nequivalent widths of $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 100$~\\AA\\ in the HV~SNe~Ia compared with\n$125$~\\AA\\ for SN~1994D. The fact that the \\ion{Si}{2} line is significantly\nstronger in the HV~SNe~Ia sample, and the \\ion{O}{1} line relatively weaker, is\nconsistent with the scenario in which a greater fraction of C and O is burned\nto IMEs in HV~SN~Ia than in more typical events; it also follows naturally from\nthe GCD model, since the obscuring pancake is formed from the products of\noxygen burning.\n\nWhile HV~SNe~Ia share many spectral characteristics, it is clear that they also\nexhibit spectral diversity. For instance, whereas the absorption trough of the\n\\ion{Ca}{2} near-IR triplet is significantly blueshifted for SN~2002bf and\nSN~2002bo relative to SN~1994D ($16,300\\ {\\rm km\\ s}^{-1}$ and $14,500\\ {\\rm km\\ s}^{-1}$ for SN~2002bf\nand SN~2002bo, respectively, compared with $\\sim$10,900~km s$^{-1}$\\ for SN~1994D,\nwhere we have assumed $\\lambda_0 = 8579$ for the \\ion{Ca}{2} near-IR triplet, a\nvalue derived using the prescription given by \\citealt{Leonard5}), it has a\nmore normal blueshift ($v = 10,900$~km s$^{-1}$) in SN~2004dt. The equivalent widths\nof the \\ion{Ca}{2} near-IR lines also show a suggestive trend, with both\nSN~2002bf and SN~2002bo having widths of $> 200$~\\AA, while both SN~2004dt and\nSN~1994D have equivalent widths of $\\sim$100~\\AA. This may indicate that the\nexplosive nucleosynthesis did not proceed up to Ca as far out in the atmosphere\n(or in the bubble in the GCD scenario) of SN~2004dt as it did in the other two\nHV~SNe.\n\nFinally, it is clear from Figure~\\ref{fig:12} that not all lines share the\nextreme velocities seen in the \\ion{Si}{2} $\\lambda 6355$ feature. For\ninstance, as first noticed by \\citet{Wang05} in a pre-maximum spectrum of\nSN~2004dt, the \\ion{S}{2} ``W'' feature in the spectra of SN~2002bo and SN~2004dt\nindicates velocities comparable to those in SN~1994D; for SN~2002bf they\nare somewhat larger, but still not near the velocity of the \\ion{Si}{2}\n$\\lambda 6355$ line. \\citet{Wang05} propose that this may indicate that sulfur\nis more confined to the lower-velocity, inner region.\n\n\\subsection{Photometry}\n\\label{sec:4.2}\n\n\\subsubsection{Ground-Based Photometry}\n\\label{sec:4.2.1}\n\nThe results of the MLCS2k2 application to the photometry of SN~2002bf and\nSN~2003du are given in Table~\\ref{tab:7}. For details of the MLCS2k2 procedure\nused, see S. Jha et al. (in preparation); an overview of the technique is\nprovided by \\citet{Riess05}. The MLCS2k2 analysis finds SN~2002bf to be of\ntypical luminosity. SN~2003du is slightly overluminous, although its\npre-maximum spectral evolution \\citep{Anupama05} demonstrates that it is {\\it\nnot} a SN~1991T-like event, as the strength of the \\ion{Si}{2} $\\lambda 6355$\nfeature is comparable to that seen in spectra of normal SNe~Ia.\n\nFor SN~2002bf, $B_{\\rm max}$ occurred on 2002 March $4.37 \\pm 0.50$, which is\n$10$ days after discovery and 5 days before our filtered observations\ncommenced. For SN~2003du, $B_{\\rm max}$ occurred on 2003 May $6.12 \\pm 0.50$,\nwhich is $14$ days after discovery, and the same day that our \\bvri\\\nobservations began. Our derived date of maximum light for SN~2003du agrees\nwith those found by \\citet{Anupama05} and \\citet{Gerardy04} using independent\ndata sets.\n\n\\subsubsection{{\\it HST} Photometry of SN~2003du}\n\\label{sec:C1c}\n\nLate-time photometry (e.g., $t > 200 {\\rm\\ d}$) of SNe~Ia exists for only a\nhandful of objects \\citep[see, e.g., ][and references therein]{Milne01}. From\nthe small sample, there are two main features of note. First, SN~Ia decline\nrates are typically much faster than the decay slope of $^{56}{\\rm Co}\n\\rightarrow$ $^{56}{\\rm Fe}$ of $0.98\\ {\\rm mag\\ (100\\ d)}^{-1}$ predicts.\nThis decay mechanism is thought to be primarily responsible for powering the\nluminosity from the early nebular phase out to $\\sim$1000 days for SNe of all\ntypes. Essentially, the $^{56}{\\rm Co}$ decays release most of their energy in\nthe form of $\\gamma$-rays, which, given enough optical depth, can become\ntrapped in the ejecta and Compton scatter off free electrons. The energetic\nelectrons generate optical photons primarily through ionization and excitation\nof atoms, and the ejecta are transparent to these photons. A small portion\n($\\sim$3.5\\%, see \\citealt*{Arnett79}) of the total $^{56}{\\rm Co}$ decay\nenergy comes in the form of positrons, which may deposit their kinetic energy\nin the ejecta and then annihilate with electrons, producing two $\\gamma$-ray\nphotons of energy $E_\\gamma = m_e c^2$. The steeper decline seen in the\nlate-time photometry of SNe~Ia has been explained by significant transparency\nof the ejecta to $\\gamma$-ray photons \\citep{Milne99} and positron escape\n\\citep{Milne01}.\n\nA quantitative measure of the late-time decline is given by\n\\citet{Cappellaro97}, who investigate how the decline from maximum to 300 days\nafter peak $V$ brightness, denoted $\\Delta m_{300}(V)$, correlates with\nintrinsic SN brightness as derived from the $\\Delta m_{15}(B)$ parameter.\nAlthough limited by small sample size (5 objects), they find a convincing\ncorrelation, with $\\Delta m_{300}(V)$ going from 6.7 mag to 8.4 mag as the\nsample runs from overluminous (SN~1991T) to subluminous (SN~1991bg) events,\nwith the normal-brightness SN~1994D characterized by $\\Delta m_{300}(V) = 7.3$\nmag. A second feature that has thus far been seen in only two SN~Ia events\n(SN~1991T and SN~1998bu; see \\citealt*{Schmidt94a} and \\citealt*{Cappellaro01},\nrespectively) is a sudden flattening of the late-time optical light curves,\nwhich has been attributed to the contribution of a light echo from foreground\ndust clouds.\n\nOur photometric data from {\\it HST}, taken $436$ days after $B_{\\rm max}$,\nallow us to investigate the late-time photometric behavior of SN~2003du. From\nthe inset of Figure~\\ref{fig:4}, it is clear that SN~2003du, like all SNe~Ia\nanalyzed before it, declines significantly faster than the $^{56}{\\rm Co}\n\\rightarrow$ $^{56}{\\rm Fe}$ decay rate. In fact, we measure the average decay\nrate in $V$ (shown by \\citealt*{Milne01} to track the bolometric luminosity of\nan SN~Ia quite accurately) of SN~2003du between our last two photometric epochs\non days $113$ and $436$ to be $\\Delta V = 1.47 \\pm 0.02 {\\rm\\ mag\\ (100\\\nd)}^{-1}$. The slope also appears to have been rather constant throughout the\nperiod between our two last epochs, as indicated by the good agreement of the\nlate-time data taken from \\citet{Anupama05} near day 300 and the decay slope\ndetermined from our data alone. Using the \\citet{Anupama05} data point and our\nestimate of $V_{\\rm max}$ (Table~\\ref{tab:7}), we derive $\\Delta m_{300}(V) =\n6.74$, which is most consistent with the values found by \\citet{Cappellaro97}\nfor overluminous events. That SN~2003du may be somewhat overluminous was also\nsuggested by its peak $V$ magnitude of $-19.67 \\pm 0.02$ mag, which is 0.17 mag\nbrighter than the fiducial template used in the MLCS2k2 procedure\n(Table~\\ref{tab:2}). There is no evidence from our data of any contribution to\nthe SN brightness from a light echo, although we note that the major\nindications of additional contributions to the apparent brightness of SNe~1991T\nand 1998bu did not become obvious until epochs $\\gtrsim 500 {\\rm\\ d}$.\n\n\\subsection{Reddening}\n\\label{sec:4.3}\n\n\\subsubsection{Techniques to Estimate SN Ia Reddening}\n\\label{sec:4.3.1}\n\nAccurate determination of SN reddening is crucial both for deriving intrinsic\nSN properties as well as interpreting spectropolarimetry, since the same dust\nthat reddens SN light can also polarize it as discussed in \\S~\\ref{sec:2.3.2}.\nWhen multi-band photometry is available, the MLCS2k2 technique can accurately\nestimate the total extinction (see \\citealt{Riess05} for discussion). When\nthis is lacking, other methods must be used.\n\nGalactic extinction along the l-o-s is accurately estimated by the dust maps of\n\\citet[][hereafter SFD]{Schlegel98} to an estimated precision of $\\sim$15\\%.\nFor host-galaxy extinction, a rather crude approach, to which many\ninvestigators resort in the absence of photometry, is to employ the rough\ncorrelation found between the total equivalent width ($W_\\lambda$) of the\ninterstellar (IS) \\ion{Na}{1} D doublet ($\\lambda\\lambda 5890, 5896$) and\nreddening \\citep{Barbon90}. The \\citeauthor{Barbon90} correlation has\nbeen subsequently improved upon by \\citet{Munari97}, who derive a more precise\nrelation that uses just the equivalent width of the \\ion{Na}{1} D2 ($\\lambda\n5890$) line. Both relations, however, warrant healthy degrees of skepticism\nsince sodium is known to be only a fair tracer of the hydrogen gas column\n(especially in dense environments, where sodium may be heavily depleted; e.g.,\n\\citealt*{Cohen73}), from which the dust column is then estimated. The\ndust-to-gas ratio also varies significantly among galaxies (e.g.,\n\\citealt*{Issa90}), and it could well be that in the case of SNe,\ncircumstellar, rather than interstellar, dust is present, for which the\ndust-to-gas ratio (or the extinguishing properties of the dust itself) could be\nunusual. A final complication is that, at the resolution typical of most\noptical spectra, individual absorption components along the l-o-s, whose\ncontributions should be considered separately to determine the total reddening,\nare blended into a single profile. In such situations, the \\citet{Munari97}\nrelation formally yields an {\\it upper limit} to the reddening, and caution\nmust be used since the derived value could seriously overestimate the actual\nreddening. Nonetheless, the \\citet{Munari97} relation is still frequently used\nto get approximate reddening values, or upper limits, especially in the case of\nnull detections of IS \\ion{Na}{1} D lines.\n\n\n\\subsubsection{The Reddening of the Four SNe Ia}\n\\label{sec:4.3.2}\n\nFor SN~2002bf and SN~2003du we shall adopt the MLCS2k2 values of $E\\BminusV = 0.08\n\\pm 0.04$ mag and $E\\BminusV = 0.01 \\pm 0.01$ mag (respectively) reported in\nTable~\\ref{tab:7} (for $R_V = 3.1$). For SN~1997dt, \\citet{Tonry03} report a\nhost-galaxy reddening value of $E\\BminusV_{\\rm Host} = 0.15$ mag (assuming $R_V =\n3.1$), the median of the values derived from MLCS2k2 and ``Bayesian Adapted\nTemplate Match Method'' analyses. The SFD dust maps predict $E\\BminusV_{\\rm\nSFD}^{\\rm MW} = 0.06$ mag for SN~1997dt, for a total estimate of $E\\BminusV = 0.21$\nmag.\n\nOur spectrum of SN~1997dt also exhibits strong \\ion{Na}{1} D IS absorption at\nboth the redshift of the host galaxy and the MW. The resolution of our\nspectrum, $\\sim$5~\\AA\\ (Table~\\ref{tab:5}), while sufficient to deblend the\n\\ion{Na}{1} D doublet, is not fine enough to resolve the individual absorption\ncomponents that likely contribute to the D1 and D2 profiles. For host-galaxy\nabsorption, we measure $W_\\lambda^{\\rm tot} = 0.77$~\\AA, with $W_\\lambda^{\\rm\nD2} = 0.42$~\\AA\\ and $W_\\lambda^{\\rm D1}= 0.35$~\\AA. This translates to\n$E\\BminusV_{\\rm NaID}^{\\rm Host} \\leq 0.21$ mag from the \\citet{Munari97} relation.\nFor the MW, we measure $W_\\lambda^{\\rm tot} = 0.76$~\\AA, with $W_\\lambda^{\\rm\nD2} = 0.44$~\\AA\\ and $W_\\lambda^{\\rm D1} = 0.32$~\\AA, yielding $E\\BminusV_{\\rm\nNaID}^{\\rm MW} \\leq 0.23$ mag from the \\citeauthor{Munari97} relation. The\nvalues given by SFD and \\citet{Tonry03} are consistent with the upper limits\nderived from the sodium relation. The purported accuracy of the SFD Galactic\nreddening value and the upper limit set by the \\citeauthor{Munari97} relation\nfor the host reddening would indicate an upper reddening limit of $E\\BminusV_{\\rm\ntotal} \\leq 0.27$ mag. This limit will prove to have important implications\nwhen we examine the spectropolarimetry of SN~1997dt in \\S~\\ref{sec:4.4.2}. The\nfact that the \\citeauthor{Munari97} relation overpredicts the reddening for\nboth the host galaxy and, especially, the MW, may indicate that multiple,\nunresolved components make up the IS line profiles. Given the upper reddening\nlimit, we conclude $E\\BminusV = 0.21 \\pm 0.06$ mag for SN~1997dt.\n\nFor SN~2004dt, $E\\BminusV_{\\rm SFD}^{\\rm MW} = 0.03$ mag, and \\citet{Wang05} report\na range of $E\\BminusV_{\\rm total} = 0.14$ to 0.2 mag from analysis of the color of\ntheir pre-maximum spectrum. However, they also note that SN~2004dt does not\nshow noticeable \\ion{Na}{1} D IS lines at the redshift of NGC~799. We confirm\nthe lack of \\ion{Na}{1} D lines in our spectrum as well, although its poor\nresolution ($\\sim$18~\\AA, see Table~\\ref{tab:5}) makes deriving even an upper\nlimit to the strength of the \\ion{Na}{1} D IS lines difficult. Formally, our\nprocedure yields $W_\\lambda {\\rm (3\\sigma)} = 0.1$~\\AA\\ for host-galaxy\n\\ion{Na}{1} D, which translates to a predicted upper limit of $E\\BminusV_{\\rm\nNaID}^{\\rm Host} < 0.02$ mag. We thus confirm the discrepancy noted by\n\\citet{Wang05} between the lack of IS sodium absorption and reddening inferred\nfrom other methods. It could be that the spectroscopic peculiarities of HV\nSNe~Ia produce photometric irregularities that affect the general reddening\nrelations. As mentioned in \\S~\\ref{sec:2.2}, \\citet{Benetti04} do find\nphotometric peculiarities, albeit minor, in their study of another HV~SN~Ia,\nSN~2002bo. However, since the \\ion{Na}{1} D relation, especially for the host\ngalaxy, is also prone to error, we have no convincing way to decide between the\ntwo estimates. We thus take the simple average of the low and high reddening\nestimates, and incorporate the disparity into our final estimate's uncertainty.\nThis yields $E\\BminusV = 0.11 \\pm 0.06$ mag as our best reddening estimate for\nSN~2004dt.\n\n\\subsection{Spectropolarimetry of Four SNe Ia}\n\\label{sec:4.4}\n\nTo summarize: SN~1997dt is likely a somewhat subluminous event that is thought\nto be reddened by $E\\BminusV = 0.21 \\pm 0.06$ mag, with an upper limit of $0.27$\nmag. SN~2002bf and SN~2004dt are HV~SNe~Ia, with SN~2002bf being the most\nextreme example yet observed. Finally, SN~2003du is a slightly overluminous\nevent that is minimally reddened. At maximum light, a previous subluminous\nevent has been found to be moderately polarized ($p \\approx 0.8\\%$), whereas\nnormal to overluminous examples have been less so ($p \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 0.4\\%$). The\ntwo HV~SNe~Ia that have reported polarization measurements show the greatest\npolarization features yet observed for any SN~Ia type, with values approaching\n$2\\%$ in the strongest lines of SN~2004dt (\\S~\\ref{sec:2.3.3}).\n\nWe now examine our single-epoch spectropolarimetry of SN~1997dt, SN~2002bf, SN\n2003du, and SN~2004dt obtained on days 21, 3, 18, and 4, respectively, after\nmaximum light. For each SN, we shall first discuss the observed polarization\nand then attempt to remove the ISP through the technique of \\citet{Wang05},\nwhich assumes that spectral regions lacking strong individual flux or\npolarization features at blue wavelengths ($\\lambda < 5000$~\\AA) are\nintrinsically unpolarized. Since the choice of the specific spectral regions\nto use for ISP determination is admittedly somewhat subjective, we shall be\ncareful to point out how our conclusions would change with other ISP choices.\n\n\n\nFollowing ISP removal and examination of the data in the $q$--$u$ plane, we\ncalculate the intrinsic polarization of the SN by rotating the $q$--$u$ axes\nthrough a single angle, $\\theta$, that places the greatest degree of\npolarization change across the spectrum along the rotated $q$ axis. The angle\nby which the axes are rotated is determined through a uniform-weight,\nleast-squares fit to the ISP-subtracted data in the $q$--$u$ plane. The\npolarization degree measured along the rotated $q$ axis is then referred to as\nthe rotated Stokes parameter \\citep[RSP;][]{Tran95,Leonard3}, whereas that\nmeasured along the rotated $u$ axis is denoted URSP.\\footnote{Note that RSP\nand URSP, as defined here, are identical to the \\citet{Wang01} definitions of\nthe polarization along the ``dominant'' and ``orthogonal'' axes, $P_d$ and\n$P_o$, respectively. That is, by definition, RSP shows polarization strength\nvariation along the chosen polarization angle (i.e., the dominant axis),\nwhereas the URSP shows polarization strength variation along the axis that is\northogonal to the dominant axis in the $q$--$u$ plane. The main difference\nbetween our construction of the RSP in the {\\it observed} polarization plots\n(Figures \\ref{fig:5}--\\ref{fig:8}) and the {\\it ISP-subtracted} data\n(Figures \\ref{fig:15}, \\ref{fig:17}, and \\ref{fig:19}) is that in\nthe former the polarization angle about which the RSP is determined is a\nsmoothly varying function of wavelength (i.e., it is the polarization angle,\n$\\theta$, smoothed over many bins; see \\citet{Leonard3} for more details on the\nprocedure), whereas in the latter, RSP and URSP are determined with respect to\na single, unchanging polarization angle. For consistency with\nspectropolarimetry work in other fields, as well as our own prior studies, we\nshall continue to use the RSP and URSP designations, rather than the $P_d$ and\n$P_o$ designations of \\citet{Wang01}.}\n\nWe shall then analyze the resulting spectropolarimetry within the context of\nthe models described in \\S~\\ref{sec:2.3.1}. By construction, the greatest\ndegree of polarization change, especially from the continuum regions, should\noccur in the RSP. Examination of URSP is especially important in the line\nfeatures, however, as it can be used to discriminate between the predictions of\nthe ellipsoidal and clumpy-ejecta models. In particular, the simple\nellipsoidal model demands the existence of a single value of $\\theta$ capable\nof placing all line polarization changes along the RSP (i.e., URSP should show\nno polarization features).\n\n\\subsubsection{Two HV~SNe~Ia: SN 2002bf and SN 2004dt}\n\\label{sec:4.4.1}\n\nWe begin by considering the spectropolarimetry of the two HV~SNe~Ia, SN~2002bf\nand SN~2004dt, which are displayed in Figures~\\ref{fig:6} and \\ref{fig:8},\nrespectively. Both objects show very low levels of observed polarization, with\n$p_V = 0.03\\%$, $\\theta_V = 62^\\circ$ for SN~2002bf and $p_V = 0.25\\%$,\n$\\theta_V = 146\\ensuremath{^{\\circ}}$ for SN~2004dt, where $p_V$ and $\\theta_V$ approximate\nthe rest-frame $V$-band polarization and polarization angle derived by\ncalculating the debiased, flux-weighted averages of $q$ and $u$ over the\ninterval 5050--5950~\\AA\\ \\citep[see][]{Leonard3}. The polarization of\nSN~2004dt may exhibit an upward trend with wavelength, increasing from about\n$0.3\\%$ at blue wavelengths to $\\sim$1.0\\% at the red edge ($\\lambda =\n9600$~\\AA); a similar but smaller trend exists in the data for SN~2002bf.\nNeither object shows significantly different polarization in the ``continuum''\nregion 6800--7800~\\AA\\ from that observed in the heavily line-blanketed\nregion below 5000~\\AA\\ (\\S~\\ref{sec:2.3.1}), suggesting that the intrinsic\ncontinuum polarization is quite low for both objects.\n\nThe salient features of both data sets are the extraordinarily large\npolarization modulations ($\\sim$2\\%) across certain P-Cygni lines, most\nnotably the \\ion{Ca}{2} near-IR triplet for SN~2002bf and the \\ion{Si}{2}\n$\\lambda 6355$ line for SN~2004dt. Smaller features may be discerned across\nother lines in both objects.\n\nBefore attempting ISP removal, it is instructive to compare our SN~2004dt data\nwith those obtained by \\citet{Wang05} taken eleven days earlier, when the SN\nwas about seven days before maximum light. The interval separating the two\nobservations is one marked by rapid spectral and photometric evolution. For\nSN~2004dt, the velocity of the minimum of the high-velocity \\ion{Si}{2}\n$\\lambda 6355$ line recedes by about $3700\\ {\\rm km\\ s}^{-1}$ (from $17,200$ to $13,500\\\n{\\rm km\\ s}^{-1}$), and the weaker, lower-velocity lines such as \\ion{S}{2} $\\lambda\\lambda\n5612, 5654$ ($\\lambda_0 = 5635$~\\AA) decrease by about $2100\\ {\\rm km\\ s}^{-1}$ (from\n$11,000$ to $8900\\ {\\rm km\\ s}^{-1}$). Remarkably, in spite of the great changes that\noccur in the flux spectrum between the two epochs, we find the\nspectropolarimetry to be virtually unchanged. The overall polarization, in\nboth magnitude and polarization angle, is consistent with the earlier epoch,\nand the detected line-polarization features, most notably those at \\ion{Si}{2}\n$\\lambda 6355$ and the \\ion{Ca}{2} near-IR triplet, are also very similar. As\nwith the \\citeauthor{Wang05} data, our observations do not show significant\npolarization modulation across the important \\ion{O}{1} $\\lambda 7774$ line,\nalthough this line is now much weaker in the total-flux spectrum than it was\nearlier. Weaker polarization features, which are clearly detected in the\nearlier epoch, are difficult to confirm in our lower S\/N data, but our data are\nnot inconsistent with the earlier measurements, within the errors.\n\nAs discussed in \\S~\\ref{sec:2.3.3}, \\citet{Wang05} explain the\nspectropolarimetry of SN~2004dt in terms of optically thick bubbles of IMEs\nthat are asymmetrically distributed within an essentially spherical oxygen\nsubstrate that remains from the progenitor material. \\citet{Wang05} further\nnote that the polarization behavior of the high-velocity lines is similar to\nthe low-velocity lines in the early epoch, implying that the structures that\nobscure the photosphere have great radial extent. Within this context, then, it\nmay not be surprising that the polarization features have not evolved\nsignificantly between the two epochs, despite the great evolution of line\nvelocity.\n\nIn order to directly compare the spectropolarimetry of SN~2002bf and SN~2004dt\nwith each other, we must attempt to remove the ISP. To establish the ISP, we\napply the technique of \\citet{Wang05} and choose the spectral regions indicated\nin Figures~\\ref{fig:6}a and \\ref{fig:8}a. For SN~2002bf, this yields $(q_{\\rm\nISP}, u_{\\rm ISP}) = (0.01\\%, 0.05\\%)$, or ISP$_{\\rm max} = 0.05\\%$ at\n$\\theta_{\\rm ISP} = 39\\ensuremath{^{\\circ}}$ at an assumed peak wavelength of the ISP of\n$\\lambda_{\\rm max} = 5500$~\\AA. For SN~2004dt we derive $(q_{\\rm ISP}, u_{\\rm\nISP}) = (0.3\\%, -0.2\\%)$, which is within the uncertainty of the ISP found by\n\\citet{Wang05} through this same approach, $(q_{\\rm ISP}, u_{\\rm ISP}) =\n(0.2\\%, -0.2\\%)$. In order to facilitate direct comparisons between our data\nand those of \\citet{Wang05}, we shall adopt their ISP value for our study of\nSN~2004dt as well.\n\nAfter removal of the small amount of ISP for SN~2002bf, we derive a\nbest-fitting axis with ${\\rm PA\\ } = 123\\ensuremath{^{\\circ}}$ (Fig.~\\ref{fig:13}), about\nwhich we calculate the intrinsic RSP and URSP shown in Figure~\\ref{fig:15}. In\na similar way, for SN~2004dt we derive a best-fitting axis of ${\\rm PA\\ } =\n146\\ensuremath{^{\\circ}}$, which may be compared with the PA of $\\sim$150$\\ensuremath{^{\\circ}}$ found by\n\\citet{Wang05}. Again, since the two values do not differ significantly, we\nadopt the \\citeauthor{Wang05} direction for ease of comparison between the two\ndata sets. The axis and ISP choices for SN~2004dt are indicated in\nFigure~\\ref{fig:14}, and the resulting RSP and URSP are shown in\nFigure~\\ref{fig:15}.\n\nFrom examination of Figure~\\ref{fig:15}, it is clear that the two events have\nspectropolarimetric similarities. Both show low overall polarization and\nmodulations across the \\ion{Si}{2} $\\lambda 6355$ and \\ion{Ca}{2} near-IR\ntriplet absorptions. The detailed character of the line polarizations do\ndiffer, however. For SN~2002bf there is a $2\\%$ polarization change across the\n\\ion{Ca}{2} feature, with a more modest ($\\sim$0.4\\%) feature detected in the\n\\ion{Si}{2} line. (When discussing overall ``polarization change'' across a\nline feature without specific reference to either the RSP or URSP, we\neffectively mean the quadrature sum of the changes seen in the two parameters.)\nFor SN~2004dt, the situation is reversed. In the case of the \\ion{Ca}{2} line,\nthis may be related to its relative strength and velocity in the two spectra,\nas both the equivalent width and velocity of the feature in the flux spectrum\nare much greater in SN~2002bf than in SN~2004dt. As discussed earlier\n(\\S~\\ref{sec:4.1}), it is possible that burning to Ca occurred more extensively\nin SN~2002bf than it did in SN~2004dt. The explanation for the \\ion{Si}{2}\nline disparity, however, is not so obvious, as the lines have similar strengths\nand velocities.\n\nClearly, strong line and weak continuum polarization levels disfavor the simple\nellipsoidal model. Applying Eq.~(\\ref{eqn:1}) to the data for SN~2002bf and\nSN~2004dt yields lower bounds on the expected continuum polarization of $p_{\\rm\ncont} \\geq 0.7\\%$ and $p_{\\rm cont} \\geq 1.3\\%$, respectively. For both\nobjects, however, we measure $p < 0.4\\%$ in both the observed and\nISP-subtracted data for the spectral region 6800--7800~\\AA, which is\nlargely devoid of line opacity in SNe~Ia near maximum (\\S~\\ref{sec:2.3.1}).\nThe ellipsoidal model therefore appears to be ruled out as the explanation for\nthe polarization characteristics of these objects.\n\nThe clumpy-ejecta model, on the other hand, provides a natural explanation for\nhigh line and low continuum polarization levels (\\S~\\ref{sec:2.3.1}). In\nparticular, the GCD scenario can successfully explain (1) the high line\nvelocities, (2) the large polarization change in the line troughs, (3) the\ngreat radial extent of the obscuring material (i.e., for $m_{\\rm atm} = 0.08\nM_\\odot$, the obscuring pancake spans the range 10,000--21,000 km s$^{-1}$), and (4)\nthe lack of significant continuum polarization. A remaining challenge is to\nexplain the lack of any polarization change across the \\ion{O}{1} $\\lambda\n7774$ line, especially in the earlier \\citet{Wang05} data for SN~2004dt, when\nthis line is extremely strong in the total-flux spectrum. Recent\nnucleosynthesis calculations based on multi-dimensional (2D and 3D)\nhydrodynamical simulations of the thermonuclear burning phase in SNe~Ia show\nthat as much as $40\\%$ to $50\\%$ of the ejected matter in SNe~Ia is unburned\ncarbon and oxygen \\citep{Travaglio04}. This has led \\citet{Wang05} to propose\nthat the primordial oxygen is nearly spherically distributed, within which\nasymmetrically distributed IME clumps, or an absorbing pancake in the GCD\nmodel, are embedded.\n\nWith this in mind, an important prediction of the clumpy-ejecta model\n(including the GCD scenario) is that the \\ion{O}{1} $\\lambda 7774$ line should,\nin fact, show a polarization change with a polarization angle that differs by\n$90^\\circ$ from what is observed in the \\ion{Si}{2} and \\ion{Ca}{2}\nlines since its distribution is essentially the inverse of these IMEs.\nHowever, some oxygen is probably also contained in the clumps or pancake, since\nit is produced by explosive carbon burning in small quantities and should be\npresent when such burning products as magnesium exist, whose spectral signature\nis unequivocally seen in the spectra. This would tend to reduce the\npolarization level in the \\ion{O}{1} $\\lambda 7774$ line. Neither our data,\nnor those of \\citet{Wang05}, are of sufficiently high S\/N to detect such a\nchange, and it must be left to future, higher-S\/N studies focused especially at\nearly times when the \\ion{O}{1} $\\lambda 7774$ feature is strong. Finding such\na PA change would further strengthen the case for the clumpy-ejecta and\/or GCD\nscenario.\n\nWe note that the ejecta-hole model of \\citet{Kasen04} is also capable of\nproducing large line polarization with weak continuum polarization for\nsight-lines near to the hole (\\S~\\ref{sec:2.3.1}). Arguing against this\nmodel in these cases, though, is that it offers no natural explanation for why\nhigh line velocities should be associated with sight lines near to the hole.\nIn fact, the \\citeauthor{Kasen04} models predict {\\it lower} absorption\nvelocities when viewing down the hole.\n\nIn all, then, our study finds that HV~SNe~Ia are, as a group, characterized by\nmuch stronger line-polarization features than are seen in other SN~Ia\nvarieties. The ellipsoidal model is incapable of explaining the polarization\ncharacteristics of these objects, whereas the clumpy-ejecta and ejecta-hole\nmodels are more successful. On balance, the case for clumpy ejecta appears to\nbe the most convincing explanation, with the GCD model investigated by\n\\citet{Kasen05} able to reproduce many of the observed spectral and\nspectropolarimetric features.\n\n\\subsubsection{SN 1997dt}\n\\label{sec:4.4.2}\n\nWe next turn to the likely subluminous SN~1997dt, which was observed about\nthree weeks past maximum light. Figure~\\ref{fig:5} reveals an extraordinarily\nhigh level of observed polarization, $p_V = 3.46\\%$ at $\\theta_V = 112^\\circ$.\nThis is by far the largest polarization yet observed for an SN~Ia. A distinct\npolarization feature is detected in the \\ion{Fe}{2} $\\lambda 4555$ trough and\nprobably also in the \\ion{Si}{2} $\\lambda 5972$ + \\ion{Na}{1} D and \\ion{Si}{2}\n$\\lambda 6355$ lines. The degree of change in the \\ion{Fe}{2} $\\lambda 4555$\nfeature reaches nearly $1\\%$ in the $q$ parameter.\n\nThis amount of observed continuum polarization is surprising, given that our\nbest total reddening estimate predicts an upper bound on the ISP of only\n$1.89\\%$ from the \\citet{Serkowski75} relation (see \\S~\\ref{sec:2.3.2}). If we\ntrust both the reddening estimate and the upper ISP bound, then an intrinsic SN\npolarization of at least $1.57\\%$ must exist to explain the observed\npolarization, far higher than has been indicated for any previous SN~Ia.\nHowever, the technique of \\citet{Wang05} suggests a much higher ISP level:\n$(q_{\\rm ISP}, u_{\\rm ISP}) = (2.53\\%, 2.68\\%)$, or ISP$_{\\rm max} = 3.60\\%$,\n$\\theta_{\\rm ISP} = 113\\ensuremath{^{\\circ}}$, for $\\lambda_{\\rm max} = 6500$~\\AA, the\nwavelength of maximum ISP that yields the most convincing Serkowski-law fit to\nthe observed polarization (Fig.~\\ref{fig:5}d). If this truly is the ISP, then\nit implies an extraordinarily high polarization efficiency for the dust along\nthe l-o-s to SN~1997dt: ${\\rm ISP} \/ E_{B-V} \\approx 18\\%\\ {\\rm mag}^{-1}$,\nwhich is double the observed Galactic limit of $9\\%\\ {\\rm mag}^{-1}$. Of\ncourse, some of this discrepancy could be removed if the true reddening were\ngreater than we suspect. However, even allowing the reddening to equal the\nupper limit of $E\\BminusV_{\\rm total} < 0.27$ mag set in \\S~\\ref{sec:4.3.2} still\nrequires the dust-polarization efficiency to exceed the Galactic limit.\n\nIt thus appears that we face a stark choice: either SN~1997dt has the highest\nintrinsic polarization of any SN~Ia yet observed, or the dust along the l-o-s\nhas an exceptionally high polarization efficiency. The epoch of our\nobservation of SN~1997dt is unique for a subluminous SN~Ia, so there is no\nempirical database from which to draw expectations and help decide between\nthese two options.\n\nThe situation for SN~1997dt, while perplexing, in fact is not unique: a similar\npalette of possibilities presented themselves in our previous study of a single\nepoch of spectropolarimetry of SN~1999gi, an SN~II-P that also had a low\nreddening and large observed polarization \\citep{Leonard4,Leonard6}. Like\nSNe~Ia, SNe~II-P as a group have historically shown very low intrinsic\ncontinuum polarization. After considering many polarization production\nmechanisms, including polarization due to newly formed dust in the SN ejecta\nand dust reflection by one or more off-center dust blobs external to the SN, we\nconcluded that the most likely explanation for the polarization of SN~1999gi\nwas that the host-galaxy dust along the l-o-s possesses a very high\npolarization efficiency, ${\\rm ISP}\/E\\bminusv = 31^{+22}_{-9}\\% {\\rm\\ mag}^{-1}$,\nwhich remains the largest value yet inferred for a single sight line in either\nthe MW or an external galaxy \\citep{Leonard7}.\n\nThere are arguments favoring a similar explanation here. First, a Serkowski\nlaw reasonably fits the observed continuum polarization of SN~1997dt\n(Fig.~\\ref{fig:5}d). Further, at 21 days past maximum, SN~1997dt may be\nnearing the end of its photospheric phase, a time when spectropolarimetry may\nbe losing its efficacy as an asymmetry probe due to a lack of electrons\navailable to scatter the light. If we believe this to be the case, then we\nshould not expect large polarization, even if the SN is highly aspherical. If\nwe accept the large ISP, then we naturally would like to know whether it is due\nto dust in the MW or NGC~7448. Unfortunately, we have not observed any distant\nGalactic ``probe'' stars (\\S~\\ref{sec:2.3.2}) near to the l-o-s to estimate the\nGalactic ISP. There are, however, reasons to suspect that it is low. First,\n20 stars within $10\\ensuremath{^{\\circ}}$ of the l-o-s are listed in the agglomeration of\nstellar polarization catalogs by \\citet{Heiles00}, and the greatest observed\npolarization is only $0.3\\%$. Second, the great majority of the reddening is\ndue to host-galaxy dust, since $E\\BminusV_{\\rm Host} = 0.15$ mag while $E\\BminusV_{\\rm\nMW} = 0.06$ mag (\\S~\\ref{sec:4.3.2}). As was the case with SN~1999gi, we would\nagain conclude that it is the dust within the host galaxy that must have the\nextraordinarily high polarization efficiency.\n\nThere are arguments to oppose this, however. Previous SN~Ia polarization\nstudies have found intrinsic polarizations increasing toward red wavelengths\n(\\S~\\ref{sec:2.3.3}), which could certainly mimic a Serkowski law over the\nlimited wavelength band covered by our observations. Further, the line\npolarization features demonstrate that the SN must possess at least some\nintrinsic polarization. Finally, the relatively late phase of the observation,\ninvoked previously to argue {\\it against} high intrinsic polarization, can also\nbe used to argue {\\it in favor} of it: at the stage immediately before an SN\nbegins the transition to the nebular phase, the deepest layers of the ejecta\nare revealed. Although in need of confirmation by detailed modeling, for this\nepoch one can plausibly argue that an optical photosphere still exists with\nsufficient optical depth to electron scattering, perhaps even reaching the\nsingle-scattering limit, which is the most polarizing atmosphere\n\\citep{Hoflich91}. If the explosion mechanism itself is asymmetric, the\nlargest imprint of the asymmetry would presumably be in the innermost layers,\nwhich could lead to a very large intrinsic polarization that reveals itself\njust at this late phase. \n\nIn fact, such an effect is seen in the\nspectropolarimetry of the SN~II-P 2004dj as it transitions to the nebular phase\n(D. C. Leonard et al., in preparation). However, in the case of SN~2004dj, the\nstrong increase is observed primarily in the ``continuum'' region\n6800--7800~\\AA\\ (and in a few strong line troughs), not in the overall level\nacross the whole spectrum. Although in need of confirmation by detailed\nmodeling, depolarizing line blanketing may also be significant for SNe~Ia at\nblue wavelengths at this epoch, which would again argue for a large ISP as the\nexplanation of the high observed polarization. One possibility that\ncircumvents this difficulty is that asymmetrically distributed concentrations\nof radioactive Ni, recently uncovered in the thinning ejecta at this relatively\nlate epoch, are responsible for the large polarization.\n\nCuriously, the single, high-polarization observation of SN~1999gi\noccurred at a similar stage of its evolution, right at the end of the\noptical plateau that characterizes the photospheric phase in SNe~II-P.\nIt is thus unfortunate that no other spectropolarimetric epochs were\nobtained for either event (SN 1999gi and SN 1997dt) to serve as a\nbasis for comparison. Multi-epoch data covering the transition from\nthe photospheric to the nebular phases of SNe of all types will\ncertainly help reveal more definitively the physical explanation for\nsuch high observed polarizations at these late times.\n\nIt may be possible to gain insight into the cause of the observed polarization\nfrom examination of the spectropolarimetry data in the $q$--$u$ plane, shown in\nFigure~\\ref{fig:16}. ISP originating from a single source (i.e., characterized\nby a single PA) will spread intrinsically unpolarized data points along a line\nin a direction that intersects the origin in the $q$--$u$ plane. Finding that\ndata lie predominantly along such a line, and exhibit a Serkowski law spectral\nshape, provides compelling evidence that a large, single source of ISP\ndominates the observed signal. Such was the case for SN~II-P 1999gi\n\\citep{Leonard7}. Similarly, for SN~1997dt, an elongation of the data points\nfrom blue to red wavelengths in a direction that roughly points back toward the\norigin also exists. \n\nWhen the ISP derived earlier (ISP$_{\\rm max} = 3.60\\%$, $\\theta_{\\rm ISP} =\n113\\ensuremath{^{\\circ}}$, for $\\lambda_{\\rm max} = 6500$~\\AA) is removed, the wavelength\ndependence of the polarization largely disappears (Fig.~\\ref{fig:16}),\nstrengthening the argument for a single, dominant ISP source. The data are\nthen seen to lie along an axis with a fairly well-defined PA of $58\\ensuremath{^{\\circ}}$\n(Fig.~\\ref{fig:16}), against which we calculate the RSP and URSP shown in\nFigure~\\ref{fig:17}. Although of individually low to moderate significance,\nthe polarization features in the \\ion{Fe}{2} $\\lambda 4555$ and \\ion{Si}{2}\n$\\lambda 6355$ troughs, seen in both RSP and URSP, do seem to prefer the same\ngeneral direction, with most of the modulation occurring along the URSP axis\n(e.g., the direction perpendicular to the main axis in the $q$--$u$ plane).\nTaken at face value, consistent line-trough polarization changes in a direction\ndiffering from that preferred by the continuum favors an origin in the\nselective blocking of photospheric light by clumpy and asymmetrically\ndistributed intermediate and iron-peak elements overlying the photosphere as\nopposed to an ellipsoidal scenario. \n\nOther plausible ISP choices, however, yield different conclusions. For\ninstance, a smaller ISP of $2.6\\%$ at the same PA yields a more nearly\nspherical constellation of points centered near $(q, u) = (-0.7\\%, -0.07\\%)$,\nwith the line excursions now pointing back toward the origin. Since some\nmodelers predict that, in SNe~Ia with an ellipsoidal asphericity, polarization\n{\\it decreases} may exist in absorption troughs (\\S~\\ref{sec:2.3.1}), the\nellipsoidal model cannot therefore be ruled out in this case as the cause of\nthe polarization. Given the marginal significance of the weaker features,\nadditional interpretation of the line-trough polarization degrees and\ndirections is probably not warranted with these data.\n\nIn conclusion, we find evidence for a large ISP contribution to the observed\npolarization of SN~1997dt, probably in the range $2.6\\% \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} {\\rm ISP}\n\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 3.6\\%$. This implies a polarization efficiency for the dust along the\nl-o-s in NGC~7448 that exceeds the Galactic limit. We do, however, also find\nevidence for polarization intrinsic to the object, most convincingly in\nspecific line features and, perhaps, in the continuum as well. A combination\nof asymmetrically distributed radioactive Ni and synthesized IMEs overlying the\nphotosphere may provide the simplest explanation for the line and potential\ncontinuum polarization at this late phase, although the ellipsoidal model \ncannot be definitively ruled out.\n\n\\subsubsection{SN 2003du}\n\\label{sec:4.4.3}\n\nOur spectropolarimetry of SN~2003du represents the highest S\/N data of our\nstudy, and permits a more detailed analysis of line features than has been\npossible for the other objects. SN~2003du presents a low level of observed\npolarization across most of the spectrum, with $p_V = 0.04\\%$, $\\theta_V =\n17^\\circ$ (Fig.~\\ref{fig:7}). The polarization exhibits an increasing trend\nwith wavelength, rising from nearly zero at blue wavelengths to $\\sim$0.2\\% at\nthe red edge of the spectrum. There are distinct and significant polarization \nchanges across several absorption features, including the \\ion{Ca}{2} near-IR \ntriplet, the \\ion{Si}{2} $\\lambda 6355$ line, probably a few weaker lines such \nas \\ion{Fe}{2} $\\lambda 4924$, and very tentatively the \\ion{Ca}{2} H \\& K\nabsorption.\n\nApplying the technique of \\citet{Wang05}, and choosing the regions indicated in\nFigure~\\ref{fig:7}a to estimate the ISP, yields $(q_{\\rm ISP}, u_{\\rm ISP}) =\n(-0.02\\%, 0.0\\%)$, or ISP$_{\\rm max} = 0.02\\%$, $\\theta_{\\rm ISP} = 90\\ensuremath{^{\\circ}}$,\nfor an assumed $\\lambda_{\\rm max} = 5500$~\\AA. This very low ISP is\nconsistent with the negligible reddening found earlier. Furthermore, on the\nsame night the SN~2003du data were taken, we observed the\ndistant Galactic star BD~$+50^\\circ1593$ (spectral type F8, $V = 10.64$ mag),\nlocated just $0.28^\\circ$ from the l-o-s of SN~2003du, and found it to be null\nto within $0.1\\%$. From its spectroscopic parallax, we estimate BD~$+50^\\circ1593$\nto be at least $190$ pc away which, at a Galactic latitude of $53^\\circ$,\nsatisfies the criterion of \\citet{Tran95} that a good MW ``probe'' star be more\nthan 150 pc from the Galactic plane. We thus have multiple reasons to suspect\nlittle ISP contaminating the data.\n\nAfter removal of the minimal ISP, a well-defined axis with ${\\rm PA\\ } =\n107\\ensuremath{^{\\circ}}$ is derived (Fig.~\\ref{fig:18}), about which we calculate the\nintrinsic RSP and URSP, shown in Figure~\\ref{fig:19}. Compared with what was\nseen in the HV~SNe~Ia, and even SN~1997dt, the line-polarization features are\nnot large, amounting to no more than $0.3\\%$. However, the very high S\/N of\nthese data makes the detections unequivocal, and establishes intrinsic\npolarization in a spectroscopically and photometrically normal SN~Ia at the\nlatest phase yet observed.\n\nThe behavior of the line-trough polarization for the \\ion{Ca}{2} near-IR\ntriplet and \\ion{Si}{2} $\\lambda 6355$ are very similar. Both show sharp\nincreases of $\\sim$0.2\\% in RSP, as well as overall increases in URSP. Modest\nRSP increases of $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 0.1\\%$ may also be discerned in the \\ion{Fe}{2}\n$\\lambda 4924$ and \\ion{Ca}{2} H \\& K absorptions. At this phase, the\n\\ion{O}{1} $\\lambda 7774$ feature in SNe~Ia is quite weak and, coupled with\ntelluric A-band contamination, makes definitive detection of polarization\nmodulation in this important region difficult; the observed changes are at\nabout the level of the statistical noise.\n\nThe commonality between the polarization behavior of the Si and Ca lines argues\nfor similar origins. Given the high S\/N of the data, we can examine rather\nclosely the basic predictions of the ellipsoidal model that (a) the\npolarization angle should be independent of wavelength, and (b) the overall\npolarization should increase from blue to red wavelengths, with an expectation\nof $p \\rightarrow 0\\%$ at $\\lambda \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 5000$ \\AA\\ (if the line opacity\nremains strong in these regions at the epoch of observation;\n\\S~\\ref{sec:2.3.1}). \n\nWith our initial choice of ISP, the first criterion is clearly not satisfied,\nas there are obvious excursions in URSP throughout the spectrum, with especially\nsharp modulations occurring across the strongest lines. In fact, for ISP\nvalues constrained to lie along the symmetry axis we are unable to find any\nvalue that convincingly satisfies both criteria of the ellipsoidal model:\nvalues in the upper-right quadrant (e.g., $[q_{\\rm ISP}, u_{\\rm ISP}] = [0.3\\%,\n0.18\\%]$) produce a polarization {\\it decrease} with wavelength across the\nspectrum, while ISP values in the lower-left quadrant that satisfy the\ncriterion of $p \\rightarrow 0\\%$ at blue wavelengths (e.g., $[q_{\\rm ISP},\nu_{\\rm ISP}] = [-0.1\\%, -0.07\\%]$) result in strong PA changes across the\nspectral lines. Choosing an ISP point far beyond the constellation of data\npoints, which may serve to make the PA changes less objectionable (although\nstill statistically significant), has the unfortunate consequence of straining\nthe limits implied from the very low reddening, at least for dust with normal\npolarizing efficiency. \n\nIn addition, examining the URSP behavior of the \\ion{Si}{2} $\\lambda 6355$ and\n\\ion{Ca}{2} near-IR lines in Figure~\\ref{fig:19} more closely, we see that the\ngenerally increasing trends in URSP across the lines show sharp decreases right\nat the locations of peak RSP modulation, although the statistical significance\nof the modulations, especially for \\ion{Ca}{2}, is not high. If the abrupt\nchanges in URSP are real, such structure is readily explained under the\nclumpy-ejecta model by variations in the distribution of the IMEs as a function\nof radius in the expanding ejecta. \\citet{Chugai92} also demonstrates that\nsuch sharp changes in line features can be produced by excitation asymmetry.\nOn the other hand, there is no obvious mechanism to produce such an effect in\nthe simple ellipsoidal models. We thus conclude that it is difficult to\nreconcile the basic predictions of the ellipsoidal model with the data for\nSN~2003du.\n\nWe therefore suspect either clumps in the ejecta overlying the photosphere or\nionization asymmetry as the cause of the inferred line and, perhaps, continuum\npolarization, and are led again to disfavor the ellipsoidal model as the cause\nof the intrinsic polarization of this SN.\n\n\n\\section{Conclusions}\n\\label{sec:5}\n\nWe present post-maximum single-epoch spectropolarimetry of four SNe~Ia,\nbringing to six the number of SNe~Ia thus far examined in detail with\nspectropolarimetry during the early phases. The four objects span a range of\nspectral and photometric properties, yet all are demonstrated to be\nintrinsically polarized. This suggests that asphericity and\/or asymmetry may\nbe a ubiquitous characteristic of SNe~Ia in the first weeks after maximum\nlight. The nature and degree of the polarization varies considerably within\nthe sample, but in a way that is consistent with, and extends, previously\nsuspected trends. Our main spectropolarimetry results are as follows:\n\n\\begin{enumerate}\n\n\\item SN~2002bf and SN~2004dt, both HV~SNe~Ia observed shortly after maximum\nbrightness, exhibit the largest polarization features yet seen definitively for\nany subtype of SN~Ia, with modulations of up to $\\sim$2\\% in the troughs of the\nstrongest lines. The overall polarization level of both objects is minimal,\nat least at blue wavelengths; there is a possible trend of increasing\npolarization with wavelength for both objects, though neither shows significant\npolarization in the ``continuum'' region 6800--7800~\\AA. The\nISP contamination is not thought to be large in either object.\n\n\\item SN~1997dt, believed to be a somewhat subluminous event, has the highest\nobserved overall polarization of any SN~Ia yet studied, $p_V = 3.46\\%$, at 21\ndays past maximum light. This demands either an extraordinarily large\npolarization efficiency for the dust along the l-o-s in NGC~7448, the largest\nintrinsic SN~Ia polarization thus far found, or perhaps some combination of the\ntwo. The observed polarization rises by about $0.5\\%$ from blue ($\\lambda =\n4300$~\\AA) to red ($\\lambda = 6700$~\\AA) wavelengths, approximating a\nSerkowski-law ISP curve rather convincingly, albeit one with a somewhat unusual\npeak wavelength ($\\lambda_{\\rm max} \\approx 6500$~\\AA). A polarization\nmodulation of nearly $1\\%$ in the strong \\ion{Fe}{2} $\\lambda 4555$ absorption,\nand a more modest change of $\\sim 0.3\\%$ in the \\ion{Si}{2} $\\lambda 6355$\nline, demonstrate that the SN does possess intrinsic polarization features.\nHowever, we conclude that ISP is responsible for the bulk of the overall\npolarization that is observed, with $2.6\\% \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} {\\rm ISP}\n\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 3.6\\%$, and that the polarization efficiency of the dust along the\nl-o-s in NGC~7448 likely exceeds the empirical Galactic limit.\n\n\\item SN~2003du is a slightly overluminous SN~Ia. Our spectropolarimetry,\n taken 18 days after maximum light, is the highest S\/N data obtained for our\n sample of objects. It reveals a low continuum polarization that increases by\n $\\sim$0.3\\% from blue to red wavelengths, with distinct changes of $\\sim$0.2\\% \n detected in the \\ion{Si}{2} $\\lambda 6355$ and \\ion{Ca}{2} near-IR\n triplet lines; smaller changes may be detected in weaker lines. The very\n similar behavior of the polarization in the two strongest lines, in both\n magnitude and direction in the $q$--$u$ plane, suggests a common polarization\n origin. ISP is thought to be minimal.\n\n\n\\end{enumerate}\n\nOrdered by increasing strength of line-polarization features in SNe~Ia, we find\nas follows: ordinary\/overluminous $<$ subluminous $<$ HV~SNe~Ia, with the\nstrength of the line-polarization features increasing from $0.2\\%$ in the\nslightly overluminous SN~2003du to $2\\%$ in both HV~SNe~Ia in our study.\nAbsolute continuum polarization levels are more difficult to establish, due\nlargely to uncertainties in the ISP, but there are compelling reasons to\nbelieve that at least three of our objects possess very little intrinsic\npolarization in spectral regions outside of specific line features. The\n\\citet{Howell01} study of SN~1999by and our data on SN~1997dt provide some\nevidence that continuum polarization may be higher in subluminous objects than\nin other types.\n\nThere are a number of alternatives from which to choose for the origin of\npolarization in SNe~Ia, including global asphericity (e.g., the ellipsoidal\nmodel), ionization asymmetry, and clumps in the ejecta overlying the\nphotosphere. The small, redward rise in the {\\it overall} polarization level\nthat is discerned in at least three of our objects can be reproduced by models\npossessing either global asphericity or an ionization asymmetry. Under the\nellipsoidal model, the levels of continuum polarization for SN~2002bf,\nSN~2003du, and SN~2004dt imply minor to major axis ratios of around $0.9$ if\nviewed equator-on \\citep{Hoflich91,Wang03}. This level of asphericity would\nproduce a luminosity dispersion of about 0.1 mag for random viewing\norientations \\citep{Hoflich91}, which could explain some of the dispersion seen\nin the brightness-decline relation of SNe~Ia. If the proposed global\nasphericity is more complicated, then the luminosity of a single SN~Ia may\ndepend on viewing angle in a non-trivial way such that, even for a large sample\nof objects, an overall bias to slightly higher or lower values may result.\n\nThe potential for significantly greater continuum polarization, perhaps of\n$\\sim$1\\%, in the likely subluminous SN~1997dt observed $\\sim$21 days after\nmaximum would imply a more severe distortion, of at least $20\\%$, from the\nmodels of \\citet{Hoflich91}. Since most cosmological applications of SNe~Ia\nrely on data acquired closer to maximum light, such a late-time asphericity,\neven if common, would not seriously affect the utility of SNe~Ia as distance\nindicators.\n\nTo explain the ubiquitous line polarization, the simple ellipsoidal model is\neffectively ruled out for three of our objects, including, most convincingly,\nthe two HV events. From a number of lines of reasoning, the most convincing\nexplanation is partial obscuration of the photosphere by clumps of newly\nsynthesized IMEs forged in the explosion.\n\nFor the HV~SNe~Ia, in particular, the GCD model studied by \\citet{Kasen05}\nprovides a plausible explanation for many of the observed spectral and\nspectropolarimetric characteristics. It predicts the existence of an optically\nthick pancake of material with significant radial extent that partially\nobscures the optical photosphere, producing larger line velocities and\nequivalent widths for many spectral features, and stronger line-trough\npolarization than is seen in more typical events. These qualitative\nexpectations are borne out by our data: The line features of SN~2002bf and\nSN~2004dt possess the strongest polarization modulations and greatest\nequivalent widths of our sample. The astonishing similarity between our epoch\nof spectropolarimetry of SN~2004dt, $\\sim$4 days after maximum, and that\npresented by \\citet{Wang05} from 11 days earlier, provides compelling evidence\nthat the obscuring material also possesses great radial extent in the thinning\nejecta.\n\nThat SNe~Ia may be separable into different groups based on their {\\it\nspectropolarimetric} characteristics yields one more clue to assist in\nnarrowing down progenitor possibilities and\/or models for the physics of the\nexplosion. The assertion that some fraction of the IMEs in the ejecta of\nSNe~Ia may be confined to bubbles or filaments is, however, a rather blunt\ndiscriminatory tool: At the present stage of theoretical modeling,\ndeflagration, delayed-detonation, off-center delayed-detonation, and GCD models\ncan all plausibly be argued to produce clumps in the ejecta \\citep[e.g.,][and\nreferences therein]{Wang05}. The specific predictions of the GCD model in\nparticular need to be further examined, preferably in full three-dimensional\nsimulations, to test whether the explosion mechanism itself is viable, and, if\nso, whether it can quantitatively reproduce the observed characteristics of at\nleast some SNe~Ia in detail. On the observational front, higher S\/N data,\npreferably obtained at multiple epochs, will help to narrow down the\npossibilities as well. With the steady advances being made in the theoretical\nunderstanding of these events, and the growing rate of SNe~Ia studied in detail\nwith spectropolarimetry, prospects for improving our understanding of these\nevents are bright.\n\n\\acknowledgments \n\nWe thank Aaron Barth, Louis-Benoit Desroches, Mohan Ganeshalingam, Deborah\nHutchings, Ed Moran, and Karin Sandstrom for assistance with some of the\nobservations and data reduction, and Saurabh Jha for producing MLCS2k2 fits for\ntwo of our objects. We thank an anonymous referee for useful suggestions that\nresulted in an improved manuscript. This research has made use of the\nNASA\/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion\nLaboratory, California Institute of Technology, under contract with NASA. The\nwork of A.V.F.'s group at UC Berkeley is supported by National Science Foundation\n(NSF) grant AST-0307894. D.C.L. is supported by an NSF Astronomy and\nAstrophysics Postdoctoral Fellowship under award AST-0401479. Additional\nfunding was provided by NASA grants GO-9155, GO-10182, and GO-10272 from the\nSpace Telescope Science Institute, which is operated by the Association of\nUniversities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.\nA.V.F. is grateful for a Miller Research Professorship at UC Berkeley, during\nwhich part of this work was completed. Some of the data presented herein were\nobtained at the W. M. Keck Observatory, which is operated as a scientific\npartnership among the California Institute of Technology, the University of\nCalifornia, and NASA; the Observatory was made possible by the generous\nfinancial support of the W. M. Keck Foundation. KAIT was made possible by\ngenerous donations from Sun Microsystems, Inc., the Hewlett-Packard Company,\nAutoScope Corporation, Lick Observatory, the NSF, the\nUniversity of California, and the Sylvia \\& Jim Katzman Foundation. The\nassistance of the staffs at Lick and Keck Observatories is greatly appreciated.\n\n\\clearpage\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{0pt}{18pt plus 4pt minus 2pt}{18pt plus 2pt minus 2pt}\n\tThe intricate interplay between magnetism and nematicity in different families of iron-based superconductors has attracted great interest in the past few years \\cite{Fernandes2014,PDai2015,Si2016}. In iron pnictides, magnetism and nematicity are tightly coupled; the antiferromagnetic transition is always coincidental with, or closely preceded by, a tetragonal-to-orthorhombic structural transition. The proximity of the two transitions can be naturally explained within the spin-nematic scenario, where the structural transition is driven by a vestigial nematic order arising from fluctuations associated with the antiferromagnetic stripe transition (see Fig. \\ref{fig:Fig1}(b))\\cite{CFang2008,Xu08,Fernandes2012}. In iron chalcogenides, the coupling between magnetism and nematicity is less obvious. FeSe undergoes a nematic phase transition without any long-range magnetic order \\cite{McQueen2009,Bohmer2017}, which has been interpreted as evidence that the nematic order in FeSe is of orbital origin \\cite{Baek2015}. Nevertheless, spin stripe fluctuations do develop below the nematic transition \\cite{Wang2016}, and static stripe order can be induced by hydrostatic pressure \\cite{Kothapalli16,Matsuura2017}.\n\t\\begin{figure}\n\t\t\\includegraphics[trim={0 0.1cm 0cm 0.2cm},clip,width=0.5\\textwidth]{Fig1.png}\n\t\t\\caption{(a-b) Schematic spin configurations of the (a) double-stripe phase, with wave-vector $\\mathbf{Q} = (\\pi, 0)$, and (b) single-stripe phase, with $\\mathbf{Q} = (\\pi, \\pi)$. (c-d) Schematic diagrams of the Montgomery method for the elastoresistivity measurement in (c) $B_{1g}$ and (d) $B_{2g}$ configuration. (e-f) The anisotropic resistivity $(\\rho_{xx} - \\rho_{yy})$ as a function of anisotropic strain $(\\epsilon_{xx} - \\epsilon_{yy})$ for Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ ($0 \\leq x \\leq 0.53$) at $T = 20$K in (e) $B_{1g}$ and (f) $B_{2g}$ channels. The $B_{1g}$ elastoresistivity coefficient $m_{11} - m_{12}$ and $B_{2g}$ elastoresistivity coefficient $2m_{66}$ can be obtained by fitting the linear slope of resistivity versus strain. The samples with high doping concentrations ($x = 0.38, 0.45, 0.53$) show predominantly a $B_{2g}$ response while the low doping ones ($x = 0, 0.12$) show comparable $B_{1g}$ and $B_{2g}$ responses.}\n\t\t\\label{fig:Fig1}\n\t\\end{figure}\n\t\\begin{figure*}\n\t\\includegraphics[width=1\\textwidth]{FIG2.eps}\n\t\\caption{Temperature and doping dependence of nematic fluctuations of annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ in the (a) $B_{1g}$ channel, with elastoresistivity coefficient $m_{11} - m_{12}$, and (b) $B_{2g}$ channel, with $2m_{66}$. For clarity, the elastoresistivity data for each doping are offset by 15 and 20 in (a) and (b), respectively. (c-e) Temperature dependence of (c) $m_{11} - m_{12}$ for $x = 0$, (d) $2m_{66}$ for $x = 0.45$ and (e) $-2m_{66}$ for Ba(Fe$_{0.93}$Co$_{0.07}$)$_{2}$As$_{2}$. Lower panels show the inverse. Solid black curves are Curie-Weiss fits. The optimal fitting range is determined by the greatest corresponding adjusted R-square value. Shaded gray regions indicate the range of temperatures where the elastoresistivity coefficients follow a Curie-Weiss law. }\n\t\\label{fig:fig2}\n\t\\end{figure*}\n\t\\\\\n\t\\indent While there are ongoing debates on the mechanism by which nematicity forms without static magnetism in FeSe \\cite{Glasbrenner2015,Wang2015,Yu2015,Khodas16,Kontani17}, Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ provides another platform to approach this problem. As selenium is replaced by tellurium (i.e. $x$ is changed from $1$ to $0$), the nematic phase transition is suppressed \\cite{Terao2019}, and inelastic neutron scattering experiments revealed a complex evolution of the spin correlations associated with different magnetic patterns \\cite{Lumsden2010,Liu2010,Zaliznyak10316,Xu2018}. In particular, close to optimal doping ($x \\sim 0.5$), the wave-vector of spin fluctuations at low temperatures is $(\\pi, \\pi)$ [in the crystallographic Brillouin zone], identical to the antiferromagnetic order in the iron pnictides. As the tellurium concentration increases, both superconductivity and the $(\\pi, \\pi)$ spin fluctuations disappear. The latter are replaced by short-range magnetic correlations near $(0, \\pi)$ that eventually condense into the static double-stripe phase in Fe$_{1+y}$Te \\cite{Li2009} (Fig. \\ref{fig:Fig1}(a)). Previous elastoresistivity measurements revealed a diverging $B_{2g}$ nematic susceptibility in optimally doped Fe$_{1+y}$Te$_{1-x}$Se$_{x}$, consistent with the existence of $(\\pi, \\pi)$ spin fluctuations \\cite{Kuo958}. This finding suggests that nematic and magnetic fluctuations remain strongly intertwined even in the absence of static nematic and magnetic orders. Nevertheless, in contrast to the magnetic sector, the behavior of nematic fluctuations for doping concentrations beyond optimal is still poorly characterized. The compositional dependence of the nematic susceptibility in Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ would therefore constitute an important step in the effort to elucidate the relationship between nematicity and magnetism.\n\t\\\\\n\t\\indent Another motivation to study Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ is to understand the influence of orbital selectivity on the nematic instability. Orbital selectivity (or orbital differentiation) refers to the fact that different orbitals are renormalized differently by electronic correlations, a characteristic property of Hund's metals that appears to be much more prominent in the iron chalcogenides in comparison with the pnictides \\cite{Yin2011,Lanata2013,Medici2014}. Experimentally, recent scanning tunneling microscopy (STM) measurements revealed the impact of orbital differentiation on the superconducting state \\cite{Sprau75}. Theoretically, it has been suggested that orbitally selective spin fluctuations may be the origin of nematicity without magnetism in FeSe \\cite{Fanfarillo2018}. Nematic order was also proposed to enhance orbital selectivity by breaking the orbital degeneracy, leading to asymmetric effective masses in different $d$-orbitals \\cite{Yu2018}. The effect of orbital differentiation becomes even more extreme as selenium is replaced by tellurium. In Fe$_{1+y}$Te$_{1-x}$Se$_{x}$, angle resolved photoemission spectroscopy (ARPES) revealed a strong loss of spectral weight of the $d_{xy}$ orbital at high temperatures, which was interpreted in terms of proximity to an orbital-selective Mott transition \\cite{Yi2015}. Similar drastic changes were also observed as a function of doping \\cite{Liu2015}, mimicking the evolution of spin fluctuations. Nevertheless, the impact of orbital incoherence on nematicity remains little explored \\cite{Bascones17}.\n\t\\\\\n\t\\indent In this report, we present systematic measurements of both the $B_{1g}$ and $B_{2g}$ nematic susceptibilities of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ ($0 \\leq x \\leq 0.53$) using the elastoresistivity technique. We demonstrate that the doping dependence of the two nematic susceptibilities closely track the evolution of the corresponding spin fluctuations. In particular, a diverging $B_{1g}$ nematic susceptibility is observed in the parent compound Fe$_{1+y}$Te, suggesting that the spin-nematic paradigm also applies to the double-stripe AFM order \\cite{Zhang2017, Moreo17, Borisov2019}. A diverging $B_{2g}$ nematic susceptibility is observed over a wide range of doping ($0.17 \\leq x \\leq 0.53$), and its magnitude is strongly enhanced by both Se doping and annealing. In addition, the temperature dependence of the $B_{2g}$ nematic susceptibility shows significant deviation from Curie-Weiss behavior above 50K. This is in sharp contrast to the iron pnictides, where the Curie-Weiss temperature dependence extends all the way to 200K. This unusual temperature dependence is captured by a theoretical calculation that includes the loss of spectral weight of the $d_{xy}$ orbital, revealing its importance for $B_{2g}$ nematic instability.\n\t\\\\\n\t\\indent Single crystals of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ were grown by the modified Bridgeman method. The electrical, magnetic and superconducting properties of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ are known to sensitively depend on $y$, the amount of excess iron. To study these effects, crystals were annealed in selenium vapor to reduce the amount of excess iron. By symmetry, the $B_{1g}$ and $B_{2g}$ nematic susceptibilities are proportional to the elastoresistivity coefficients $m_{11} - m_{12}$ and $2m_{66}$, respectively. We performed the elastoresistivity measurements in the Montgomery geometry, which enables simultaneous determination of the full resistivity tensor, hence the precise decomposition into different symmetry channels, as illustrated in Fig. \\ref{fig:Fig1}(c) and (d). Details of the Montgomery elastoresistivity measurements can be found elsewhere \\cite{Kuo958}. The crystal orientation is determined by polarization resolved Raman spectroscopy. Representative data of anisotropic resistivity as a function of anisotropic strain at 20K in $B_{1g}$ and $B_{2g}$ channels are shown in Fig. \\ref{fig:Fig1}(e) and (f). The $B_{1g}$ elastoresistivity coefficient $m_{11} - m_{12}$ and the $B_{2g}$ one, $2m_{66}$, can be obtained by fitting the linear slope of resistivity versus strain. Samples with high doping concentrations ($x = 0.38, 0.45, 0.53$) show predominantly a $B_{2g}$ response while the low doping ones ($x = 0, 0.12$) show comparable $B_{1g}$ and $B_{2g}$ responses.\n\t\\\\\n\t\\indent Figs. \\ref{fig:fig2}(a) and (b) show the temperature dependence of $m_{11} - m_{12}$ and $2m_{66}$ of annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ for $0 \\leq x \\leq 0.53$. For $0.28 \\leq x \\leq 0.53$, $2m_{66}$ shows a strong temperature dependence that grows continuously as temperature decreases. For $x = 0.45$, $2m_{66}$ reaches a value of $\\sim$ 100, comparable to optimally doped pnictides. While preserving a similar diverging temperature dependence, the maximum absolute value of $2m_{66}$ decreases rapidly as selenium concentration decreases, from 100 for $x = 0.45$ to 8 for $x = 0.17$. On the other hand, $m_{11} - m_{12}$ shows a diverging response when $x$ is below $0.17$, which is in the vicinity of the double-stripe AFM order. As selenium concentration increases, $m_{11} - m_{12}$ evolves to a temperature independent response, with small kinks at low temperatures likely coming from contamination of $2m_{66}$ due to misalignment. Overall, our observation of the doping dependence of $2m_{66}$ and $m_{11} - m_{12}$ is consistent with the evolution of low-temperature spin fluctuations from predominantly $(\\pi,0)$ at small $x$ to predominantly $(\\pi,\\pi)$ at optimal doping $x\\sim 0.5$ \\cite{Lumsden2010,Liu2010,Zaliznyak10316,Xu2018}.\n\t\\\\\n\t\\indent To gain more insight, we fit the $2m_{66}$ and $m_{11} - m_{12}$ to a Curie-Weiss temperature dependence:\n\t\\begin{eqnarray}\n\tm=m^{0}+\\frac{\\lambda}{a(T-T^{*})}\n\t\\end{eqnarray}\n\tFor the parent compound Fe$_{1+y}$Te, $m_{11} - m_{12}$ can be well fitted to a Curie-Weiss behavior in the temperature range just above the double stripe AFM ordering temperature $T_{\\rm mag}$ = 71.5K (Fig. \\ref{fig:fig2}(c)). The fitted Curie-Weiss temperature $T^{*}$ is slightly smaller than $T_{\\rm mag}$. Despite the smaller absolute value ($\\sim$ 10 at maximum), the behavior of $m_{11} - m_{12}$ is reminiscent of the $2m_{66}$ in the parent phase of iron pnictides, suggesting that the spin-nematic mechanism is still at play here, in agreement with theoretical expectations \\cite{Zhang2017, Moreo17, Borisov2019}. \n\t\\\\\n\t\\begin{figure}\n\t\t\\includegraphics[trim={0 1cm 0 0cm},clip,width=0.48\\textwidth]{FIG3.eps}\n\t\t\\caption{Comparison with the Hall coefficient $R_{H}$. (a) Doping dependence of the absolute value of the $B_{2g}$ elastoresistivity coefficient $2m_{66}$ (red squares) and of $R_{H}$ (black diamonds) at 16K. Dashed lines are guide to the eyes. (b) Colormap of the negative Hall coefficient -$R_{H}$, (c) of $m_{11} - m_{12}$ (d) and of $|2m_{66}|$ as a function of temperature and doping. The double-spin stripe and the superconducting transition temperatures are denoted as blue squares and yellow triangles, respectively.}\n\t\t\\label{fig:fig3}\n\t\\end{figure}\n\t\\indent Fig. \\ref{fig:fig2}(d) shows the Curie-Weiss fitting of $2m_{66}$ for the $x = 0.45$ sample. The fitting of $2m_{66}$ only works at low temperatures, as can be seen in the linear temperature dependence of $|2m_{66} -2m_{66}^{0}|^{-1}$ below 50K. It shows a significant deviation for temperature greater than 50K. The $T^{*}$ obtained from the low-temperature fitting is close to 0K. Intriguingly, the $T^{*}$ extracted from the Curie-Weiss fitting is approximately zero for all $0.17 \\leq x \\leq 0.53$, while the Curie constant $\\lambda\/a$ decreases with $x$ (SOM). While the number of doping concentrations studied in the current work is insufficient to support a power law analysis, $2m_{66}$ at constant $T = 16$K appears to be diverging as $x$ increases from 0.17 to 0.45 (Fig. \\ref{fig:fig3}(a)). Both the doping dependence and the near zero $T^{*}$ are consistent with the existence of a putative nematic quantum critical point at $x \\sim 0.5$. Interestingly, recent work doping FeSe with Te suggests that the $90$K nematic transition of FeSe is continuously suppressed and extrapolates to $0$ at $x \\sim 0.5$ \\cite{Terao2019}.\n\t\\\\\n\t\\indent This deviation from Curie-Weiss at high temperatures is very unusual. In the iron pnicitides, such a deviation was only observed at low temperatures in transition-metal doped BaFe$_{2}$As$_{2}$ (Fig. \\ref{fig:fig2}(e)) and LaFeAsO. This unusual temperature dependence of $2m_{66}$ appears to echo the coherent-incoherent crossover observed by ARPES \\cite{Yi2015}, where the spectral weight of the $d_{xy}$ orbital is strongly suppressed as the temperature increases or as the selenium concentration decreases. To further confirm this correlation, we measured the Hall coefficient $R_H$, which has been demonstrated to be a good indicator of this incoherent-to-coherent crossover \\cite{Ding2014,Otsuka2019,Liu2015}. The recovery of the $d_{xy}$ spectral weight is generally correlated with a sign-change of $R_H$ \\cite{Otsuka2019} from positive to negative. Fig. \\ref{fig:fig3}(a) shows the low-temperature $R_H$ and $2m_{66}$ as a function of doping, whereas Fig. \\ref{fig:fig3}(b-d) contain the full temperature and doping dependence of $R_H$, $m_{11} - m_{12}$, and $|2m_{66}|$, respectively. These plots reveal the strong correlation between a negative $R_H$ and an enhancement of $2m_{66}$.\n\t\\\\\n\t\t\\begin{figure}\n\t\t\\includegraphics[width=0.38\\textwidth]{fig4.eps}\n\t\t\\caption{The effect of annealing on the nematic susceptibility of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$. (a-c) Temperature dependence of (a) normalized in-plane resistivity ($R\/R_{300 \\rm{K}}$), (b) Hall coefficient $\tR_{H}$ and (c) elastoresistivity coefficient $2m_{66}$ of as-grown (black) and annealed (red) samples for $x = 0.45$. The vertical grey line marks the temperature below which the behavior of annealed and as-grown samples starts to deviate from each other. Inset of (a) shows the temperature dependence of the zero field cooling (ZFC) magnetic susceptibility measured at 100Oe (H $\\parallel$ ab). The superconducting volume fraction is significantly enhanced for the annealed sample. }\n\t\t\\label{fig:Fig4}\n\t\\end{figure}\n\t\\indent The properties of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ also depend on the amount of excess iron, which can only be removed by annealing \\cite{Sun_2019}. Taking $x = 0.45$ as an example, the resistance of the annealed sample is metallic for temperatures below 150K (Fig. \\ref{fig:Fig4}(a)). As Fig. \\ref{fig:Fig4}(b) shows, at around 40K the Hall coefficient of the annealed sample turns from positive to negative, which is a signature of incoherent to coherent crossover. In contrast, the resistance of the as-grown sample shows a weakly insulating upturn at low temperatures (Fig. \\ref{fig:Fig4}(a) black dashed curve), and the Hall coefficient remains positive at all temperatures (Fig. \\ref{fig:Fig4}(b) black circles), indicating that the $d_{xy}$ orbital is still incoherent at low temperatures. Interestingly, at the same temperature where the resistance and the Hall coefficient of the as-grown and annealed samples depart from each other, the elastoresistivity coefficient $2m_{66}$ shows a pronounced enhancement for the annealed sample (Fig. \\ref{fig:Fig4}(c)). Such an enhancement was observed in all annealed samples (SOM), providing further evidence of the correlation between the enhancement of the nematic susceptibility and the coherence of the $d_{xy}$ orbital.\n\t\\\\\n\t\\indent The doping and annealing dependences of $2m_{66}$ presented above strongly suggest that the $B_{2g}$ nematic susceptibility also have an orbitally-selective character.\n\tIndeed, previous theoretical works have highlighted the impact of orbital degrees of freedom on spin-driven nematicity \\cite{Fanfarillo15, Khodas16, Kontani16, Yu2018, Chubukov18, Fanfarillo2018}. Using a slave-spin approach, Ref. \\cite{Bascones17} found a suppression of the orbital-nematic susceptibility due to orbital incoherence. To model our data, we employ the generalized random phase approximation (RPA) of Ref. \\cite{Christensen2016} to compute the spin-driven nematic susceptibility for the five-orbital Hubbard-Kanamori model (details in the SM). For fully coherent orbitals, it was found that the largest contribution to the nematic susceptibility $\\chi_{\\rm nem}$ comes from the $d_{xy}$ orbital. Thus, one expects that $\\chi_{\\rm nem}$ would be suppressed if the $d_{xy}$ orbital were to become less coherent.\n\t\\\\\n\t\t\\begin{figure}\n\t\t\\includegraphics[trim={0 0cm 0cm 0cm},clip,width=0.5\\textwidth]{FIG5.eps}\n\t\t\\caption{Comparison with theoretical calculations. (a-b) Temperature dependence of (a) $|2m_{66}|$ and (b) $|2m_{66} - 2m_{66}^{0}|^{-1}$ of optimally doped Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ (red circles) and BaFe$_{2}$(As$_{0.66}$P$_{0.34}$)$_{2}$ (black squares). The red and black lines show Curie Weiss fittings. The data for BaFe$_{2}$(As$_{0.66}$P$_{0.34}$)$_{2}$ follows a Curie-Weiss behavior all the way up to $200$K, whereas for Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$, it deviates from Curie-Weiss behavior above $\\sim$ 50K. (c-d) Theoretical calculation of the normalized nematic susceptibility $\\chi_{\\rm nem}$ and its inverse, plotted as a function of the relative temperature with respect to the theoretical nematic transition temperature ($T-T_{\\rm nem}$) for different spectral weight $0.7 \\leq Z_{xy} \\leq 1$.}\n\t\t\\label{fig:fig5}\n\t\\end{figure}\n\t\\indent To verify this scenario, we calculated how $\\chi_{\\rm nem}$ changes upon suppressing the spectral weight $Z_{xy}$ of the $d_{xy}$ orbital. For our purposes, the reduction in $Z_{xy}$ acts phenomenologically as a proxy of the incoherence of this orbital, similarly to \\cite{Sprau75}, but its microscopic origin is not important. Fig. \\ref{fig:fig5}(c)-(d) contrasts the nematic susceptibility for $ 0.7 \\leq Z_{xy} \\leq 1$. We note two main trends arising from the suppression of $d_{xy}$ spectral weight: first, as anticipated, the nematic susceptibility (and the underlying nematic transition temperature, which is non-zero in the model) is reduced (Fig. \\ref{fig:fig5}(c)). Second, its temperature dependence changes from a Curie-Weiss-like behavior over an extended temperature range to a behavior in which the inverse nematic susceptibility quickly saturates and strongly deviates from a linear-in-T dependence already quite close to the nematic transition (Fig. \\ref{fig:fig5}(d)). These behaviors are remarkably similar to those displayed by the elastoresistance data shown in Fig. \\ref{fig:fig5}(a)-(b), with $Z_{xy} = 1$ mimicking the behavior of optimally P-doped BaFe$_{2}$As$_{2}$ and $Z_{xy} < 1$, of optimally doped Fe$_{1+y}$Te$_{1-x}$Se$_x$. Interestingly, the susceptibility associated with $(\\pi, \\pi)$ fluctuations is also suppressed by the decrease in $Z_{xy}$, in qualitative agreement with the neutron scattering experiments \\cite{Xu2012} (for a more detailed discussion, see SM). Of course, since $Z_{xy}$ in our model is an input, and not calculated microscopically, our model is useful to capture tendencies, but not to extract the experimental value of $Z_{xy}$. Furthermore, note that in our calculation $Z_{xy}$ is temperature-independent, while in the experiment it changes with temperature.\n\t\\\\\n\t\\indent In summary, our results reveal the close connection between nematic fluctuations and spin fluctuations in Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ for both $B_{1g}$ and $B_{2g}$ channels. Additionally, the unusual temperature dependence of the $B_{2g}$ nematic susceptibility can be attributed to the coherent-to-incoherent crossover experienced by the $d_{xy}$ orbital, providing direct evidence for the orbital selectivity of the nematic instability. Our work presents Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ as an ideal platform to study the physics of intertwined orders in a strongly correlated Hund's metal.\n\n\\begin{acknowledgments}\nWe thank Ming Yi for fruitful discussion. The work at UW was supported by NSF MRSEC at UW (DMR-1719797). The material synthesis was supported by the Northwest Institute for Materials Physics, Chemistry, and Technology (NW IMPACT) and the Gordon and Betty Moore Foundation's EPiQS Initiative, Grant GBMF6759 to J.-H.C. J.H.C. acknowledge the support of the David and Lucile Packard Foundation, the Alford P. Sloan Foundation and the State of Washington funded Clean Energy Institute. Theory work (M.H.C. and R.M.F.) was supported by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Award No. DE-SC0020045. The Raman measurement is partially supported by Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division (DE-SC0012509).\n\\end{acknowledgments}\n\n\n\\section{0pt}{18pt plus 4pt minus 2pt}{18pt plus 2pt minus 2pt}\n\t\\section{S1 Polarized Raman spectroscopy on Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$}\n\tPolarized Raman spectroscopy was used to determine the crystal orientation. Representative data of the Raman spectra are shown in FIG. S\\ref{fig:FigS1}. Measurements were made on freshly cleaved surfaces of annealed Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ single crystals in a backscattering configuration at 20K using a 532nm laser, a 1200 groove mm$^{-1}$ grating spectrometer (Princeton Acton 2500i) and a liquid-nitrogen cooled charge-coupled device (CCD) detector. The laser power was set to 100$\\mu$W and the laser spot was focused on a $\\sim$ 2 $\\times$ 2 $\\mu m^{2}$ surface. The (ab) configuration corresponds to the polarization of the incident (scattered) light along the crystallographic a (b) axis, while the (xy) configuration corresponds to the polarization of the incident (scattered) light along the x (y) direction, which is the direction rotated by 45$^{\\circ}$ from the a (b) axis in the ab plane. The presence and absence of $B_{1g}$ phonon mode at 208 cm$^{-1}$ in the (xy) and (ab) configuration unambiguously confirm the determination of crystal orientation. A weak $A_{1g}$ phonon mode at 159 cm$^{-1}$ was observed in both (ab) and (xy) configurations due to a leakage of other polarization components as observed elsewhere. \n\t\\begin{figure*}\n\t\\renewcommand{\\figurename}{Fig. S}\n\t\\includegraphics[trim={0 0.1cm 0cm 0.2cm},clip,width=1\\textwidth]{FigS1.eps}\n\t\\caption{Polarized Raman spectra on Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ in the (ab) and (xy) configurations at 20K. The $B_{1g}$ mode at 208 cm$^{-1}$ distinguishes the (xy) configuration from the (ab) configuration.}\n\t\\label{fig:FigS1}\n\\end{figure*}\n\\section{S2 Annealing effects of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$}\nAs grown crystals are known to host excess iron which can be removed by annealing in Te or Se vapor. Samples with and without excess Fe show drastically different electrical, magnetic, and superconducting properties. To study these effects, crystals were cleaved into thin slices ($\\sim$ 1 mm), loaded in a crucible with another crucible of an appropriate amount of selenium powder beneath it, sealed in quartz tubes, and annealed at 500$^\\circ$C for a week. Further annealing in a Se vapor causes significant hardship in cleaving sizable samples. The average compositions of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ samples were determined on at least 4 regions of the crystals by the energy-dispersive x-ray (EDXS) method using the Sirion XL30 scanning electron microscope. We only use EDS measurement to determine x (Se\/Te) ratio. For the measurement of y, even though we consistently obtained a smaller value of y after annealing, the determination of the absolute value of y is inconclusive. Similar difficulty to accurately determined y has also been reported previously \\cite{Rinott17}. By comparing the transport data and the position and the width of transition, we estimate the upper limit of y is below 0.06 for x $<$ 0.4 and and 0.02 for x $>$ 0.4. \n\\\\\n\\indent The annealing significantly changed the electrical transport behavior, as shown in FIG. S\\ref{fig:figS3} insets. After annealing, the resistivity changes from a weakly insulating behavior for as-grown crystals (black curves) to a metallic behaivor for annealed samples (red curves). The superconducting transition also becomes sharper with an increased T$_{c}$ after annealing. Fig. S\\ref{fig:figS3} shows the elastoresistivity coefficient in the B$_{2g}$ channel, $2m_{66}$, for both as-grown and annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (x = 0.17 - 0.45). The elastoresistivity coefficient $2m_{66}$ is enhanced for the annealed samples (red squares), suggesting the correlation between the enhanced nematic susceptibility and the coherence of the $d_{xy}$ orbital as discussed in the main text. We notice that the $2m_{66}$ changes sign as x decreases, and the x = 0.28 even shows a sign changing before and after annealing. This behavior might be related to the sensitivity of the sign of resistivity anisotropy to the shape of Fermi surfaces and spin fluctuations, which warrants future investigation.\n\\begin{figure*}\n\t\\includegraphics[trim={0 0cm 0 0.2cm},clip,width=1\\textwidth]{FigS3.eps}\n\t\\renewcommand{\\figurename}{Fig. S}\n\t\\caption{Temperature dependence of $2m_{66}$ for as-grown (black squares) and annealed (red squares) (a) Fe$_{1+y}$Te$_{0.83}$Se$_{0.17}$ (b) Fe$_{1+y}$Te$_{0.72}$Se$_{0.28}$ (c) Fe$_{1+y}$Te$_{0.62}$Se$_{0.38}$ and (d) Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$. The black and red lines underneath the square data show the Curie Weiss fittings for the $2m_{66}$ of as-grown and annealed samples respectively. The fitting parameters are listed in Table S\\ref{tab:tabs2}. For as-grown crystals, $2m_{66}$ for x = 0.17 and 0.28 are negative. After annealing, $2m_{66}$ for x = 0.28 flips sign from negative to positive and the $2m_{66}$ for x = 0.17 remains negative. Insets of (a)-(d) are the temperature dependences of normalized resistances for as-grown (black) and annealed (red) Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (x = 0.17 0.28 0.38 0.45) respectively.}\n\t\\label{fig:figS3}\n\\end{figure*}\n\\section{S3 Elastoresistivity measurement}\n\\begin{figure}\n\t\\includegraphics[trim={0 0cm 0cm 0cm},clip,width=0.45\\textwidth]{FIGS4.png}\n\t\\renewcommand{\\figurename}{Fig. S}\n\t\\caption{Temperature dependence of (a) the isotropic ($A_{1g}$) elastoresistivity response to the tensile strain and (b) the anisotropic ($B_{2g}$) elastoresistivity response to the anisotropic strain, of a single crystal of Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ prepared along the Fe-Fe bonding direction. The zero-strain point is determined by the intersection of the strain dependence of $\\rho_{xx}$ and $\\rho_{yy}$ at each temperature. }\n\t\\label{fig:FigS4}\n\\end{figure}\n\\indent The elastoresistivity measurement is conducted by gluing the square shape samples on a piezoelectric stack with the Montgomery contact geometry. Square edge of the samples are cut along the Fe-Fe (Fe-Ch) bonding direction for $B_{2g}$ ($B_{1g}$) elastoresistivity measurements, with electrical contacts at the four corners of the square. With this method, we are able to measure the resistivity along two perpendicular directions on the same piece of sample, and therefore eliminate the problems caused by variations of the samples and the transmitted strain. By gluing the sample on the sidewall of the piezoelectric stack and applying voltages, we apply an in situ tunable anisotropic strain on the sample. The magnitude of the strain is calibrated by measuring the resistance change of a strain gauge glued on the other side of the piezo stack. In this approach, one can decompose the shear strain into different symmetry channels and precisely determine the elastoresistivity coefficients. In the configuration shown in the inset of FIG. S\\ref{fig:FigS4}(a), the strain can be decomposed into the anisotropic $B_{2g}$ strain and the isotropic $A_{1g}$ strain. FIG. S\\ref{fig:FigS4}(a) and (b) show the elastoresistivity responses in the $A_{1g}$ and $B_{2g}$ symmetry channels. The linear slope at zero strain point of the elastoresistivity in the $A_{1g}$ and $B_{2g}$ channels are the first order elastoresistivity coefficients $m_{A_{1g}}^{A_{1g}}$ and $m_{B_{2g}}^{B_{2g}}$, respectively. One may also notice nonlinear responses in the elastoresistivity of annealed Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ are present in both the $B_{2g}$ and $A_{1g}$ channels. According to the symmetry constraints, the nonlinear term in the $A_{1g}$ channel is most likely due to the second-order $B_{2g}$ strain:\n\\begin{eqnarray}\n(\\frac{\\Delta\\rho}{\\rho_{0}})_{A_{1g}}=m_{A_{1g}}^{A_{1g}}\\epsilon_{A_{1g}}+m_{A_{1g}}^{B_{2g},B_{2g}}(\\epsilon_{B_{2g}})^{2}+m_{A_{1g}}^{A_{1g},A_{1g}}(\\epsilon_{A_{1g}})^{2}+O(\\epsilon^{3})\n\\end{eqnarray}\n\\indent While for the $B_{2g}$ response (FIG. S\\ref{fig:FigS4} (b)), the nonlinearity in the $B_{2g}$ channel can be caused by either a mixed-in isotropic resistivity due to the sample's deviation from a perfect square along the Fe-Fe bonding direction, or a second-order strain $\\epsilon_{B_{2g}}\\epsilon_{A_{1g}}$. \n\\begin{eqnarray}\n(\\frac{\\Delta\\rho}{\\rho_{0}})_{B_{2g}}^{m}=m_{B_{2g}}^{B_{2g}}\\epsilon_{B_{2g}}+m_{A_{1g}}^{B_{2g},B_{2g},mixed}(\\epsilon_{B_{2g}})^{2}+m_{B_{2g}}^{B_{2g},A_{1g}}\\epsilon_{B_{2g}}\\epsilon_{A_{1g}}+O(\\epsilon^{3})\n\\end{eqnarray}\t\n\\indent Here, $(\\frac{\\Delta\\rho}{\\rho_{0}})_{B_{2g}}^{m}$ is the measured value of resistivity anisotropy, which contains the mixed-in isotropic component. In this measurement, we mainly focus on the anisotropy term $m_{B_{2g}}^{B_{2g}}$, also known as $2m_{66}$ in the Voigt notation, which is proportional to the nematic susceptibility in the $B_{2g}$ symmetry channel of $D_{4h}$ symmetry group. By finding the anisotropic strain neutral point and fitting the anisotropic elastoresistivity response quadratically, we are able to remove the mixed-in error and extract the first order elastoresistivity coefficient $2m_{66}$. \n\\section{S4 Curie-Weiss fitting of elastoresistivity coefficient $2m_{66}$ and $m_{11} - m_{12}$}\n\\begin{figure*}\n\t\\includegraphics[trim={0cm 2cm 0cm 1cm},clip,width=1\\textwidth]{FIGS5.eps}\n\t\\renewcommand{\\figurename}{Fig. S}\n\t\\caption{Curie Weiss fitting of the $B_{2g}$ elastoresistivity coefficient $2m_{66}$ of both as-grown and annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (x = 0.28, 0.38, 0.45, 0.53). (a) - (d) as-grown Fe$_{1+y}$Te$_{1-x}$Se$_{x}$. (e) - (g) annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$. Upper panels show $|2m_{66}|$, whereas lower panels show $|2m_{66} - 2m_{66}^{0}|^{-1}$ (left axes of lower panels, blue symbols) and $|2m_{66} - 2m_{66}^{0}|(T-T^{*})$ (right axes of lower panels, yellow circles). Black (upper panels) and red (lower panels) lines show the fits to Curie-Weiss behavior of $|2m_{66}|$ and $|2m_{66} - 2m_{66}^{0}|^{-1}$respectively. Grey horizontal lines (lower panels) shows the average values of $|2m_{66} - 2m_{66}^{0}|(T-T^{*})$ in the fitting temperature range. The optimal fitting range (grey shaded) is determined by the greatest corresponding adjusted R-Square value. For (a), $2m_{66}$ is negative. For (b) - (h), $2m_{66}$ is positive.}\n\t\\label{fig:FIGS5}\n\\end{figure*}\n\\indent FIG. S\\ref{fig:FIGS5} shows the Curie Weiss fitting of the $B_{2g}$ elastoresistivity coefficient $2m_{66}$ for both as-grown and annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (x = 0.28 - 0.53). The optimal fitting range is illustrated by the grey shaded area. The divergent temperature dependence of $2m_{66}$ can be fitted by the Curie Weiss law well below 50K but deviates from the Curie Weiss law at higher temperatures. Other fitting parameters are listed in Table S\\ref{tab:tabs2}. The bare nematic critical temperature $T^{*}$ is negative for all as-grown samples. With the Se concentration increases, the bare nematic critical temperature $T^{*}$ increases and approaches zero for x = 0.53. For annealed samples, the $T^{*}$ extracted from the Curie-Weiss fitting is approximately zero, while the Curie constant $\\lambda\/a$ decreases with x. Near x = 0, in the proximity of the double spin stripe order, a diverging $B_{1g}$ nematic susceptibility was also observed. The Curie Weiss fitting parameters from the fit of $B_{1g}$ elastoresistivity coefficient $m_{11} - m_{12}$ are listed in Table S\\ref{tab:tabs3}. \n\\squeezetable\n\\begin{table}\n\t\t\\renewcommand{\\tablename}{TABLE S}\n\t\\caption{\\label{tab:tabs2}Curie-Weiss fitting parameters from the Fit of $2m_{66}$ for as-grown and annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (x = 0.17 - 0.53)}\n\t\\begin{ruledtabular}\n\t\t\\begin{tabular}{ccccccc}\n\t\t\tMaterials& Fitting Range (K) & $2m_{66}^{0}$ & $\\lambda\/a_{0} (K)$ & $T^{*} (K)$ & Sample Dimension ($\\mu m$) & Adjust R-square\\\\\n\t\t\t\\hline\n\t\t\t&&&As-grown&&&\\\\\n\t\t\t\\hline\n\t\t\tFe$_{1+y}$Te$_{0.83}$Se$_{0.17}$ & $11-140$ & $3.2\\pm0.4$ & $-397\\pm16$ & $-44.6\\pm7.7$ & $530\\times480\\times20$ & $0.9904$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.72}$Se$_{0.28}$ & $17-96$ & $7.0\\pm0.6$ & $-811\\pm10$ & $-24.5\\pm2.9$ & $1210\\times1150\\times10$ & $0.9982$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.62}$Se$_{0.38}$ & $15-50$ & $-6.8\\pm1.0$ & $592\\pm55$ & $-9.0\\pm2.1$ & $870\\times740\\times40$ & $0.9984$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ & $20-68$ & $-15.22\\pm3.0$ & $1594\\pm222$ & $-9.3\\pm1.2$ & $1080\\times1070\\times30$ & $0.9966$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.47}$Se$_{0.53}$ & $20-93$ & $-16.7\\pm2.0$ & $1752\\pm159$ & $-0.8\\pm1.7$ & $670\\times570\\times60$ & $0.9981$\\\\\n\t\t\t\\hline\n\t\t\t&&&Annealed&&&\\\\\n\t\t\t\\hline\n\t\t\tFe$_{1+y}$Te$_{0.83}$Se$_{0.17}$ & $25-200$ & $2.2\\pm0.2$ & $-243\\pm10$ & $-1.9\\pm1.9$ & $460\\times440\\times10$ & $0.9965$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.72}$Se$_{0.28}$ & $17-49$ & $-5.3\\pm0.6$ & $359\\pm32$ & $-0.4\\pm1.2$ & $840\\times800\\times50$ & $0.9991$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.62}$Se$_{0.38}$ & $19-46$ & $-24.4\\pm2.3$ & $1583\\pm135$ & $-2.9\\pm1.5$ & $380\\times360\\times30$ & $0.9995$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ & $18-56$ & $-28.6\\pm3.1$ & $2004\\pm186$ & $1.5\\pm1.2$ & $410\\times360\\times30$ & $0.9988$\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table}\n\\squeezetable\n\\begin{table}\n\t\\renewcommand{\\tablename}{TABLE S}\n\t\\caption{\\label{tab:tabs3}Curie-Weiss fitting parameters from the Fit of $m_{11} - m_{12}$ for Fe$_{1+y}$Te}\n\t\\begin{ruledtabular}\n\t\t\\begin{tabular}{ccccccc}\n\t\t\tMaterials& Fitting Range (K) & $m_{11}^{0}-m_{12}^{0}$ & $\\lambda\/a_{0} (K)$ & $T^{*} (K)$ & Sample Dimension ($\\mu m$) & Adjust R-square\\\\\n\t\t\t\\hline\n\t\t\tFe$_{1+y}$Te & $71-104$ & $0.29\\pm0.06$ & $13\\pm4$ & $70.2\\pm0.1$ & $443\\times440\\times24$ & $0.9992$\\\\\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table}\n\n\n\\section{S5 RPA calculation of the nematic susceptibility}\n\nTo capture the effect of orbital degrees of freedom on the nematic susceptibility, we begin by introducing the multi-orbital Hubbard-Kanamori Hamiltonian (momentum sums are implicit):\n\\begin{eqnarray}\n\t\\mathcal{H} &=& \\sum_{\\mu} \\left[\\epsilon_{\\mu\\nu}(\\mbf{k}) - \\mu \\delta_{\\mu\\nu}\\right]c^{\\dagger}_{\\mbf{k}\\mu\\sigma}c_{\\mbf{k}\\nu\\sigma} \\nonumber + U \\sum_{\\mu} n_{\\mbf{q}\\mu\\uparrow} n_{-\\mbf{q}\\mu\\downarrow} + U' \\sum_{\\substack{\\mu < \\nu \\\\ \\sigma \\sigma'}}n_{\\mbf{q}\\mu\\sigma}n_{-\\mbf{q}\\nu\\sigma'} \\\\ &+& \\frac{J}{2}\\sum_{\\substack{\\mu\\neq\\nu \\\\ \\sigma\\sigma'}}c^{\\dagger}_{\\mbf{k}+\\mbf{q}\\mu\\sigma}c_{\\mbf{k}\\nu\\sigma}c^{\\dagger}_{\\mbf{k}'-\\mbf{q}\\nu\\sigma'}c_{\\mbf{k}'\\mu\\sigma} + \\frac{J'}{2}\\sum_{\\substack{\\mu\\neq\\nu \\\\ \\sigma}}c^{\\dagger}_{\\mbf{k}+\\mbf{q}\\mu\\sigma}c^{\\dagger}_{\\mbf{k}'-\\mbf{q}\\mu\\bar{\\sigma}}c_{\\mbf{k}'\\nu\\bar{\\sigma}}c_{\\mbf{k}\\nu\\sigma}\\,.\n\\end{eqnarray}\nHere $\\mu$ and $\\nu$ are orbital indices, $\\sigma$ labels spin and we assume $U'=U-2J$ and $J'=J$. $\\epsilon^{\\mu\\nu}(\\mbf{k})$ denotes the dispersion, in this case obtained from a tight-binding fit to DFT. In the results shown in the main text, we used the band structure parameters presented in Ref.~[\\onlinecite{ikeda10}], which give three hole-like Fermi pockets and two electron-like Fermi pockets. While this tight-binding parametrization is not intended to model specifically optimally doped FeTe$_{1-x}$Se$_x$, it offers a solid framework to elucidate, on general grounds, the tendencies of how the nematic susceptibility is affected by the suppression of $d_{xy}$ orbital spectral weight.\n\nThe interactions can be conveniently expressed as elements of the same tensor:\n\\begin{eqnarray}\n\tU^{\\mu\\mu\\mu\\mu} = U\\,, \\quad U^{\\mu\\nu\\nu\\mu} = U'\\,, \\quad U^{\\mu\\nu\\mu\\nu} = J'\\,, \\quad U^{\\mu\\mu\\nu\\nu} = J\\,.\n\\end{eqnarray}\nThe RPA expression for the spin-driven nematic susceptibility of this model was derived previously by two of us in Ref. \\cite{christensen16} (see also Ref. \\cite{fanfarillo18}). Here, we simply quote the result from the former paper, derived in the approximation that the antiferromagnetic order parameter is diagonal in orbital space (as shown for instance in Ref. \\cite{christensen17}). The nematic susceptibility in this case is a rank-4 tensor given by:\n\\begin{eqnarray}\n\t\\chi_{\\rm nem}^{\\mu\\nu\\rho\\lambda}\/N = \\left( \\int_q \\chi^{\\rho \\iota}(q)\\chi^{\\lambda \\kappa}(q) \\right) \\left(\\delta_{\\mu \\kappa}\\delta_{\\nu \\iota} - g^{\\mu\\nu \\gamma \\phi} \\int_q \\chi^{\\gamma \\iota }(q)\\chi^{\\phi \\kappa}(q) \\right)^{-1}\\,,\\label{eq:nem_susc}\n\\end{eqnarray}\nwhere $\\int_q \\equiv T \\sum_{\\omega_m}\\int \\mathrm{d}^2 \\mbf{q}$, $N=3$ is the number of components of the magnetic order parameter, and the Einstein summation convention is used. Here\n\\begin{eqnarray}\n\t\\chi^{\\mu\\nu}(q) = \\left[ (U_{\\mu\\nu})^{-1} - \\chi_0^{\\mu\\nu}(q) \\right]^{-1}\n\\end{eqnarray}\nis the RPA magnetic susceptibility, which is a rank-2 tensor. The bare magnetic susceptibility is given by:\n\\begin{eqnarray}\n\t\\chi^{\\mu\\nu}_{0}(q) = - \\sum_k \\mathcal{G}^{\\mu\\nu}(k+q)\\mathcal{G}^{\\nu\\mu}(k)\\,,\n\\end{eqnarray}\nwhere repeated indices are not summed and $\\mathcal{G}^{\\mu\\nu}$ denotes the bare multi-orbital Green's function:\n\\begin{eqnarray}\n\t\\mathcal{G}^{\\mu\\nu}(k) = \\sum_{m} \\frac{a^{m}_{\\mu}(\\mbf{k})a^{m}_{\\nu}(\\mbf{k})^{\\ast}}{i\\omega_n - \\xi^{m}(\\mbf{k})}\\,,\n\\end{eqnarray}\nwith $k=(i\\omega_m,\\mbf{k})$. For notational convenience, we introduce:\n\\begin{eqnarray}\n\t\\mathcal{G}_X^{\\mu\\nu} \\equiv \\mathcal{G}^{\\mu\\nu}(\\mbf{k}+\\mbf{Q}_1,i\\omega_m)\\,,\\quad \n\t\\mathcal{G}_Y^{\\mu\\nu} \\equiv \\mathcal{G}^{\\mu\\nu}(\\mbf{k}+\\mbf{Q}_2,i\\omega_m)\\,.\n\\end{eqnarray}\nHere, $\\mbf{Q}_1 = (\\pi, 0)$ and $\\mbf{Q}_2 = (0, \\pi)$ (in the 1-Fe unit cell). Finally, the nematic coupling constant $g$ in Eq. (\\ref{eq:nem_susc}) is also a rank-4 tensor obtained from convolutions of the Green's functions:\n\\begin{eqnarray}\n\tg^{\\rho\\nu\\eta\\mu} &=& -\\frac{1}{16}\\sum_{k}\\Big( 2\\mathcal{G}^{\\mu\\rho}\\mathcal{G}^{\\rho\\nu}_{X} \\mathcal{G}^{\\nu\\eta}\\mathcal{G}^{\\eta\\mu}_{X} - \\mathcal{G}^{\\mu\\rho}\\mathcal{G}^{\\rho\\eta}_{X} \\mathcal{G}^{\\eta\\nu}\\mathcal{G}^{\\nu\\mu}_{X} - \\mathcal{G}^{\\mu\\rho}\\mathcal{G}^{\\rho\\nu}_{X} \\mathcal{G}^{\\nu\\eta}\\mathcal{G}^{\\eta\\mu}_{Y} \\nonumber \\\\ && \\qquad - \\mathcal{G}^{\\nu\\rho}\\mathcal{G}^{\\rho\\mu}_{X} \\mathcal{G}^{\\mu\\eta}_{X+Y}\\mathcal{G}^{\\eta\\nu}_{X} + \\mathcal{G}^{\\mu\\rho}\\mathcal{G}^{\\rho\\eta}_{X} \\mathcal{G}^{\\eta\\nu}_{X+Y}\\mathcal{G}^{\\nu\\mu}_{Y} \\Big) + (X \\leftrightarrow Y)\\,,\n\\end{eqnarray}\nwhere repeated indices are not summed.\n\nTo elucidate how orbital differentiation affects the nematic susceptibility within our framework, we follow the approach outlined in Ref.~\\cite{kreisel17} and modify the Green's function by:\n\\begin{eqnarray}\n\t\\mathcal{G}^{\\mu \\nu} \\rightarrow \\sqrt{Z_{\\mu}Z_{\\nu}}\\mathcal{G}^{\\mu\\nu}\\,,\n\\end{eqnarray}\nwhere we stress that, once again, the Einstein convention is not assumed. Since both $g^{\\mu\\nu\\rho\\lambda}$ and $\\chi^{\\mu\\nu}(q)$ depend on the Green's functions, a reduced $Z$ factor will have an intricate impact on the nematic susceptibility. As discussed in the main text, this is a phenomenological way to mimic the complicated effect of incoherence and loss of spectral weight on the response function, whose validity requires that the system remains in a metallic state.\n\nThe nematic susceptibility plotted in the main text for different values of $Z_{xy}$ corresponds to the largest eigenvalue $\\lambda_{\\rm nem}$ of the equation:\n\\begin{equation}\n\\chi_{\\rm nem}^{\\mu\\nu\\rho\\lambda} \\Phi^{(n)}_{\\mu \\nu} = \\lambda^{(n)}_{\\rm nem} \\Phi^{(n)}_{\\rho \\lambda} \n\\end{equation}\nIn Fig. S\\ref{fig:nematic_susc_weight}, we show how the corresponding eigenvector $\\Phi^{(n)}_{\\mu \\nu}$ changes for decreasing $Z_{xy}$. Here we used $U=1.2$ eV and $J=U\/6$. To highlight the impact of $Z_{xy}$ we fixed the filling to $5.9$ electrons per site. As expected, the main contribution to the nematic susceptibility, signaled by the brightest squares in the figure, shifts from intra $xy$-orbital processes to intra $xz\/yz$-orbital processes. \n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{nem_weight.pdf}\n\\renewcommand{\\figurename}{Fig. S}\n\\caption{\\label{fig:nematic_susc_weight} $\\Phi^{(1)}_{\\mu\\nu}$ for $Z_{xy}=1$ and $Z_{xy}=0.7$ at a temperature immediately prior to the nematic instability of each case. For $Z_{xy}=1$, the dominant contribution to the nematic susceptibility arises from the $xy$ orbital. As $Z_{xy}$ is reduced, the $xz$ and $yz$ orbitals become the dominant ones.}\n\\end{figure}\n\n\nA direct consequence of the reduction of the nematic susceptibility is a suppression of the nematic transition temperature, which in our model is manifested as a divergence of the largest eigenvalue $\\lambda_{\\rm nem}$. In Fig. S\\ref{fig:temp_as_Z}, we plot both the nematic transition temperature $T_{\\rm nem}$ and the bare (i.e. non-renormalized by nematic order) magnetic transition temperature $T_{\\rm mag}$ as a function of $Z_{xy}$. Note that not only are both transition temperatures strongly suppressed, but their separation also decreases significantly for decreasing $Z_{xy}$. As noted in Ref. \\cite{christensen16}, these RPA transition temperatures are, not surprisingly, overestimated with respect to the actual transition temperatures. For this reason, and to be able to compare the temperature dependencies of the nematic susceptibilities of systems with very different values of $T_{\\rm nem}$, in the main text we plot $\\lambda_{\\rm nem}$ as a function of $T - T_{\\rm nem}$.\n\\begin{figure}\n\\includegraphics[width=0.4\\textwidth]{Temp_as_Z.pdf}\n\\renewcommand{\\figurename}{Fig. S}\n\\caption{\\label{fig:temp_as_Z} Nematic (blue) and bare magnetic (red) transition temperatures as a function of $1-Z_{xy}$. Both are reduced by reducing $Z_{xy}$, along with their relative separation. At $Z_{xy}=0.7$ the separation between the two vanishes within our temperature resolution ($\\delta T < 0.2$ meV).}\n\\end{figure}\n\nThe suppression of $T_{\\rm mag}$ for decreasing $Z_{xy}$ is also manifested in the suppression of the overall magnitude of the bare magnetic susceptibility calculated at the spin-stripe wave-vector, $\\chi_{\\rm mag}$, as shown in Fig. S\\ref{fig:magnetic_susceptibility}.\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{inverse_mag_susc_abs_T.png}\n\\renewcommand{\\figurename}{Fig. S}\n\\caption{\\label{fig:magnetic_susceptibility} Inverse magnetic susceptibility for $ 0.7 \\leq Z_{xy}\\leq 1$ as a function of temperature. $Z_{xy}$ has a pronounced effect on the intensity of the magnetic susceptibility for a fixed temperature, and reducing $Z_{xy}$ reduces the overall intensity. On the other hand, the temperature dependence of the susceptibility is little affected by the change in $Z_{xy}$, and the magnetic susceptibility remains Curie-Weiss like, albeit the slope changes. Here we show the leading eigenvalue $\\lambda_{\\rm mag}$ as defined in Eq.~\\eqref{eq:mag_susc_def}.}\n\\end{figure}\nHere, $\\chi_{\\rm mag}$ was calculated in a manner similar to the nematic susceptibility:\n\\begin{equation}\n\t\\chi^{\\mu\\nu}(\\mathbf{q}) \\Psi^{(n)}_{\\nu}(\\mathbf{q}) = \\lambda_{\\rm mag}^{(n)}(\\mathbf{q})\\Psi^{(n)}_{\\mu}(\\mathbf{q})\\,,\\label{eq:mag_susc_def}\n\\end{equation}\nwhere, in Fig. S\\ref{fig:magnetic_susceptibility}, $\\mathbf{q}=\\mathbf{Q}_1$ or $\\mathbf{q}=\\mathbf{Q}_2$ and we show the leading eigenvalue. Note that, in contrast to the nematic susceptibility, the temperature dependence of $\\chi_{\\rm mag}$ is not strongly affected by the reduction of $d_{xy}$ spectral weight, as seen clearly from Fig. S\\ref{fig:magnetic_susceptibility}. However, for a fixed temperature, there is a strong suppression of $\\chi_{\\rm mag}$ with decreasing $Z_{xy}$. This last feature is in qualitative agreement with neutron scattering experiments, which showed that, in optimally-doped FeTe$_{1-x}$Se$_x$, the low-energy stripe-type magnetic fluctuations are suppressed with increasing temperature \\cite{xu12}. According to our analysis in the main text, increasing the temperature promotes a less coherent $d_{xy}$ orbital. Note, however, that Ref. \\cite{xu12} also reported that, as temperature was increased, besides a suppression of intensity at the stripe wave-vector, an incommensurate peak appeared in the momentum-resolved magnetic susceptibility at low energies. We did not observe such incommensurate peaks in our energy integrated magnetic susceptibility, i.e. the peak remains at $\\mathbf{q}=\\mathbf{Q}_1$ (and at $\\mathbf{q}=\\mathbf{Q}_2$), suggesting that this effect cannot be captured phenomenologically by a constant $Z_{xy}$ factor.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}