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+{"text":"\\section{Appendix}\n\\setcounter{equation}{0}\n\nIn reference \\cite{BN} it was shown that a version of the Floreanini Jackiw\naction, coupled to gravity can be obtained beginning with a scalar field\ncoupled covariantly to gravity then writing it as a first order action\nand imposing a (non covariant) constraint that selects one of the\n(chiral) solutions of the classical equation of motion. Here we will\nderive the same result using the Mandelstan\n\\cite{Mandelstan}\ndecomposition for a scalar field into chiral bosons.  Let us begin,\nas \\cite{BN}, with a scalar\nfield coupled covariantly to gravity:\n\n\\begin{equation}\n\\label{scalar}\n{\\cal L} = {1 \\over 2} \\sqrt{-g} g^{\\mu\\nu} \\partial_{\\mu} \\phi\n\\partial_{\\nu}\\phi\n\\end{equation}\n\n\\noindent The introduction of the auxiliary variable $p$ makes it possible to\nwrite this Lagrangian density in a first order form\n\n\n\\begin{equation}\n\\label{scalarfirst}\n{\\cal L} = {\\sqrt{-g}\\over 2} \\left[ -g^{00} p^2\n+ 2 g^{00} p \\dot\\phi\n+ 2g^{01} \\dot\\phi \\phi^{\\prime} +  g^{11} \\phi^{\\prime}\\phi^{\\prime}\n\\right]\n\\end{equation}\n\n\\noindent Decomposing the scalar field in its (Mandelstan) components:\n\n\\begin{equation}\n\\label{mandelstan}\n\\phi = \\phi_+ + \\phi_-\n\\end{equation}\n\n\\noindent one is able to associate each of this fields with one\nof the (chiral) solutions of the classical equation of motion by\nimposing\\cite{remark}:\n\n\\begin{equation}\n\\label{mandelstan2}\np ={g^{01}\\over g^{00}} ( \\phi_+^{\\prime} + \\phi_-^{\\prime} )\n+ {1\\over \\sqrt{-g} g^{00}} ( \\phi_-^{\\prime} - \\phi_+^{\\prime})\n\\end{equation}\n\n\\noindent Inserting (\\ref{mandelstan}) and (\\ref{mandelstan2}) in\n(\\ref{scalarfirst}) we get\n\n\\begin{eqnarray}\n\\label{scalarmandelstan}\n{\\cal L}  & = & -\\dot\\phi_+ \\phi_+^{\\prime} -\n{\\cal G_+} \\phi_+^{\\prime} \\phi_+^{\\prime} \\nonumber\\\\\n& &\\mbox{} + \\dot\\phi_- \\phi_-^{\\prime} -\n{\\cal G_-} \\phi_-^{\\prime} \\phi_-^{\\prime}\n\\end{eqnarray}\n\n\\noindent showing explicitly the decomposition of the scalar field in\nchiral components, each of them coupling with the metric exactly\nas in \\cite{BN}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. Our original motivation was to understand the possible obstructions to the third Lie theorem for algebraic Lie pseudogroups. This article is concerned with the simply transitive case. These obstructions should appear in the Galois group of certain connection associated to a Lie algebroid. However, we have written the article in the language of regular and rational parallelisms of algebraic varieties and their symmetries.\n\nA theorem of P.~Deligne says that any Lie algebra can be realized as a~parallelism of an algebraic variety. This is a sort of algebraic version of the third Lie theorem. Notwithstanding, there is one main problem: given an algebraic variety with a parallelism, how far is it from being an algebraic group? Is it possible to conjugate this parallelism with the canonical parallelism of invariant vector f\\\/ields on an algebraic group?\n\nIn the analytic context, from the Darboux--Cartan theorem \\cite[p.~212]{sharpe}, a $\\mathfrak{g}$-parallelized complex manifold $M$ has a natural $(G,G)$ structure where $G$ is a Lie group with $\\mathfrak{lie}(G) = \\mathfrak{g}$. The obstruction to be a covering of~$G$, as manifold with a~$(G,G)$ structure, is contained in a mo\\-no\\-dromy group \\cite[p.~130]{sharpe}. In~\\cite{Wang}, Wang proved that parallelized compact complex manifolds are biholomorphic to quotients of complex Lie groups by discrete cocompact subgroups. This result has been extended by Winkelmann in \\cite{Winkelmann1, Winkelmann2} for some open complex manifolds.\n\nIn this article we address the problem of classif\\\/ication of rational parallelisms on algebraic varieties up to birational transformations. Such a classif\\\/ication seems impossible in the algebraic category but we prove a criterion to ensure that a parallelized algebraic variety is isogenous to an algebraic group. Summarizing, we pursue the following plan: We regard inf\\\/initesimal symmetries of a rational parallelism as horizontal sections of a connection that we call the reciprocal Lie connection. This connection has a Galois group which is represented as a group of internal automorphisms of a Lie algebra. The obstruction to the algebraic conjugation to an algebraic group, under some assumptions, appear in the Lie algebra of this Galois group.\n\nIn Section \\ref{section_parallelisms} we introduce the basic def\\\/initions; several examples of parallelisms are given here. In Section \\ref{section_lie} we study the properties of connections on the tangent bundle whose local analytic horizontal sections form a sheaf of Lie algebras of vector f\\\/ields. We call them {\\it Lie connections}. They always come by pairs, and they are characterized by having vanishing curvature and constant torsion (Proposition~\\ref{prop_Lie_char}). We see that each rational parallelism has an attached pair of Lie connections, one of them with trivial Galois group. We compute the Galois groups of some parallelisms given in examples (Proposition \\ref{prop_example}), and prove that any algebraic subgroup of ${\\rm PSL}_2(\\mathbf C)$ appears as the dif\\\/ferential Galois group of a $\\mathfrak{sl}_2(\\mathbf C)$-parallelism (Theorem~\\ref{thm_SL2}). Section~\\ref{section_DC} is devoted to the construction of the isogeny between a $\\mathfrak g$-parallelized variety and an algebraic group $G$ whose Lie algebra is~$\\mathfrak g$. In order to do this, we introduce the Darboux--Cartan connection, a $G$-connection whose horizontal sections are parallelism conjugations. We prove that if $\\mathfrak g$ is centerless then the Darboux--Cartan connection and the reciprocal Lie connection have isogenous Galois groups. We prove that the only centerless Lie algebras $\\mathfrak{g}$ such that there exists a $\\mathfrak{g}$-parallelism with a trivial Galois group are algebraic Lie algebras, i.e., Lie algebras of algebraic groups. In particular this allows us to give a criterion for a parallelized variety to be isogenous to an algebraic group. The vanishing of the Lie algebra of the Galois group of the reciprocal connection is a necessary and suf\\\/f\\\/icient condition for a parallelized variety to be isogenous to an algebraic group:\n\n\\begin{Theorem*\nLet $\\mathfrak g$ be a centerless Lie algebra. An algebraic variety $(M,\\omega)$ with a rational parallelism of type $\\mathfrak g$ is isogenous to an algebraic group if and only if $\\mathfrak{gal}(\\nabla^{\\rm rec}) = \\{0\\}$.\n\\end{Theorem*}\n\nThe notion of {\\it isogeny} can be extended beyond the simply-transitive case. Let us consider a~complex Lie algebra $\\mathfrak g$. An {\\it infinitesimally homogeneous variety} of type $\\mathfrak g$ is a pair $(M,\\mathfrak s)$ consisting of a complex smooth irreducible variety $M$ and a f\\\/inite-dimensional Lie algebra \\smash{$\\mathfrak s \\subset \\mathfrak X(M)$} isomorphic to $\\mathfrak g$ that spans the tangent bundle of $M$ on the generic point.\n\nWe are interested in conjugation by rational or by algebraic maps, so that, whenever necessary, we replace $M$ by a suitable Zariski open subset. In this context, we say that a dominant rational map $f\\colon M_1 \\dasharrow M_2$ between varieties of the same dimension conjugates the inf\\\/initesimally homogeneous varieties $(M_1,\\mathfrak s_1)$ and $(M_2,\\mathfrak s_2)$ if $f^*(\\mathfrak s_2) = \\mathfrak s_1$. We say that $(M_1,\\mathfrak s_1)$ and $(M_2,\\mathfrak s_2)$ are {\\it isogenous} if they are conjugated to the same inf\\\/initesimally homogeneous space of type~$\\mathfrak g$.\n\nUnder some hypothesis on the Lie algebra $\\mathfrak s\\subset \\mathfrak X(M)$ one can prove that $(M,\\mathfrak s)$ is isogenous to a homogeneous space $(G\/H,\\mathfrak{lie}(G)^{\\rm rec})$ with the action of right invariant vector f\\\/ields. These hypothesis are satisf\\\/ied by transitive actions of $\\mathfrak{sl}_{n+1}(\\mathbf C)$ on $n$-dimensional varieties. As a particular case of Theorem~\\ref{homogeneous} one has\n\\begin{Theorem**}\nLet $(M,\\mathfrak s)$ be an infinitesimally homogeneous variety of complex dimension $n$ such that $\\mathfrak s$ is isomorphic to $\\mathfrak{sl}_{n+1}(\\mathbf C)$. Then there exists a dominant rational map $M \\dasharrow \\mathbf{CP}_n$ conjugating $\\mathfrak s$ with the Lie algebra $\\mathfrak{sl}_{n+1}(\\mathbf C)$ of projective vector fields in $\\mathbf{CP}_n$.\n\\end{Theorem**}\n\nAppendix~\\ref{ApA} is devoted to a geometrical presentation of Picard--Vessiot theory for linear and principal connections. Finally, Appendix~\\ref{apB} contains a detailed proof of Deligne's theorem of the realization of a regular parallelism modeled over any f\\\/inite-dimensional Lie algebra. This includes also a computation of the Galois group that turns out to be, for this particular construction, an algebraic torus.\n\n\\section{Parallelisms}\\label{section_parallelisms}\n\nLet $M$ be a smooth connected af\\\/f\\\/ine variety over $\\mathbf C$ of dimension~$r$. We denote by $\\mathbf C[M]$ its ring of regular functions and by $\\mathbf C(M)$ its f\\\/ield of rational functions. Analogously, we denote by~$\\mathfrak X[M]$ and~$\\mathfrak X(M)$ respectively the Lie algebras of regular and rational vector f\\\/ields in $M$, and so on.\n\nLet $\\mathfrak g$ be a Lie algebra of dimension $r$. We f\\\/ix a basis $A_1,\\ldots,A_r$ of $\\mathfrak g$, and the following notation for the associated structure constants $[A_i,A_j] = \\sum_{k}\\lambda_{ij}^kA_k$.\n\nA parallelism of type $\\mathfrak g$ of $M$ is a realization of the Lie algebra $\\mathfrak g$ as a Lie algebra of pointwise linearly independent vector f\\\/ields in $M$. More precisely:\n\n\\begin{Definition}A regular parallelism of type $\\mathfrak g$ in $M$ is a Lie algebra morphism, $\\rho\\colon \\mathfrak g \\to \\mathfrak X[M]$ such that $\\rho A_1(x), \\ldots, \\rho A_r(x)$ form a basis of $T_xM$ for any point $x$ of $M$.\n\\end{Definition}\n\n\\begin{Example}\\label{ex:AGP}Let $G$ be an algebraic group and $\\mathfrak g$ be its Lie algebra of left invariant vector f\\\/ields. Then the natural inclusion $\\mathfrak g\\subset \\mathfrak X[G]$ is a regular parallelism of~$G$. The Lie algebra $\\mathfrak g^{\\rm rec}$ of right invariant vector f\\\/ields is another regular parallelism of the same type. Let invariant and right invariant vector f\\\/ields commute, hence, an algebraic group is naturally endowed with a~pair of commuting parallelisms of the same type.\n\\end{Example}\n\nFrom Example \\ref{ex:AGP}, it is clear that any \\emph{algebraic} Lie algebra is realized as a parallelism of some algebraic variety. On the other hand, Theorem~\\ref{TDeligne} due to P.~Deligne and published in~\\cite{Malgrange}, ensures that any Lie algebra is realized as a regular parallelism of an algebraic variety. Analogously, we have the def\\\/initions of rational and local analytic parallelism. Note that a~rational parallelism in~$M$ is a regular parallelism in a Zariski open subset $M^\\star \\subseteq M$.\n\nThere is dual def\\\/inition, equivalent to that of parallelism. This is more suitable for calculations.\n\n\\begin{Definition}A regular parallelism form (or coparallelism) of type $\\mathfrak g$ in $M$ is a $\\mathfrak g$-valued\n$1$-form $\\omega\\in\\Omega^1[M]\\otimes_{\\mathbb C} \\mathfrak g$ such that:\n\\begin{itemize}\\itemsep=0pt\n\\item[(1)]For any $x\\in M$, $\\omega_x\\colon T_x M \\to \\mathfrak g$ is a linear isomorphism.\n\\item[(2)]If $A$ and $B$ are in $\\mathfrak g$ and $X$, $Y$ are vector f\\\/ields such that $\\omega(X) = A$ and $\\omega (Y) = B$ then $\\omega[X,Y] = [A,B]$.\n\\end{itemize}\n\\end{Definition}\n\nAnalogously, we def\\\/ine local analytic and rational coparallelism of type $\\mathfrak g$ in $M$. It is clear that each coparallelism induces a parallelism, and reciprocally, by the relation $\\omega (\\rho (A)) = A$. Thus, there is a natural equivalence between the notions of parallelism and coparallelism. From now on we f\\\/ix $\\rho$ and $\\omega$ equivalent parallelism and coparallelism of type $\\mathfrak g$ on $M$.\n\nThe Lie algebra structure of $\\mathfrak g$ forces $\\omega$ to satisfy Maurer--Cartan structure equations\n\\begin{gather*}{\\rm d}\\omega + \\frac{1}{2}[\\omega,\\omega] = 0.\\end{gather*}\nTaking components $\\omega = \\sum_{i}\\omega_iA_i$ we have\n\\begin{gather*}{\\rm d}\\omega_i +\\sum_{j,k=1}^r \\frac{1}{2}\\lambda_{jk}^i\\omega_j\\wedge \\omega_k = 0.\\end{gather*}\n\n\\begin{Example}\\label{ex:group} Let $G$ be an algebraic group and $\\mathfrak g$ be the Lie algebra of left invariant vector f\\\/ields in $G$. Then the structure form $\\omega$ is the coparallelism corresponding to the parallelism of Example~\\ref{ex:AGP}.\n\\end{Example}\n\n\\begin{Example}\\label{ex:B}Let $\\mathfrak g = \\langle A_1,A_2\\rangle$ be the 2-dimensional Lie algebra with commutation relation\n\\begin{gather*}[A_1, A_2]= A_1.\\end{gather*}\nThe vector f\\\/ields\n\\begin{gather*}X_1 = \\frac{\\partial}{\\partial x},\\qquad X_2 = x\\frac{\\partial}{\\partial x}+ \\frac{\\partial}{\\partial y},\\end{gather*}\ndef\\\/ine a regular parallelism via $\\rho (A_i)= X_i$ of $\\mathbf C^2$. The associated parallelism form is\n\\begin{gather*}\\omega = A_1{\\rm d}x + (A_2 - x A_1){\\rm d}y.\\end{gather*}\n\\end{Example}\n\n\\begin{Example}[Malgrange]\\label{ex:MD} Let $\\mathfrak g = \\langle A_1,A_2,A_3 \\rangle$ be the 3-dimensional Lie algebra with commutation relations\n\\begin{gather*}[A_1, A_2]= \\alpha A_2,\\qquad [A_1,A_3]=\\beta A_3, \\qquad [A_2,A_3] = 0,\\end{gather*}\nwith $\\alpha$, $\\beta$, non zero complex numbers. In particular, if $\\alpha\/\\beta$ is not rational then $\\mathfrak g$ is not the Lie algebra of an algebraic group. The vector f\\\/ields\n\\begin{gather*}X_1 = \\frac{\\partial}{\\partial x} + \\alpha y \\frac{\\partial}{\\partial y} + \\beta z\\frac{\\partial}{\\partial z},\\qquad\nX_2 = \\frac{\\partial}{\\partial y},\\qquad X_3 = \\frac{\\partial}{\\partial z},\\end{gather*}\ndef\\\/ine a regular parallelism via $\\rho (A_i)= X_i$ of $\\mathbf C^3$. The associated parallelism form is\n\\begin{gather*}\\omega = (A_1 - A_2\\alpha y - A_3\\beta z){\\rm d}x + A_2{\\rm d}y + A_3{\\rm d}z.\\end{gather*}\n\\end{Example}\n\n\\begin{Definition} \\label{isogenous} Let $(M,\\omega)$ and $(N,\\theta)$ be algebraic manifolds with coparallelisms of type~$\\mathfrak g$. We say that they are isogenous if there is an algebraic manifold $(P,\\eta)$ with a coparallelism of type~$\\mathfrak g$ and dominant maps $f\\colon P\\to M$ and $g\\colon P \\to N$ such that $f^*(\\omega) = g^*(\\theta) = \\eta$.\n\\end{Definition}\n\nClearly, the notion of isogeny of parallelized varieties extends that of isogeny of algebraic groups.\n\n\\begin{Example}\\label{ex:cover} Let $f\\colon M\\dasharrow G$ be a dominant rational map with values in an algebraic group with $\\dim_{\\mathbf{C}}M =\\dim_{\\mathbf{C}}G$. Then $\\theta =f^*(\\omega)$ is a rational parallelism form in $M$.\n\\end{Example}\n\n\\begin{Example}\\label{ex:finite}Let $H$ be a f\\\/inite subgroup of the algebraic group $G$ and\n\\begin{gather*}\\pi\\colon \\ G\\to M = H\\setminus G = \\{Hg \\colon g\\in G\\}\\end{gather*} be the quotient by the action of $H$ on the left side. The structure form $\\omega$ in $G$ is left-invariant and then it is projectable by $\\pi$. Then, $\\theta = \\pi_*(\\omega)$ is a regular parallelism form in~$M$.\n\\end{Example}\n\n\\begin{Example}\\label{ex:coverfinite} Combining Examples \\ref{ex:cover} and \\ref{ex:finite}, let $H\\subset G$ be a f\\\/inite subgroup and $f\\colon M\\to H \\setminus G$ be a dominant rational map between manifolds of the same dimension. Then $\\theta = f^*(\\pi_*(\\omega))$ is a rational parallelism form in $M$.\n\\end{Example}\n\n\\begin{Example}By application of Example \\ref{ex:coverfinite} to the case of the multiplicative group we obtain rational multiples of logarithmic forms in $\\mathbf{CP}_1$, $\\frac{p}{q}\\frac{{\\rm d}f}{f}$ where $f\\in\\mathbf C(z)$. Thus, rational multiples of logarithmic forms in $\\mathbf{CP}_1$ are the rational coparallelisms isogenous to that of the multiplicative group.\n\\end{Example}\n\n\\begin{Example}By application of Example \\ref{ex:coverfinite} to the case of the additive group we obtain the exact forms in $\\mathbf{CP}_1$, ${\\rm d}F$ where $F\\in\\mathbf C(z)$. Thus, the exact forms in $\\mathbf{CP}_1$ are the rational coparallelisms isogenous to that of the additive group.\n\\end{Example}\n\n\\begin{Example}\\label{ex:quotient_c}Let $H$ be a subgroup of the algebraic group $G$, with Lie algebra $\\mathfrak h\\subset \\mathfrak g$. Let us\nassume that $\\mathfrak h$ admits a supplementary Lie algebra $\\mathfrak h'$\n\\begin{gather*}\\mathfrak g = \\mathfrak h \\oplus \\mathfrak h' \\qquad \\mbox{(as vector spaces).}\\end{gather*}\nWe consider the left quotient $M= H \\setminus G$ of $G$ by the action of $H$ and the quotient map $\\pi\\colon G\\to M$. It turns out that $\\mathfrak h'$ is a Lie algebra of vector f\\\/ields in $G$ projectable by $\\pi$, and thus $\\pi_*|_{\\mathfrak h'} \\colon \\mathfrak h' \\to \\mathfrak X[M]$ gives a parallelism of $M$ that is regular in the open subset\n\\begin{gather*}\\{Hg\\in M \\colon \\operatorname{Adj}_g(\\mathfrak h) \\cap \\mathfrak h' = \\{0\\} \\}.\\end{gather*}\nIt turns out to be regular in $M$ if $H\\lhd G$. Examples~\\ref{ex:B} and~\\ref{ex:MD} are particular cases where $G$ is $\\operatorname{Af\\\/f}(2,\\mathbf C)$ and $\\operatorname{Af\\\/f}(3,\\mathbf C)$ respectively.\n\\end{Example}\n\n\\begin{Remark}We can see also Example~\\ref{ex:quotient_c} as a coparallelism. Let $\\pi'\\colon\\mathfrak g\\to \\mathfrak h'$ be the projection given by the vector space decomposition $\\mathfrak g = \\mathfrak h \\oplus \\mathfrak h'$. Since $\\pi'\\circ \\omega$ is left invariant form in $G$, it is projectable by~$\\pi$. Hence, there is a form $\\omega'$ in $M$ such that $\\pi^*\\omega' = \\pi'\\circ \\omega$. This form $\\omega'$ is the corresponding coparallelism.\n\\end{Remark}\n\n\\section{Associated Lie connection}\\label{section_lie}\n\n\\subsection{Reciprocal connections}\n\nLet $\\nabla$ be a linear connection (rational or regular) on $TM$. The reciprocal connection is def\\\/ined as\n\\begin{gather*}\\nabla^{\\rm rec}_{\\vec X}\\vec Y = \\nabla_{\\vec Y} \\vec X + \\big[\\vec X,\\vec Y\\big].\\end{gather*}\nFrom this def\\\/inition it is clear that the dif\\\/ference $\\nabla - \\nabla^{\\rm rec} = \\operatorname{Tor}_{\\nabla}$ is the torsion tensor, $\\operatorname{Tor}_{\\nabla} = -\\operatorname{Tor}_{\\nabla^{\\rm rec}}$ and $(\\nabla^{\\rm rec})^{\\rm rec} = \\nabla$.\n\n\\subsection{Connections and parallelisms}\n\nLet $\\omega$ be a coparallelism of type $\\mathfrak g$ in $M$ and $\\rho$ its equivalent parallelism. Denote by $\\vec X_i$ the basis of vector f\\\/ields in $M$ such that $\\omega(\\vec X_i)=A_i$ is a basis of $\\mathfrak g$.\n\n\\begin{Definition}The connection $\\nabla$ associated to the parallelism $\\omega$ is the only linear connection in $M$ for which $\\omega$ is a $\\nabla$-horizontal form.\n\\end{Definition}\n\nClearly $\\nabla$ is a f\\\/lat connection and the basis $\\{\\vec X_i\\}$ is a basis of the space of $\\nabla$-horizontal vector f\\\/ields. In this basis $\\nabla$ has\nvanishing Christof\\\/fel symbols\n\\begin{gather*}\\nabla_{\\vec X_i} \\vec X_j = 0.\\end{gather*}\n\nLet us compute some inf\\\/initesimal symmetries of $\\omega$. A vector f\\\/ield $\\vec Y$ is an inf\\\/initesimal symmetry of $\\omega$ if ${\\rm Lie}_{\\vec Y}\\omega = 0$, or equivalently, if it commutes with all the vector f\\\/ields of the parallelism\n\\begin{gather*}\\big[\\vec X_i, \\vec Y\\big] = 0, \\qquad i = 1,\\ldots, r.\\end{gather*}\n\n\\begin{Lemma}\\label{Lemma1}Let $\\nabla$ be the connection associated to the parallelism $\\omega$. Then for any vector field~$\\vec Y$ and any $j=1,\\ldots,r$\n\\begin{gather*}\\big[\\vec X_j, \\vec Y\\big] = \\nabla^{\\rm rec}_{\\vec X_j}{\\vec Y}.\\end{gather*}\nThus, $\\vec Y$ is an infinitesimal symmetry of $\\omega$ if and only if it is a~horizontal vector field for the reciprocal connection $\\nabla^{\\rm rec}$.\n\\end{Lemma}\n\n\\begin{proof}A direct computation yields the result. Take $\\vec Y = \\sum\\limits_{k=1}^r f_k\\vec X_k$, for each $j$ we have\n\\begin{gather*}\\nabla^{\\rm rec}_{\\vec X_j} \\vec Y = \\sum_{k=1}^r\\big( \\big(\\vec X_jf_k\\big)\\vec X_k + f_k\\big[\\vec X_j, \\vec X_k\\big]\\big) = \\big[\\vec X_j,\\vec Y\\big]. \\tag*{\\qed}\\end{gather*}\\renewcommand{\\qed}{}\n\\end{proof}\n\nThe above considerations also give us the Christof\\\/fel symbols for $\\nabla^{\\rm rec}$ in the basis $\\{\\vec X_i\\}$\n\\begin{gather*}\\nabla^{\\rm rec}_{\\vec X_i}\\vec X_j = \\big[\\vec X_i, \\vec X_j\\big] = \\sum_{k=1}^r \\lambda_{ij}^k \\vec X_k,\\end{gather*}\ni.e., the Christof\\\/fel symbols of $\\nabla^{\\rm rec}$ are the structure constants of the Lie algebra $\\mathfrak g$.\n\n\\begin{Lemma}\\label{Lemma2} Let $\\nabla$ be the connection associated to a coparallelism in $M$. Then, $\\nabla^{\\rm rec}$ is flat, and the Lie bracket of two $\\nabla^{\\rm rec}$-horizontal vector fields is a $\\nabla^{\\rm rec}$-horizontal vector field.\n\\end{Lemma}\n\n\\begin{proof}The f\\\/latness and the preservation of the Lie bracket by $\\nabla^{\\rm rec}$ are direct consequences of the Jacobi identity. Let us compute the curvature\n\\begin{gather*}R\\big(\\vec X_i,\\vec X_j,\\vec X_k\\big) = \\nabla^{\\rm rec}_{\\vec X_i}\\big(\\nabla^{\\rm rec}_{\\vec X_j} X_k\\big)\n- \\nabla^{\\rm rec}_{\\vec X_j}\\big(\\nabla^{\\rm rec}_{\\vec X_i} \\vec X_k\\big)\n- \\nabla^{\\rm rec}_{[\\vec X_i,\\vec X_j]}\\vec X_k\\\\\n\\hphantom{R\\big(\\vec X_i,\\vec X_j,\\vec X_k\\big)}{} = \\rho ([A_i,[A_j,A_k]] - [A_j,[A_i, A_k]] - [[A_i,A_j],A_k]) = 0.\\end{gather*}\nLet us compute the Lie bracket for $\\vec Y$ and $\\vec Z$ $\\nabla^{\\rm rec}$-horizontal vector f\\\/ields\n\\begin{gather*}\\nabla^{\\rm rec}_{\\vec X_i}\\big[\\vec Y, \\vec Z\\big]\\! = \\big[\\vec X_i,\\big[\\vec Y, \\vec Z\\big]\\big]\\! =\n\\big[\\big[\\vec X_i, \\vec Y\\big], \\vec Z\\big]\\! + \\big[ \\vec Y, \\big[\\vec X_i, \\vec Z \\big]\\big]\\! =\n\\big[\\nabla^{\\rm rec}_{\\vec X_i}\\vec Y, \\vec Z\\big]\\! +\\big[\\vec Y, \\nabla^{\\rm rec}_{\\vec X_i}\\vec Z\\big]\\! = 0.\\!\\!\\!\\!\\!\\!\\tag*{\\qed}\\end{gather*}\\renewcommand{\\qed}{}\n\\end{proof}\n\n\\begin{Lemma}\\label{Lemma3} Let $x\\in M$ be a regular point of the parallelism form $\\omega$. The space of germs at~$x$ of horizontal vector fields for $\\nabla^{\\rm rec}$ is a Lie algebra isomorphic to $\\mathfrak g$. Moreover, let $\\vec Y_1,\\ldots, \\vec Y_r$ be horizontal vector fields with initial conditions $\\vec Y_i(x) = \\vec X_i(x)$, then $\\big[\\vec Y_i, \\vec Y_j\\big] = - \\sum\\limits_{k=1}^r \\lambda_{ij}^k \\vec Y_k$, where the $\\lambda_{i,j}$ are the structure constants of the Lie algebra generated by the $\\vec X_i$.\n\\end{Lemma}\n\n\\begin{proof} We can write the vector f\\\/ields $\\vec Y_i$ as linear combinations of the vector f\\\/ields $\\vec X_i$: $\\vec Y_i = \\sum\\limits_{j=1}^r a_{ji}\\vec X_j$. The matrix $(a_{ij})$ satisf\\\/ies the dif\\\/ferential equation\n\\begin{gather*}\\vec X_{k}a_{ij} = - \\sum_{\\alpha = 1}^r \\lambda_{k \\alpha}^i a_{\\alpha j}, \\qquad a_{ij}(x) = \\delta_{ij}.\\end{gather*}\nOn the other hand, we have $\\big[\\vec Y_i, \\vec Y_j\\big](x) = \\sum\\limits_{k=1}^r \\hat\\lambda_{ij}^k \\vec Y_k (x)$, for certain unknown structure cons\\-tants~$\\hat\\lambda_{ij}^k$. Let us check that $\\hat\\lambda_{ij}^k = \\lambda_{ji}^k = -\\lambda_{ij}^k$,\n\\begin{gather*}\\big[\\vec Y_i,\\vec Y_j\\big] = \\left[\\sum_{\\alpha=1}^r a_{\\alpha i} \\vec X_\\alpha, \\sum_{\\beta = 1}^r a_{\\beta j} \\vec X_{\\beta} \\right] \\\\\n\\hphantom{\\big[\\vec Y_i,\\vec Y_j\\big]}{} = \\sum_{\\alpha, \\beta, \\gamma =1}^r -a_{\\alpha i}\\lambda_{\\alpha \\gamma}^\\beta a_{\\gamma j}\\vec X_\\beta\n+ \\sum_{\\alpha, \\beta, \\gamma =1}^r a_{\\beta j}\\lambda_{\\beta\\gamma}^\\alpha a_{\\gamma i} \\vec X_{\\alpha}\n+ \\sum_{\\alpha, \\beta, \\gamma =1}^r a_{\\beta j} a_{\\alpha i} \\lambda_{\\alpha \\beta}^\\gamma \\vec X_\\gamma.\\end{gather*}\nTaking values at $x$, we obtain\n\\begin{gather*}\\big[\\vec Y_i, \\vec Y_j\\big](x) = \\sum_{\\beta =1}^r - \\lambda_{i j}^\\beta \\vec Y_\\beta(x)\n+ \\sum_{\\alpha =1}^r \\lambda_{j i}^\\alpha \\vec Y_{\\alpha}(x) + \\sum_{\\gamma =1}^r \\lambda_{i j}^\\gamma \\vec Y_\\gamma(x) =\n\\sum_{\\alpha =1}^r \\lambda_{j i}^k \\vec Y_{k}(x).\\tag*{\\qed}\\end{gather*}\\renewcommand{\\qed}{}\n\\end{proof}\n\n\\begin{Example}Let $G$ be an algebraic group with Lie algebra $\\mathfrak g$. As seen in Example~\\ref{ex:group} the Maurer--Cartan structure form $\\omega$ is a coparallelism in~$G$. Let $\\nabla$ be the connection associated to this coparallelism. There is another canonical coparallelism, the right invariant Maurer--Cartan structure form $\\omega_{\\rm rec}$, let us consider ${\\bf i}\\colon G\\to G$ the inversion map,\n\\begin{gather*}\\omega_{\\rm rec} = - {\\bf i}^*(\\omega).\\end{gather*}\nAs may be expected, the connection associated to the coparallelism $\\omega_{\\rm rec}$ is $\\nabla^{\\rm rec}$. Right invariant vector f\\\/ields in $G$ are inf\\\/initesimal symmetries of left invariant vector f\\\/ields and vice versa. In this case, the horizontal vector f\\\/ields of $\\nabla$ and $\\nabla^{\\rm rec}$ are regular vector f\\\/ields.\n\\end{Example}\n\nAs shown in the next three examples, symmetries of a rational parallelism are not in general rational vector f\\\/ields.\n\n\\begin{Example}Let us consider the Lie algebra $\\mathfrak g$ and the coparallelism $\\omega = A_1{\\rm d}x + (A_2 - x A_1){\\rm d}y$, of Example~\\ref{ex:B}. Let $\\nabla$ be its associated connection. In cartesian coordinates, the only non-vanishing Christof\\\/fel symbol of the reciprocal connection is $\\Gamma_{21}^1 = -1$. A basis of $\\nabla^{\\rm rec}$-horizontal vector f\\\/ields is\n\\begin{gather*}\\vec Y_1 = e^y\\frac{\\partial}{\\partial x}, \\qquad \\vec Y_2 = \\frac{\\partial}{\\partial y}.\\end{gather*}\nNote that they coincide with $\\vec X_1$, $\\vec X_2$ at the origin point and $\\big[\\vec Y_1,\\vec Y_2\\big] = - Y_1$.\n\\end{Example}\n\n\\begin{Example} Let us consider the Lie algebra $\\mathfrak g$ and the coparallelism $\\omega = (A_1 - \\alpha y A_2 - \\beta z A_3){\\rm d}x + A_2{\\rm d}y + A_3{\\rm d}z$ of Example~\\ref{ex:MD}. Let $\\nabla$ be its associated connection. In cartesian coordinates, the only non-vanishing Christof\\\/fel symbols of the reciprocal connection are\n\\begin{gather*}\\Gamma_{11}^2 = -\\alpha,\\qquad \\Gamma_{11}^3 = -\\beta.\\end{gather*}\nA basis of $\\nabla^{\\rm rec}$-horizontal vector f\\\/ields is\n\\begin{gather*}\\vec Y_1 = \\frac{\\partial}{\\partial x}, \\qquad \\vec Y_2 = e^{\\alpha x}\\frac{\\partial}{\\partial y},\\qquad \\vec Y_3 = e^{\\beta x}\\frac{\\partial}{\\partial z}.\\end{gather*}\nNote that they coincide with $\\vec X_1$, $\\vec X_2$, $\\vec X_3$ at the origin point and\n\\begin{gather*}\\big[\\vec Y_1,\\vec Y_2\\big] = - \\alpha Y_2,\\qquad \\big[\\vec Y_1,\\vec Y_3\\big] = -\\beta \\vec Y_3.\\end{gather*}\n\\end{Example}\n\n\\begin{Example}Let us consider the Lie algebra $\\mathfrak g$ of Example~\\ref{ex:MD} and the coparallelism\n\\begin{gather*}\\omega = (A_1 - \\alpha y A_2 - \\beta z A_3)\\frac{{\\rm d}x}{x} + A_2{\\rm d}y + A_3{\\rm d}z.\\end{gather*}\nLet $\\nabla$ be its associated connection. In cartesian coordinates, the only non-vanishing Christof\\\/fel symbols of the reciprocal connection are\n\\begin{gather*}\\Gamma_{11}^2 = -\\alpha,\\qquad \\Gamma_{11}^3 = -\\beta.\\end{gather*}\nA basis of $\\nabla^{\\rm rec}$-horizontal vector f\\\/ields on a simply connected open subspace $U\\subset \\mathbb C^\\ast \\times \\mathbb C^2$ is\n\\begin{gather*}\\vec Y_1 = x\\frac{\\partial}{\\partial x}, \\qquad \\vec Y_2 = x^{\\alpha}\\frac{\\partial}{\\partial y}, \\qquad \\vec Y_3 = x^{\\beta}\\frac{\\partial}{\\partial z}.\\end{gather*}\n\\end{Example}\n\n\\subsection{Lie connections}\n\nThe connections $\\nabla$ and $\\nabla^{\\rm rec}$ associated to a coparallelism $\\omega$ of type $\\mathfrak g$ are particular cases of the following def\\\/inition.\n\n\\begin{Definition} A Lie connection (regular or rational) in $M$ is a f\\\/lat connection $\\nabla$ in $TM$ such that the Lie bracket of any two horizontal vector f\\\/ields is a horizontal vector f\\\/ield.\n\\end{Definition}\n\nGiven a Lie connection $\\nabla$ in $M$, there is a $r$-dimensional Lie algebra $\\mathfrak g$ such that the space of germs of horizontal vector f\\\/ields at a regular point $x$ is a~Lie algebra isomorphic to $\\mathfrak g$. We will say that $\\nabla$ is a Lie connection of type~$\\mathfrak g$. The following result gives several algebraic characterizations of Lie connections:\n\n\\begin{Proposition}\\label{prop_Lie_char} Let $\\nabla$ be a linear connection in $TM$, the following statements are equivalent:\n\\begin{itemize}\\itemsep=0pt\n\\item[$(1)$] $\\nabla$ is a Lie connection;\n\\item[$(2)$] $\\nabla^{\\rm rec}$ is a Lie connection;\n\\item[$(3)$] $\\nabla$ is flat and has constant torsion, $\\nabla \\operatorname{Tor}_{\\nabla} = 0$;\n\\item[$(4)$] $\\nabla$ and $\\nabla^{\\rm rec}$ are flat.\n\\end{itemize}\n\\end{Proposition}\n\n\\begin{proof} Let us f\\\/irst see (1)$\\Leftrightarrow$(2). Let $\\nabla$ be a Lie connection. Around each point of the domain of $\\nabla$ there is a parallelism, by possibly transcendental vector f\\\/ields, such that $\\nabla$ is its associated connection. Then, Lemma~\\ref{Lemma2} states (1)$\\Rightarrow$(2). Taking into account that $(\\nabla^{\\rm rec})^{\\rm rec} = \\nabla$ we have the desired equivalence.\n\nLet us see now that (1)$\\Leftrightarrow$(3). Let us assume that $\\nabla$ is a f\\\/lat connection. For any three vector f\\\/ields $X$, $Y$, $Z$ in $M$ we have\n\\begin{gather*}(\\nabla_X \\operatorname{Tor}_{\\nabla})(Y,Z) = - \\operatorname{Tor}_{\\nabla}(\\nabla_X Y,Z ) - \\operatorname{Tor}_{\\nabla}(Y,\\nabla_X Z) + \\nabla_X \\operatorname{Tor}_{\\nabla}(Y,Z).\n\\end{gather*}\nLet us assume that $Y$ and $Z$ are $\\nabla$-horizontal vector f\\\/ields. Then, we have\n\\begin{gather*} \\operatorname{Tor}_{\\nabla}(Y,Z) = \\nabla_Y Z - \\nabla_Z Y - [Y,Z] = - [Y,Z]\\end{gather*}\nand the previous equality yields\n\\begin{gather*}(\\nabla_X \\operatorname{Tor}_{\\nabla})(Y,Z) = - \\nabla_X[Y,Z].\\end{gather*}\nThus, we have that $\\nabla \\operatorname{Tor}_{\\nabla}$ vanishes if and only if the Lie bracket of any two $\\nabla$-horizontal vector f\\\/ields is also $\\nabla$-horizontal. This proves (1)$\\Leftrightarrow$(3).\n\nFinally, let us see (1)$\\Leftrightarrow$(4). It is clear that (1) implies (4) so we only need to see (4)$\\Rightarrow$(1). Assume $\\nabla$ and $\\nabla^{\\rm rec}$ are f\\\/lat. Then, locally, there exist a basis $\\{\\vec X_i\\}$ of $\\nabla$-horizontal vector f\\\/ields and a~basis $\\{\\vec Y_i\\}$ of $\\nabla^{\\rm rec}$-horizontal vector f\\\/ields. By the def\\\/inition of the reciprocal connection, we have that a vector f\\\/ield $\\vec X$ is $\\nabla$-horizontal if and only if it satisf\\\/ies $[\\vec X, \\vec Y_i]=0$ for $i=1,\\ldots,r$. By the Jacobi identity we have\n\\begin{gather*}\\big[\\big[\\vec X_i,\\vec X_j\\big],\\vec Y_k\\big]= 0.\\end{gather*}\nThe Lie brackets $\\big[\\vec X_i,\\vec X_j\\big]$ are also $\\nabla$-horizontal and $\\nabla$ is a Lie connection.\n\\end{proof}\n\n\\begin{Lemma} Let $\\nabla$ be a Lie connection on $M$. Let $x$ be a regular point and $\\vec X_1,\\ldots, \\vec X_r$ and $\\vec Y_1,\\ldots, \\vec Y_r$ be basis of horizontal vector field germs on $M$ for $\\nabla$ and~$\\nabla^{\\rm rec}$ respectively with same initial conditions $\\vec X_i(x) = \\vec Y_i(x)$. Then\n\\begin{gather*}\\big[\\vec X_i,\\vec X_j\\big](x) = - \\big[\\vec Y_i,\\vec Y_j\\big](x).\\end{gather*}\nIt follows that $\\nabla$ and $\\nabla^{\\rm rec}$ are of the same type $\\mathfrak g$.\n\\end{Lemma}\n\n\\begin{proof} By def\\\/inition $\\nabla$ is the connection associated to the local analytic parallelism given by the basis $\\{\\vec X_i\\}$ of horizontal vector f\\\/ields. Then we apply Lemma~\\ref{Lemma3} in order to obtain the desired conclusion.\n\\end{proof}\n\n\\subsection{Some results on Lie connections by means of Picard--Vessiot theory}\n\nDef\\\/initions and general results concerning the Picard--Vessiot theory of connections are given in Appendix~\\ref{ApA}.\n\n\\begin{Proposition} Let $\\nabla$ be a rational Lie connection in $TM$. The $\\nabla$-horizontal vector fields are the symmetries of a rational parallelism of $M$ if and only if $\\operatorname{Gal}(\\nabla^{\\rm rec}) = \\{1\\}$.\n\\end{Proposition}\n\n\\begin{proof} We will use the notations of Section~\\ref{A6}: $R^1(TM)$ is the ${\\rm GL}_n(\\mathbb{C})$-principal bundle associated to $TM$ and $\\mathcal F'$ is the ${\\rm GL}_n(\\mathbb{C})$-invariant foliation on $R^1(TM)$ given by graphs of local basis of $\\nabla$-horizontal sections. The Galois group $\\operatorname{Gal}(\\nabla^{\\rm rec})$ can be computed as soon as we know the Zariski closure $\\overline{\\mathscr{L}}$ of a leaf $\\mathscr{L}$ of the induced foliation $\\mathcal F'$ on $R^1(TM)$. $\\operatorname{Gal}(\\nabla^{\\rm rec})$ is f\\\/inite is and only if $\\overline{\\mathscr{L}} = \\mathscr{L}$ and is $\\{1\\}$ if and only if $\\mathscr{L}$ is the graph of a rational section $M \\to R^1(TM)$. This means that there exists a basis of rational $\\nabla^{\\rm rec}$-horizontal sections. These sections give the desired parallelism.\n\\end{proof}\n\n\\begin{Proposition} For any Lie connection $\\nabla$, $\\operatorname{Gal}(\\nabla)\\subseteq \\operatorname{Aut}(\\mathfrak g)$.\n\\end{Proposition}\n\n\\begin{proof}Let us choose a point $x \\in M$ regular for $\\nabla$ and a basis $A_1,\\ldots, A_r$ of $\\mathfrak{g}$, i.e., a basis $Y_1,\\ldots, Y_r$ of local $\\nabla$-horizontal section of~$TM$ at~$x$.\n\nUsing notation of Section~\\ref{A6}, we will identify $R^1(T_xM)$ with the set of isomorphisms of linear spaces $ \\sigma\\colon \\mathfrak g \\to T_xM$; now $\\operatorname{Gal}(\\nabla)\\subseteq {\\rm GL}(\\mathfrak g)$. Because of the construction of $\\mathfrak g$, we have a~canonical point in~$R^1(TM)$ corresponding to the identity $ \\sigma_o\\colon \\mathfrak g \\to T_xM$.\n\nFor $m \\in M$, if $\\sigma$ is an isomorphism from $\\mathfrak g$ to $T_{m}M$ then one def\\\/ines $H^k_{{i,j}}(\\sigma)$ to be\n\\begin{gather*}\\frac{[X_{i},X_{j}] \\wedge X_{1}\\wedge \\cdots \\wedge \\widehat{X_{k}}\\wedge \\cdots \\wedge X_{r}}{X_{k} \\wedge X_{1}\\wedge \\cdots \\wedge \\widehat{X_{k}} \\wedge \\cdots \\wedge X_{r}} \\Big|_m, \\end{gather*} where $X_i$ is the horizontal section such that $X_i(m)= \\sigma A_i$. These functions are regular functions on~$R^1(TM)$. Moreover they are constant and equal to the constant structures on the Zariski closure of the leaf passing through~$\\sigma_o$. The Galois group is the stabilizer of this leaf then the functions~$H^k_{{i,j}}$ are invariant under the action of the Galois group, i.e., the Galois group preserves the Lie bracket.\n\\end{proof}\n\n\\begin{Proposition}\\label{prop_example} Let $\\mathfrak h'$ be a Lie sub-algebra of the Lie algebra of some algebraic group and let $G$ be the smallest algebraic subgroup such that ${\\mathfrak{lie}}(G) = \\mathfrak g \\supset \\mathfrak h'$. Assume the existence of an algebraic subgroup $H$ of $G$ whose Lie algebra $\\mathfrak h$ is supplementary to $\\mathfrak h'$ in $\\mathfrak g$, $\\mathfrak g = \\mathfrak h \\oplus \\mathfrak h'$. Let us consider the following objects:\n\\begin{itemize}\\itemsep=0pt\n\\item[$(a)$] the quotient map $\\pi\\colon G \\to M$ where $M$ is the variety of cosets $H\\setminus G$, and $\\nabla$ the Lie connection associated to\nthe parallelism $\\pi_*\\colon \\mathfrak h' \\to \\mathfrak X[M]$ in $M$ $($as given in Example~{\\rm \\ref{ex:quotient_c})};\n\\item[$(b)$] its reciprocal Lie connection $\\nabla^{\\rm rec}$ on~$M$;\n\\item[$(c)$] the Lie algebras of right invariant vector fields\n\\begin{gather*}\\mathfrak g^{\\rm rec} = {\\bf i}_*(\\mathfrak g), \\qquad \\mathfrak h'^{\\rm rec} = {\\bf i}_*(\\mathfrak h'),\\end{gather*}\nwhere ${\\bf i}$ is the inverse map on $G$.\n\\end{itemize}\nThen, the following statements are true:\n\\begin{itemize}\\itemsep=0pt\n\\item[$(i)$] $\\mathfrak h'$ is an ideal of $\\mathfrak g$ $($equivalently $\\mathfrak h'^{\\rm rec}$ is an ideal of $\\mathfrak g^{\\rm rec})$;\n\\item[$(ii)$] $\\mathfrak h$ is commutative $($equivalently $H$ is virtually abelian$)$;\n\\item[$(iii)$] the adjoint action of $G$ on $\\mathfrak g^{\\rm rec}$ preserves $\\mathfrak h'^{\\rm rec}$ and thus gives, by restriction, a morphism $\\overline{\\operatorname{Adj}}\\colon G \\to \\operatorname{Aut}(\\mathfrak h'^{\\rm rec})$;\n\\item[$(iv)$] The Galois group of the connection $\\nabla^{\\rm rec}$ is ${\\overline{\\operatorname{Adj}}}(H) \\subseteq \\operatorname{Aut}(\\mathfrak h'^{\\rm rec})$ and thus virtually abelian.\n\\end{itemize}\n\\end{Proposition}\n\n\\begin{proof}We have that $\\mathfrak g$ is the algebraic hull of $\\mathfrak h'$. From Lemma~\\ref{ap2_2} in Appendix~\\ref{apB} we obtain $[\\mathfrak g,\\mathfrak g] \\subseteq \\mathfrak h'$. Statement~(i) follows straightforwardly. Let us consider $A$ and $B$ in $\\mathfrak h$. Then $[A,B]$ is in $\\mathfrak h$ and also in $\\mathfrak h'$ by the previous argument. Thus, $[A,B]=0$ and this f\\\/inishes the proof of statement~(ii). Let us denote by $H'$ the subgroup of $G$ spanned by the image of $\\mathfrak h'$ by the exponential map. For each element $h\\in H'$, the adjoint action of $h$ preserves the Lie algebra~$\\mathfrak h'$. By continuity of the adjoint action in the Zariski topology, we have that~$\\mathfrak h'$ is preserved by the adjoint action of all elements of~$G$. This proves statement~(iii). In order to prove the last statement in the proposition we have to construct a Picard--Vessiot extension for the connec\\-tion~$\\nabla^{\\rm rec}$. Let us consider a basis $\\{A_1,\\ldots, A_m\\}$ of $\\mathfrak h'$ and let $\\bar A_i$ be the projection~$\\pi_*(A_i)$. We have an extension of dif\\\/ferential f\\\/ields\n\\begin{gather*}(\\mathbf C(M), \\bar{\\mathcal D}) \\subseteq (\\mathbf C(G), \\mathcal D),\\end{gather*}\nwhere $\\bar{\\mathcal D}$ stands for the $\\mathbf C(M)$-vector space of derivations spanned by $\\bar A_1,\\ldots, \\bar A_m$ and $\\mathcal D$ stands for the $\\mathbf C(G)$-vector space of derivations spanned by $A_1,\\ldots,A_m$ (see Appendix~\\ref{ApA} for our conventions on dif\\\/ferential f\\\/ields).\n\nThe projection $\\pi$ is a principal $H$-bundle. Any rational f\\\/irst integral of $\\{A_1,\\ldots,A_m\\}$ is constant along $H'$ and thus it is necessarily a complex number. Thus, the above extension has no new constants and it is strongly normal in the sense of Kolchin, with Galois group $H$. Note that the dif\\\/ferential f\\\/ield automorphism corresponding to an element $h\\in H$ is the pullback of functions by the left translation $L_{h}^{-1}$, that is, $(hf)(g) = f\\big(h^{-1}g\\big)$.\n\nThe horizontal sections for the connection $\\nabla^{\\rm rec}$ are characterized by the dif\\\/ferential equations\n\\begin{gather}\\label{eq_proof_H}\n[\\bar A_i, X] = 0.\n\\end{gather}\nLet us consider $\\{B_1,\\ldots,B_m\\}$ a basis of $\\mathfrak h'^{\\rm rec}$. From the Zariski closedness of $H$ in~$G$ it follows that there are regular functions $f_{ij}\\in\\mathbf C[G]$ such that $B_i = \\sum\\limits_{j=1}^m f_{ij} A_j$. Thus let us def\\\/ine $\\bar B_i = \\sum\\limits_{j=1}^m f_{ij} \\bar A_j$. Those objects are vector f\\\/ields in $M$ with coef\\\/f\\\/icients in $\\mathbf C[G]$, and clearly satisfy equation~\\eqref{eq_proof_H}. Thus, the Picard--Vessiot extension of $\\nabla^{\\rm rec}$ is spanned by the functions~$f_{ij}$ and it is embedded, as a dif\\\/ferential f\\\/ield, in~$\\mathbf C(G)$. Let us denote such extension by $\\mathbf L$. We have a~chain of extensions\n\\begin{gather*}\\mathbf C(M) \\subseteq \\mathbf L \\subseteq \\mathbf C(G).\\end{gather*}\nBy Galois correspondence, the Galois group of $\\nabla^{\\rm rec}$ is a quotient $H\/K$ where $K$ is the subgroup of elements of $H$ that f\\\/ix, by left translation, the functions $f_{ij}$. In order to prove statement~(iv) we need to check that this group $K$ is the kernel of the morphism $\\overline{\\operatorname{Adj}}$.\n\nLet us note that the image under the adjoint action by $g\\in G$ of an element $B\\in \\mathfrak h'^{\\rm rec}$ is given by the left translation, $\\overline{\\operatorname{Adj}}(g)(B) = L_{g*}(B)$. This transformation makes sense for any derivation of $\\mathbf C[G]$, and thus we have an action of~$G$ on $\\mathfrak X(G)$. Let us take $h$ in the kernel of $\\overline{\\operatorname{Adj}}$, thus, $\\overline{\\operatorname{Adj}}(h)(B_j) = B_j$ for any index~$j$. Applying the transformation $L_{h*}$ to the expression of~$B_i$ as linear combination of the left invariant vector f\\\/ields~$A_j$ we obtain $B_i = \\sum\\limits_{j=1}^m L_{h*}(f_{ij} A_j) = \\sum\\limits_{j=1}^m h(f_{ij})A_j$. The coef\\\/f\\\/icients of~$B_i$ as linear combination of the $A_j$ are unique, and thus, $h(f_{ij}) = f_{ij}$ we conclude that~$h$ is an automorphism f\\\/ixing $\\mathbf L$. On the other hand, let us take $h\\in H$ f\\\/ixing $\\mathbf L$. Then $L_{h*}\\big(\\sum f_{ij}A_j\\big) = \\sum f_{ij} A_j$ thus $\\overline{\\operatorname{Adj}}(h)(B_i) = B_i$ and then $h$ is in the kernel of $\\overline{\\operatorname{Adj}}$.\n\\end{proof}\n\n\\subsection[Some examples of $\\mathfrak{sl}_2$-parallelisms]{Some examples of $\\boldsymbol{\\mathfrak{sl}_2}$-parallelisms}\\label{sl2}\n\nWe will construct some parallelized varieties as subvarieties of the arc space of the af\\\/f\\\/ine line~${\\mathbb A}^1_\\mathbf C$. This family of\nexamples show how to realize every subgroup of ${\\rm PSL}_2(\\mathbf C)$ as the Galois group of the reciprocal Lie connection.\n\n\\subsubsection{The arc space of the af\\\/f\\\/ine line and its Cartan 1-form} \\label{sl2parallelisms}\n\nIn our special case, the arc space of the af\\\/f\\\/ine line ${\\mathbb A}^1_\\mathbf C$ with af\\\/f\\\/ine coordinate $z$, is the space of all formal power series $\\widehat{z} = \\sum z^{(i)} \\frac{x^i}{i !}$. It will be denoted by ${\\mathscr{L}}$, its ring of regular functions is $\\mathbf C[ {\\mathscr{L}}] = \\mathbf C\\big[z^{(0)},z^{(1)}, z^{(2)},\\ldots \\big]$. For an open subset $U\\subset \\mathbf C$ one denotes by~${\\mathscr{L}}U$ the set of power series $\\widehat z$ with $z^{(0)} \\in U$.\n\nA biholomorphism $f\\colon U \\to V$ between open sets of $\\mathbf C$ can be lift to a biholomorphism $ {\\mathscr{L}}f\\colon {\\mathscr{L}}U \\to {\\mathscr{L}}V$ by composition $\\widehat{z} \\to f\\circ\\widehat{z}$.\n\nLet $\\widehat{\\mathfrak X}$ be the Lie algebra of formal vector f\\\/ields $\\mathbf C[[x]]\\frac{\\partial}{\\partial x}$. One can build a rational form $\\sigma\\colon T{\\mathscr{L}} \\to \\widehat{\\mathfrak X}$ in following way (see \\cite[Section~2]{guillemin-sternberg}). Let $v = \\sum a_i \\frac{\\partial}{\\partial z^{(i)}}$ be a tangent vector at the formal coordinate $\\widehat{p}$, i.e., an arc in the Zariski open subset $\\{z^{(1)} \\not = 0\\}$. The local coordinate $\\widehat{p}$ can be used to have formal coordinates $p_0,p_{1}, p_{2}, \\ldots $, on $\\mathscr{L}$ and $v$ can be written $v = \\sum b_i \\frac{\\partial}{\\partial p_{i}}$. The form~$\\sigma$ is def\\\/ined by $\\sigma(v) = \\sum b_i \\frac{x^i}{i !} \\frac{\\partial}{\\partial x} $. This form is rational and is an isomorphism between~$T_p {\\mathscr{L}}$ and $\\widehat{\\mathfrak X}$ satisfying $d\\sigma = - \\frac{1}{2}[\\sigma, \\sigma]$ and $({\\mathscr{L}}f)^\\ast \\sigma = \\sigma$ for any biholomorphism~$f$.\n\nThis means that $\\sigma$ provides an action of $\\widehat{\\mathfrak X}$ commuting with the lift of biholomorphisms. This form seems to be a~coparallelism but it is not compatible with the natural structure of pro-variety of~${\\mathscr{L}}$ and~$\\widehat{\\mathfrak X}$: $\\sigma^{-1}\\big(\\frac{\\partial}{\\partial x}\\big) = \\sum\\limits_{i \\geq 0} z^{(i+1)}\\frac{\\partial}{\\partial z^{(i)}}$ is a derivation of degree $+1$ with respect to the pro-variety structure of ${\\mathscr{L}}$. The total derivation above will be denoted by~$E_{-1}$. This gives a dif\\\/ferential structure to the ring~$\\mathbf C[ {\\mathscr{L}}]$.\n\n\\subsubsection{The parallelized varieties}\n\nLet $\\nu\\in \\mathbf C(z)$ be a rational function, $f$ be the rational function on the arc space given by the Schwarzian derivative\n\\begin{gather*} f\\big(z^{(0)},z^{(1)},z^{(2)},z^{(3)}\\big) = \\frac{z^{(3)}}{z^{(1)}}-\\frac{3}{2} \\left(\\frac{z^{(2)}}{z^{(1)}}\\right) ^2+ \\nu\\big(z^{(0)}\\big)\\big(z^{(1)}\\big)^2,\\end{gather*}\nand $I \\subset \\mathbf C[{\\mathscr L}]$ be the $E_{-1}$-invariant ideal generated by $p\\big(z^{(0)}\\big) {z^{(1)}}^2f\\big(z^{(0)},z^{(1)},z^{(2)},z^{(3)}\\big)$ where $p$ is a minimal denominator of~$\\nu$.\n\n\\begin{Lemma} The zero set $V$ of $I$ is a dimension $3$ subvariety of ${\\mathscr{L}}$ and $\\omega (TV) = {\\mathfrak{sl}}_2(\\mathbf C) \\subset \\widehat{\\mathfrak X}$. This provides a ${\\mathfrak{sl}}_2$-parallelism on~$V$.\n\\end{Lemma}\n\n\\begin{proof} One can compute explicitly this parallelism using $z^{(0)}$, $z^{(1)}$ and $z^{(2)}$ as \\'etale coordinates on a Zariski open subset of $V$. Let us f\\\/irst compute the $\\mathfrak{sl}_2$ action on ${\\mathscr{L}}$. The standard inclusion of $\\mathfrak{sl}_2$ in $\\widehat{\\mathfrak X}$ is given by $E_{-1} = \\frac{\\partial}{\\partial x}$, $E_0 = x\\frac{\\partial}{\\partial x}$ and $E_{1} = x^2\\frac{\\partial}{\\partial x}$. Their actions on ${\\mathscr{L}}$ are given by $E_{-1} = \\sum z^{(i+1)}\\frac{\\partial}{\\partial z^{(i)}}$, $E_0 = \\sum iz^{(i)}\\frac{\\partial}{\\partial z^{(i)}}$ and $E_1= \\sum i(i-1) z^{(i-1)}\\frac{\\partial}{\\partial z^{(i)}}$. The ideal $I$ is generated by the functions $E_{-1}^n\\cdot f$. By def\\\/inition $E_{-1}\\cdot f \\in I$, a direct computation gives that $E_0\\cdot f = 2f \\in I$, $E_1\\cdot f = 0 \\in I$. The relations in $\\mathfrak{sl}_{2}$ give that $E_{-1}\\cdot I \\subset I$, $E_0\\cdot I \\subset I$ and $E_1\\cdot I \\subset I$, i.e., the vector f\\\/ields $E_{-1}$, $E_0$ and $E_1$ are tangent to~$V$.\n\\end{proof}\n\nNow parameterizing $V$ by $z^{(0)}$, $z^{(1)}$ and $z^{(2)}$ one gets\n\\begin{gather*}\n E_{-1}|_{\\mathbf C ^3} = z^{(1)}\\frac{\\partial}{\\partial z^{(0)}} + z^{(2)}\\frac{\\partial}{\\partial z^{(1)}} + \\left(-\\nu\\big(z^{(0)}\\big)\\big(z^{(1)}\\big)^3 + \\frac{3}{2}\\frac{(z^{(2)})^2}{z^{(1)}}\\right) \\frac{\\partial}{\\partial z^{(2)}},\\nonumber\\\\\n E_0|_{\\mathbf C^3} = z^{(1)}\\frac{\\partial}{\\partial z^{(1)}} + 2 z^{(2)}\\frac{\\partial}{\\partial z^{(2)}}, \\qquad E_1|_{\\mathbf C^3} = 2 z^{(1)}\\frac{\\partial}{\\partial z^{(2)}}.\\label{parrallel}\n\\end{gather*}\nThey form a rational $\\mathfrak{sl}_{2}$-parallelism on $\\mathbf C^3$ depending on the choice of a rational function in one variable.\n\n\\subsubsection{Symmetries and the Galois group of the reciprocal connection}\n\n\\begin{Theorem}\\label{thm_SL2}Any algebraic subgroup of ${\\rm PSL}_{2}(\\mathbf C)$ can be realized as the Galois group of the reciprocal connection of a~parallelism of~$\\mathbf C^3$.\n\\end{Theorem}\n\n\\begin{proof}A direct computation shows that $z\\mapsto \\varphi(z)$ is an holomorphic function satisfying\n\\begin{gather*} \\frac{\\varphi'''}{\\varphi'}-\\frac{3}{2} \\left(\\frac{\\varphi''}{\\varphi'}\\right)^2 + \\nu(\\varphi)(\\varphi')^2 = \\nu(z),\n\\end{gather*}\nif and only if its prolongation ${\\mathscr{L}}\\varphi\\colon \\widehat{z} \\mapsto \\varphi (\\widehat{z})$ on the space ${\\mathscr{L}}$ preserves~$V$ and preserves each of the vector f\\\/ields~$E_{-1}$,~$E_0$ and~$E_1$.\n\nTaking inf\\\/initesimal generators of this pseudogroup, one gets for any local analytic solution of the linear equation\n\\begin{gather}\\label{lin}\na''' + 2 \\nu a' +\\nu 'a =0,\n\\end{gather}\na vector f\\\/ield $X = a(z)\\frac{\\partial}{\\partial z}$ whose prolongation on ${\\mathscr{L}}$ is\n\\begin{gather*}\n{\\mathscr{L}}X = a\\big(z^{(0)}\\big)\\frac{\\partial}{\\partial z^{(0)}} + a'\\big(z^{(0)}\\big)z^{(1)}\\frac{\\partial}{\\partial z^{(1)}} + \\big( a''\\big(z^{(0)}\\big)\\big(z^{(1)}\\big)^2 + a'\\big(z^{(0)}\\big) z^{(2)}\\big)\\frac{\\partial}{\\partial z^{(2)}} + \\cdots.\n\\end{gather*}\nThe equation (\\ref{lin}) ensures that ${\\mathscr{L}}{X}$ is tangent to $V$. The invariance of $\\sigma$, $({\\mathscr{L}}X)_{\\ast}\\sigma =0$, ensures that ${\\mathscr{L}}X$ commutes with the ${\\mathfrak{sl}}_{2}$-parallelism given above. This means that for any solution~$a$ of~(\\ref{lin}) the vector f\\\/ield\n\\begin{gather*} a\\big(z^{(0)}\\big)\\frac{\\partial}{\\partial z^{(0)}} + a'\\big(z^{(0)}\\big)z^{(1)}\\frac{\\partial}{\\partial z^{(1)}}+ \\big( a''\\big(z^{(0)}\\big)\\big(z^{(1)}\\big)^2 + a'\\big(z^{(0)}\\big) z^{(2)}\\big)\\frac{\\partial}{\\partial z^{(2)}}, \\end{gather*}\ncommutes with $E_{-1}|_{\\mathbf C ^3}$, $E_0|_{\\mathbf C ^3}$ and $E_{1}|_{\\mathbf C ^3}$.\n\nThen the linear dif\\\/ferential system of f\\\/lat section for the reciprocal connection reduces to the linear equation~(\\ref{lin}). This equation is the second symmetric power of $y'' = \\nu(z)y$. If $G \\subset {\\rm SL}_2(\\mathbf C)$ is the Galois group of $y'' = \\nu(z)y$ then the image of its second symmetric power representation $s^2\\colon G \\to \\operatorname{Sym}^2(\\mathbf C^2)$ is the Galois group of~(\\ref{lin}). The kernel of this representation is $\\{{\\rm Id}, -{\\rm Id}\\}$ then the Galois group of~(\\ref{lin}) is an algebraic subgroup of ${\\rm PSL}_{2}(\\mathbf C)$.\n\nLet us remark that, as it follows from its def\\\/inition, the Galois group of an equation contains the monodromy group. Moreover one can determine the monodromy group of classical dif\\\/ferential equations. Hypergeometric equations depend on three complex numbers $(a,b,c)$\n\\begin{gather*\nz(1-z) F'' + (c-(a+b+1)z)F' - abF = 0,\n\\end{gather*}\nand is equivalent to\n\\begin{gather*}y'' = \\nu(\\ell, n,m ; z) y,\\end{gather*}\nwith \\begin{gather*} \\nu(\\ell,m,n ;z) = \\frac{\\big(1-\\ell^2\\big)}{4z^2} +\\frac{1-m^2}{4(1-z)^2} +\\frac{1-\\ell^2-m^2+n^2}{4 z(1-z)},\\end{gather*}\nand\n\\begin{gather*}F = z^{-c\/2}(1-z)^{(c-a-b-1)\/2} y, \\qquad \\ell = 1-c,\\qquad m =c-a-b,\\qquad n=a-b.\\end{gather*}\nThese two equations have the same projectivized Galois group in ${\\rm PGL}_2(\\mathbf C)$. Any algebraic subgroup of ${\\rm PGL}_2(\\mathbf C)$ will be realized by an appropriate choice of $(a,b,c)$.\n\\subsubsection{The whole group}\n\nFor $a=b=1\/2$, $c = 1$, the hypergeometric equation is the Picard--Fuchs equation of Legendre family. Its monodromy group is $\\Gamma(2) \\subset {\\rm SL}_2(\\mathbf Z)$ and is Zariski dense in~${\\rm SL}_2(\\mathbf C)$.\n\n\\subsubsection{The triangular subgroups}\nFor $b=0$ and $a=-1$ one can compute a basis of solutions of the equation: $1$ and $\\int \\big( \\frac{1-z}{z} \\big)^c {\\rm d}z$. If $c$ is not rational, the Galois group is the group of invertible matrices $\\left[\\begin{smallmatrix} u & v\\\\ 0 & 1 \\end{smallmatrix}\\right]$. When $c$ is rational then $u$ must be a root of the unity of order the denominators of~$c$. When $c\\in \\mathbf Z$, the Galois group is the group of matrices $\\left[\\begin{smallmatrix} 1 & v\\\\ 0 & 1 \\end{smallmatrix}\\right]$.\n\nFor $b=0 $ and $c= a+1$ a basis of solution is given by $z^{-a}$ and $1$. Its Galois group is a~subgroup of the group of matrices $\\left[\\begin{smallmatrix} u& 0 \\\\ 0 & 1 \\end{smallmatrix}\\right]$. The parameter $a$ is rational if and only if it is a f\\\/inite subgroup.\n\n\\subsubsection{The dihedral subgroups}\n\nFor $c=1\/2$ and $a+b=0$, a basis of solution is given by $\\big(\\sqrt{z}+\\sqrt{1-z}\\big)^a$ and $\\big(\\sqrt{z}-\\sqrt{1-z}\\big)^a$. The monodromy group is a~dihedral group in ${\\rm GL}_2(\\mathbf C)$ whose quotients give dihedral subgroups of ${\\rm PGL}_2(\\mathbf C)$.\n\n\\subsubsection{The tetrahedral subgroup}\nThis group is the monodromy group of hypergeometric equation for $\\ell = 1\/3$, $m= 1\/2$ and $n=1\/3$. A basis of solution is given by \\begin{gather*}(z-1)^{-1\/12}\\Big(\\sqrt{3}\\big(z^{1\/3}+1\\big) \\pm 2\\sqrt{z^{2\/3} + z^{1\/3} +1} \\Big)^{1\/4}. \\end{gather*}\n\n\\subsubsection{The octahedral subgroup}\nThis group is the monodromy group of hypergeometric equation for $\\ell = 1\/2$, $m= 1\/3$ and $n=1\/4$. A basis of solution is given by\n\\begin{gather*}\n(z-1)^{-1\/24}\\Big[\\sqrt{3} \\big( \\big(\\sqrt{z}-1\\big)^{1\/3} +\\big(\\sqrt{z}+1\\big)^{1\/3}\\big)^{1\/3}\\\\\n\\qquad{} \\pm 2\\sqrt{\\big(\\sqrt{z}-1\\big)^{2\/3} + (z-1)^{1\/3} +\\big(\\sqrt{z}+1\\big)^{2\/3}}\\Big]^{1\/4}.\\\\\n\\end{gather*}\n\n\\subsubsection{The icosaedral subgroup}\nThis group is the monodromy group of hypergeometric equation for $\\ell = 1\/2$, $m= 1\/3$ and $n=1\/5$. As icosahedral group is not solvable, the solution space is not described using formulas as simple as in preceding examples.\n\\end{proof}\n\n\\section{Darboux--Cartan connections}\\label{section_DC}\n\n\\subsection{Connection of parallelism conjugations}\\label{DC conjugation}\nLet $\\omega$ be a rational coparallelism on $M$ of type $\\mathfrak g$ and $G$ an algebraic group with Lie algebra of left invariant vector f\\\/ields $\\mathfrak g$ and Maurer--Cartan form $\\theta$. Denote by $M^\\star$ the open subset of~$M$ in wich $\\omega$ is regular. We will study the contruction of conjugating maps between the paralle\\-lisms~$(M,\\omega)$ and $(G,\\theta)$.\n\nLet us consider the trivial principal bundle $\\pi\\colon P = G\\times M \\to M$. In this bundle we consider the action of $G$ by right translations $(g, x) * g' = (gg', x)$. Let $\\Theta$ be the $\\mathfrak g$-valued form $\\Theta = \\theta - \\omega$ in~$P$.\n\n\\begin{Definition}\\label{DCconnection} The kernel of $\\Theta$ is a rational f\\\/lat invariant connection on the principal bundle $\\pi\\colon P \\to M$. We call it the Darboux--Cartan connection of parallelism conjugations from~$(M,\\omega)$ to~$(G,\\theta)$.\n\\end{Definition}\n\nThe equation $\\Theta = 0$ def\\\/ines a foliation on $P$ transversal to the f\\\/ibers at regular points of~$\\omega$. The leaves of the foliation are the graphs of analytic parallelism conjugations from~$(M,\\omega)$ to~$(G,\\theta)$. By means of dif\\\/ferential Galois theory the Darboux--Cartan connection has a Galois group $\\operatorname{Gal}(\\Theta)$ with Lie algebra~$\\mathfrak{gal}(\\Theta)$. The following facts are direct consequences of the def\\\/inition of the Galois group:\n\\begin{itemize}\\itemsep=0pt\n\\item[(a)] there is a regular covering map $c\\colon (M^\\star, \\omega) \\to (U,\\theta)$ with $U$ an open subset of $G$, and $c^*(\\theta) = \\omega$ if and only if $\\operatorname{Gal}(\\Theta) = \\{1\\}$;\n\\item[(b)] there is a regular covering map $c\\colon (M^\\star, \\omega) \\to (U,q_*\\theta)$ with $U$ an open subset of~$G\/H$,~$H$ a group of f\\\/inite index, and $c^*(q_*\\theta|_U) = \\omega$ if and only if $\\mathfrak{gal}(\\Theta) = \\{0\\}$.\n\\end{itemize}\nIn any case, the necessary and suf\\\/f\\\/icient condition for $(M,\\omega)$ and $(G,\\theta)$ to be isogenous parallelized varieties is that $\\mathfrak{gal}(\\Theta) = \\{0\\}$.\n\n\\subsection{Darboux--Cartan connection and Picard--Vessiot}\n\nNote that the coparallelism $\\omega$ gives a rational trivialization of~$TM$ as the trivial bundle of f\\\/i\\-ber~$\\mathfrak g$. In $TM$ we have def\\\/ined the connection $\\nabla^{\\rm rec}$ whose horizontal vector f\\\/ields are the symmetries of the parallelism. On the other hand, $G$ acts in $\\mathfrak g$ by means of the adjoint action. The Cartan-Darboux connection induces then a connection~$\\nabla^{\\rm adj}$ in the associated trivial bundle $\\mathfrak g\\times M$ of f\\\/iber~$\\mathfrak g$.\n\n\\begin{Proposition}\\label{adj_conjugation}The map\n\\begin{gather*}\\tilde\\omega\\colon \\ \\big(TM, \\nabla^{\\rm rec}\\big) \\to \\big(\\mathfrak g \\times M, \\nabla^{\\rm adj}\\big), \\qquad X_x \\mapsto (\\omega_x(X_x), x)\\end{gather*}\nis a birational conjugation of the linear connections $\\nabla^{\\rm rec}$ and $\\nabla^{\\rm adj}$.\n\\end{Proposition}\n\n\\begin{proof}It is clear that the map $\\tilde\\omega$ is birational. Let us consider $\\{A_1,\\ldots,A_m\\}$ a basis of~$\\mathfrak g$. Let $\\rho\\colon \\mathfrak g\\to \\mathfrak X(M)$ be the parallelism associated to the parallelism $\\omega$ and let us def\\\/ine $X_i = \\rho(A_i)$. Then $\\{X_1,\\ldots, X_n\\}$ is a rational frame in $M$ and the map $\\tilde\\omega$ conjugates the vector f\\\/ield $X_i$ with the constant section $A_i$ of the trivial bundle of f\\\/iber~$\\mathfrak g$. By def\\\/inition of the reciprocal connection\n\\begin{gather*}\\nabla^{\\rm rec}_{X_i}X_j = [X_i,X_j].\\end{gather*}\nOn the other hand, by def\\\/inition of the adjoint action and application of the covariant derivative as in equation~\\eqref{covariant_associated}\nof Appendix~\\ref{ap7_associated} we obtain\n\\begin{gather*}\\nabla^{\\rm adj}_{X_i}A_j = [A_i,A_j].\\end{gather*}\nTherefore we have that $\\tilde\\omega$ is a rational morphism of linear connections that conjugates $\\nabla^{\\rm rec}$ with~$\\nabla^{\\rm adj}$.\n\\end{proof}\n\nThe following facts follow directly from Proposition~\\ref{adj_conjugation}, and basic properties of the Galois group.\n\n\\begin{Corollary}\\label{cor:rec} Let us consider the adjoint action $\\rm{ Adj}\\colon G \\to {\\rm GL}(\\mathfrak g)$ and its derivative ${\\rm adj}\\colon \\mathfrak{g}\\to \\operatorname{End}(\\mathfrak g)$. The following facts hold:\n\\begin{itemize}\\itemsep=0pt\n\\item[$(a)$] $\\operatorname{Gal}(\\nabla^{\\rm rec}) = \\operatorname{Adj}({\\rm Gal(\\Theta)})$;\n\\item[$(b)$] $\\mathfrak{gal}(\\nabla^{\\rm rec}) = {\\rm adj}(\\mathfrak{gal}(\\Theta))$;\n\\item[$(c)$] if $\\mathfrak g$ is centerless then $\\mathfrak{gal}(\\nabla^{\\rm rec})$ is isomorphic to $\\mathfrak{gal}(\\Theta)$;\n\\item[$(d)$] assume $\\mathfrak g$ is centerless, then the necessary and sufficient condition for $(M,\\omega)$ and $(G,\\theta)$ to be isogenous is that $\\mathfrak{gal}(\\nabla^{\\rm rec})=\\{0\\}$.\n\\end{itemize}\n\\end{Corollary}\n\n\\begin{proof}(a) and (b). First, by Proposition \\ref{adj_conjugation} we have that $\\operatorname{Gal}(\\nabla^{\\rm rec}) = \\operatorname{Gal}(\\nabla^{\\rm adj})$ and so $\\mathfrak{gal}(\\nabla^{\\rm rec}) = \\mathfrak{gal}(\\nabla^{\\rm adj})$. By def\\\/inition $\\nabla^{\\rm adj}$ is the associated connection induced by $\\Theta$ in the associated bundle $\\mathfrak g\\times M$. This trivial bundle is the associated bundle induced by the adjoint representation $\\operatorname{Adj}\\colon G\\to \\operatorname{End}(\\mathfrak g)$. Then, by Theorem~\\ref{associated galois}, we have $\\operatorname{Gal}(\\nabla^{\\rm rec}) = \\operatorname{Adj}({\\rm Gal(\\Theta)})$ and $\\operatorname{Gal}(\\nabla^{\\rm rec}) = \\operatorname{Adj}({\\rm Gal(\\Theta)})$.\n\n(c) It is a direct consequence of (b). The kernel of ${\\rm adj}\\colon \\mathfrak g \\to \\operatorname{End}(\\mathfrak g)$ is the center of~$\\mathfrak g$.\n\n(d) It follows from the def\\\/inition of Darboux--Cartan connection (see remarks after Def\\\/i\\-ni\\-tion~\\ref{DCconnection}) that the necessary and suf\\\/f\\\/icient condition for $(M,\\omega)$ and $(G,\\theta)$ to be isogenous is that $\\mathfrak{gal}(\\nabla^{\\rm rec})=\\{0\\}$. By point~(b) we conclude.\n\\end{proof}\n\n\\subsection{Algebraic Lie algebras}\n\nLet us consider $(M,\\omega)$ a rational coparallelism of type $\\mathfrak g$ with $\\mathfrak g$ a centerless Lie algebra. We do not assume \\emph{a priori} that $\\mathfrak g$ is an algebraic Lie algebra. The connection $\\nabla^{\\rm rec}$ is, as said in Proposition~\\ref{adj_conjugation}, conjugated to the connection in $\\mathfrak g\\times M$ induced by the adjoint action. Note that, in order to def\\\/ine this connection we do not need the group operation but just the Lie bracket in~$\\mathfrak g$. We have an exact sequence\n\\begin{gather*}0\\to \\mathfrak g' \\to \\mathfrak g \\to \\mathfrak g^{ab}\\to 0,\\end{gather*}\nwhere $\\mathfrak g'$ is the derived algebra $[\\mathfrak g, \\mathfrak g]$. Since the Galois group acts by adjoint action, we have that $\\mathfrak g'\\times M$ is stabilized by the connection $\\nabla^{\\rm rec}$ and thus we have an exact sequence of connections\n\\begin{gather*}0\\to (\\mathfrak g'\\times M, \\nabla')\\to \\big(\\mathfrak g \\times M,\\nabla^{\\rm rec}\\big)\\to \\big(\\mathfrak g^{ab}\\times M, \\nabla^{ab}\\big)\\to 0.\\end{gather*}\n\n\\begin{Lemma} The Galois group of $\\nabla^{ab}$ is the identity, therefore $\\nabla^{ab}$ has a basis of rational horizontal sections.\n\\end{Lemma}\n\n\\begin{proof} By def\\\/inition, the action of $\\mathfrak g$ in $\\mathfrak g^{ab}$ vanishes. Thus, the constant functions $M\\to \\mathfrak g^{ab}$ are rational horizontal sections.\n\\end{proof}\n\n\\begin{Lemma}\\label{th:algebraic} Let $\\omega$ be a rational coparallelism of $M$ of type $\\mathfrak g$ with $\\mathfrak g$ a centerless Lie algebra.\nIf $\\mathfrak{gal}(\\nabla^{\\rm rec}) = \\{0\\}$ then $\\mathfrak g$ is an algebraic Lie algebra.\n\\end{Lemma}\n\n\\begin{proof} Assume $\\mathfrak g$ is a linear Lie algebra and et $E$ be the smallest algebraic subgroup such that $ \\operatorname{Lie} (E) = \\mathfrak e\\supset \\mathfrak g$. We may assume that $E$ is also centerless. Let $A_{1}, \\ldots, A_{r}$ be a basis of $\\mathfrak g$, for $i=1,\\ldots,r$, $X_{i} = \\omega^{-1}(A_{i})$. Complete with $B_{1}, \\ldots, B_{p}$ in such way that $A_1,\\ldots,A_r,B_1,\\ldots,B_p$ is a basis of $\\mathfrak e$. We consider in $E \\times M$ the distribution spanned by the vector f\\\/ields $A_{i}+X_{i}$. This is a $E$-principal connection called~$\\nabla$.\n\nLet $\\overline{\\nabla}$ be the induced connection via the adjoint representation on $\\mathfrak e \\times M$ then\n\\begin{enumerate}\\itemsep=0pt\n\\item[1)] $\\overline{\\nabla}$ preserves $\\mathfrak g$ and $\\overline{\\nabla}|_{\\mathfrak g} = \\nabla^{\\rm rec}$, by hypothesis $\\mathfrak{gal}(\\overline{\\nabla}|_{\\mathfrak g}) = \\{0\\}$;\n\\item[2)] if $\\widetilde{\\nabla}$ is the quotient connection on ${\\mathfrak e}\/ \\mathfrak g$ then $\\mathfrak{gal}(\\widetilde{\\nabla}) =\\{0\\}$.\n\\end{enumerate}\nIf $\\varphi \\in \\mathfrak{gal}(\\overline{\\nabla})$ then for any $X \\in \\mathfrak g$, $[X,B_{i}] \\in \\mathfrak g$ thus $0 = \\varphi [X,B_{i}] = [ X, \\varphi B_{i}]$ and $\\varphi B_i$ commute with~$\\mathfrak g$. From the second point above $\\varphi B \\in \\mathfrak g$. By hypothesis $\\varphi B_i =0$ and $\\mathfrak{gal}(\\overline{\\nabla}) =\\{0\\}$. The projection on $E$ of an algebraic leaf of $\\nabla$ gives an algebraic leaf for the foliation of~$E$ by the left translation by~$\\mathfrak g$. This proves the lemma.\n\\end{proof}\n\n\\begin{Theorem}\\label{th_criteria} Let $\\mathfrak g$ be a centerless Lie algebra. An algebraic variety $(M,\\omega)$ with a rational parallelism of type~$\\mathfrak g$ is isogenous to an algebraic group if and only if $\\mathfrak{gal}(\\nabla^{\\rm rec}) = \\{0\\}$.\n\\end{Theorem}\n\n\\begin{proof} It follows directly from Lemma \\ref{th:algebraic} and Corollary \\ref{cor:rec}.\n\\end{proof}\n\n\\begin{Corollary}\\label{th:pair}Let $\\mathfrak g$ be a centerless Lie algebra. Any algebraic variety endowed with a pair of commuting rational parallelisms of type $\\mathfrak g$ is isogenous to an algebraic group endowed with its two canonical parallelisms of left and right invariant vector fields.\n\\end{Corollary}\n\n\\begin{proof}Just note that to have a pair of commuting parallelism is a more restrictive condition than having a parallelism with vanishing Lie algebra of the Galois group of its reciprocal connection.\n\\end{proof}\n\nThis result can be seen as an algebraic version of Wang result in \\cite{Wang}. It gives the classif\\\/ication of algebraic varieties endowed with pairs of commuting parallelisms. Assuming that the Lie algebra is centerless is not a superf\\\/luous hypothesis, note that the result clearly does not hold for abelian Lie algebras. There are rational $1$-forms in~$\\mathbf{CP}_1$ that are not exact (isogenous to $(\\mathbf C, {\\rm d}z)$) nor logarithmic (isogenous to~$(\\mathbf C^*,{\\rm d}\\log(z))$). In these examples, the pair of commuting parallelisms is given by twice the same parallelism.\n\n\\begin{Remark}Let $(M,\\omega,\\omega')$ be a manifold endowed with a pair of commuting parallelism forms of type $\\mathfrak g$, a centerless Lie algebra. From Lemma~\\ref{th:algebraic} we have that $\\mathfrak g$ is an~algebraic Lie algebra. We can construct the algebraic group enveloping $\\mathfrak g$ as follows. We consider the adjoint action\n\\begin{gather*}{\\rm adj}\\colon \\ \\mathfrak g\\hookrightarrow \\operatorname{End}(\\mathfrak g).\\end{gather*}\nThe algebraic group enveloping $\\mathfrak g$ is identif\\\/ied with the algebraic subgroup $G$ of $\\operatorname{Aut}(\\mathfrak g)$ whose Lie algebra is ${\\rm adj}(\\mathfrak g)$. From Corollary~\\ref{cor:rec}(a), we have that $\\operatorname{Gal}(\\Theta)=\\{e\\}$. Thus, there is a~rational map $f\\colon M\\to G$ such that $f^*(\\theta) = \\omega$, where~$\\theta$ is the Maurer--Cartan form of~$G$. We can express explicitly this map in terms of the commuting parallelism forms. For each $x\\in M$ in the domain of regularity of the parallelisms, $\\omega(x)$ and $\\omega'(x)$ are isomorphisms of~$T_xM$ with~$\\mathfrak g$. We def\\\/ine\n\\begin{gather*}f(x) = - \\omega(x)\\circ \\omega'(x)^{-1}.\\end{gather*}\n\\end{Remark}\n\n\\begin{Remark}In virtue of Corollary~\\ref{th:pair}, if $\\mathfrak g$ is a non-algebraic centerless Lie algebra, there is no algebraic variety endowed with a pair of regular commuting parallelisms of type $\\mathfrak g$. This limits the possible generalizations of Theorem~\\ref{TDeligne}.\n\\end{Remark}\n\n\\begin{Remark}B.~Malgrange has given in~\\cite{malgrange-P} another criterion: If $(M, \\omega)$ is a parallelized variety and $\\mathcal F$ is the foliation on $M \\times M$ given by $\\operatorname{pr}_1^\\ast \\omega - \\operatorname{pr}_2^\\ast \\omega = 0$. Then $(M,\\omega)$ is birational to an algebraic group if and only if leaves of~$\\mathcal F$ are graphs of rational maps. The relations with Theorem~\\ref{th_criteria} and Corollary~\\ref{th:pair} are the following. One can identify $TM$ with the vertical tangent (i.e., the kernel of~${\\rm d} \\operatorname{pr}_2$) along the diagonal in $M\\times M$. The diagonal is a leaf of $\\mathcal{F}$ and the linearization of $\\mathcal F$ along the diagonal def\\\/ines a connection $\\nabla_{\\mathcal F}$ on $TM$. By construction:\n\\begin{itemize}\\itemsep=0pt\n\\item $\\nabla_{\\mathcal F}$-horizontal sections commute with the parallelism, it is the reciprocal Lie connection;\n\\item if leaves of $\\mathcal F$ are algebraic then $\\nabla_{\\mathcal F}$-horizontal section are algebraic.\n\\end{itemize}\n\\end{Remark}\n\n\\section{Some homogeneous varieties}\n\nThe notion of {\\it isogeny} can be extended beyond the simply-transitive case. Let us consider a~complex Lie algebra $\\mathfrak g$. An {\\it infinitesimally homogeneous variety} of type $\\mathfrak g$ is a pair $(M,\\mathfrak s)$ consisting of a complex smooth irreducible variety $M$ and a f\\\/inite-dimensional Lie algebra \\smash{$\\mathfrak s \\subset \\mathfrak X(M)$} isomorphic to~$\\mathfrak g$.\n\nAs before, we are interested in conjugation by rational and algebraic maps so that, whenever necessary, we replace $M$ by a suitable Zariski open subset. In this context, we say that a~dominant rational map $f\\colon M_1 \\dasharrow M_2$ between varieties of the same dimension conjugates the inf\\\/initesimally homogeneous varieties $(M_1,\\mathfrak s_1)$ and $(M_2,\\mathfrak s_2)$ if $f^*(\\mathfrak s_2) = \\mathfrak s_1$. We say that $(M_1,\\mathfrak s_1)$ and $(M_2,\\mathfrak s_2)$ are {\\it isogenous} if they are conjugated to the same inf\\\/initesimally homogeneous space of type $\\mathfrak g$.\n\nLet $G$ be an algebraic group over $\\mathbf C$, $K$ an algebraic subgroup, $\\mathfrak{lie}(G)$ its Lie algebra of left invariant vector f\\\/ields and $\\mathfrak{lie}(G)^{\\rm rec}$ its Lie algebra of right invariant vector f\\\/ields. A natural example of inf\\\/initesimally homogeneous space are the homogeneous spaces $G\/H$ endowed with the induced action of the Lie algebra $\\mathfrak{lie}(G)^{\\rm rec}$. We want to recognize when a~inf\\\/initesimally homogeneous space is isogenous to an homogeneous space. We prove that if $\\mathfrak s \\subset \\mathfrak X(M)$ is a {\\it normal} Lie algebra of vector f\\\/ields then~$(M,\\mathfrak s)$ is isogenous to a homogeneous space. In particular, we prove that any $n$-dimensional inf\\\/initesimally homogeneous space of type $\\mathfrak{sl}_{n+1}(\\mathbf C)$ is isogenous to the projective space. Our answer is based on a generalization of the computations done in Section~\\ref{sl2}.\n\n\\subsection[The $\\mathfrak{sl}_2$ case]{The $\\boldsymbol{\\mathfrak{sl}_2}$ case}\n\n\\begin{Theorem}[Loray--Pereira--Touzet (private communication)] Let $\\mathscr{C}$ be a curve with~$X$,~$Y$,~$H$ three rational vector f\\\/ields such that $[X,Y] = H$, $[H,X] = -X$ and $[H,Y] = Y$. Then there exists a rational dominant map $h \\colon \\mathscr{C} \\dasharrow \\mathbf {CP}_1$ such that $X = h^\\ast\\big(\\frac{\\partial}{\\partial z}\\big)$, $H = h^\\ast \\big(z\\frac{\\partial}{\\partial z}\\big)$ and $Y = h^\\ast \\big(z^2\\frac{\\partial}{\\partial z}\\big)$.\n\\end{Theorem}\nTheir proof is elementary. We outline here a more sophisticated proof in the case $\\mathscr{C} = A^1_{\\mathbf C}$ that will be generalized in the next section.\n\\begin{proof} Notations are the ones introduced in Section~\\ref{sl2}. $\\mathscr{L}$ is the space of parameterized arcs $\\widehat{z} = \\sum_i z^{(i)} \\frac{x^i}{i!}$ on $\\mathscr{C}$. The vector space $\\mathbf C X + \\mathbf C H + \\mathbf C Y$ is denoted by $\\mathfrak g$. Let $r_o\\colon (\\mathbf C, 0) \\to A^1_\\mathbf C$ be an arc with $r_o'(0) \\not = 0$ and consider $V \\subset \\mathscr{L}$ def\\\/ined by\n\\begin{gather*}\nV =\\{ \\widehat{z} \\in \\mathscr{L} \\, | \\, \\widehat{z}^\\ast \\mathfrak g =r_o^\\ast \\mathfrak g \\}.\n\\end{gather*}\n\\begin{Claim}\nThis is a $3$-dimensional algebraic variety.\n\\end{Claim}\n\n\\begin{Claim}The prolongations $\\mathscr{L} X$, $\\mathscr{L} Y$ and $\\mathscr{L} H$ define a $\\mathfrak{sl}_2$-parallelism on $V$.\n\\end{Claim}\n\nLet us describe the canonical structure of $\\mathscr{L}$ (see \\cite[pp.~11--12]{guillemin-sternberg} or next section for a dif\\\/ferent presentation).\nFor $k$ an integer greater or equal to~$-1$, let us consider the vector f\\\/ield on~$\\mathscr{L}$\n\\begin{gather*}E_k = \\sum_{i\\geq k} \\frac{i !}{(i-k-1)!} z^{(i-k)}\\frac{\\partial}{\\partial z^{(i)}}.\\end{gather*}\nWe def\\\/ine a morphism of Lie algebra $\\rho\\colon \\widehat{\\mathfrak X} \\to \\mathfrak X(\\mathscr{L})$ by $x^{k+1}\\frac{\\partial}{\\partial x} \\mapsto E_k$ and the adic continuity.\n\n\\begin{Claim}The Cartan form $\\sigma$ $($as defined in Section~{\\rm \\ref{sl2parallelisms})} restricted to $V$ takes values in the Lie algebra $r_0^*(\\mathfrak g)$. It is the parallelism form reciprocal to the parallelism~$\\mathscr{L} X$,~$\\mathscr{L} H$ and~$\\mathscr{L} Y$ of~$V$.\n\\end{Claim}\n\nUsing Corollary \\ref{th:pair}, $V$ is isogeneous to ${\\rm PSL}_2(\\mathbf C)$ as def\\\/ined in Def\\\/inition~\\ref{isogenous}. For $p\\in M$, $V_p = \\{ \\widehat{z} \\in V \\ |\\ \\widehat{z}(0)=p\\}$ are homogeneous spaces for the action of $\\widetilde{K} = \\{\\varphi \\colon (\\mathbf C,0) \\to (\\mathbf C,0) \\, |\\, r_o\\circ \\varphi \\in V\\}$, i.e., $\\mathscr{C} = V\/\\widetilde{K}$. Let $K$ be the subgroup of ${\\rm PSL}_2(\\mathbf C)$ of upper triangular matrices.\n\n\\begin{Claim}The actions of $\\widetilde{K}$ on $V$ and the right action of $K$ on ${\\rm PSL}_2(\\mathbf C)$ are conjugated by the isogeny.\n\\end{Claim}\n\nThis induces an isogeny between $\\mathscr{C}$ and $\\mathbf {CP}_1$. Let $\\pi_1$ and $\\pi_2$ be the two maps of the isogeny. A local transformation $\\varphi$ such that $\\pi_1\\circ \\varphi = \\pi_1$ satisf\\\/ies $\\varphi^\\ast \\pi_1^\\ast(X,H,Y) = \\pi_1^\\ast(X,H,Y)$ and the same is true for the push-forward $(\\pi_2)_\\ast \\varphi$ of $\\varphi$ on $\\mathbf {CP}_1$. Then $(\\pi_2)_\\ast \\varphi$ preserves $\\frac{\\partial}{\\partial z}$ and $z\\frac{\\partial}{\\partial z}$. It is the identity. This f\\\/inishes the proof.\n\\end{proof}\n\n\\subsection{Some jet spaces}\\label{some_jet_spaces}\n\nLet $M$ be a $n$-dimensional af\\\/f\\\/ine variety. The space of parameterized subspaces of $M$ is the set of formal maps: $ M^{[n]} = \\{ r\\colon (\\mathbf C^n,0) \\to M \\}$. Like the arc space, it has a natural structure of pro-algebraic variety. We will give the construction of its coordinate ring following \\cite[Section~2.3.2, p.~80]{beilinson-drinfeld}. Let $\\mathbf C[\\partial_1, \\ldots, \\partial_n]$ be the $\\mathbf C$-vector space of linear partial dif\\\/ferential operators with constant coef\\\/f\\\/icients. The coordinate ring of $M^{[n]}$ is $\\operatorname{Sym}(\\mathbf C[M]\\otimes \\mathbf C[\\partial_1,\\ldots,\\partial_n]) \/ \\mathcal L$ where\n\\begin{itemize}\\itemsep=0pt\n \\item the tensor product is a tensor product of $\\mathbf C$-vector spaces;\n \\item $\\operatorname{Sym}( V )$ is the $\\mathbf C$-algebra generated by the vector space $V$;\n \\item $\\mathbf C[M]\\otimes \\mathbf C[\\partial_1,\\ldots, \\partial_n]$ has a structure of $\\mathbf C[\\partial_1,\\ldots,\\partial_n]$-module {\\it via}\n the right composition of dif\\\/ferential operators;\n \\item $\\operatorname{Sym}(\\mathbf C[M]\\otimes \\mathbf C[\\partial_1,\\ldots, \\partial_n])$ has the induced structure of $\\mathbf C[\\partial_1,\\ldots,\\partial_n]$-algebra;\n \\item the Leibniz ideal $\\mathcal L$ is the $\\mathbf C[\\partial_1,\\ldots, \\partial_n]$-ideal generated by $fg\\otimes 1 - (f\\otimes1)(g\\otimes1)$ for all $(f,g) \\in \\mathbf C[M]^2$ and by $1 - 1\\otimes 1$.\n \\end{itemize}\nLocal coordinates $(z_1, \\ldots, z_n)$ on $M$ induce local coordinates on $M^{[n]}$ {\\it via} the Taylor expansion of maps $r$ at $0$\n\\begin{gather*}\nr(x_1\\ldots, x_n) = \\left( \\sum_{\\alpha \\in \\mathbf N^n} r_1^{\\alpha} \\frac{x^\\alpha}{\\alpha!}, \\ldots,\\sum_{\\alpha \\in \\mathbf N^n} r_n^{\\alpha} \\frac{x^\\alpha}{\\alpha!} \\right).\n\\end{gather*}\nOne denotes by $z_i^{\\alpha}\\colon M^{[n]} \\to \\mathbf C$ the function def\\\/ined by $z_i^{\\alpha}(r) = r_i^{\\alpha}$. This function is the element $z_i\\otimes \\partial^\\alpha$ in~$\\mathbf C[M^{[n]}]$.\n\n\\subsubsection{Prolongation of vector f\\\/ields}\nAny derivation $Y$ of $\\mathbf C[M]$ can be trivially extended to a derivation of $\\operatorname{Sym}(\\mathbf C[M]\\otimes \\mathbf C[\\partial_1, \\ldots, \\partial_n])$. It preserves the ideal generated by $fg\\otimes 1 - (f\\otimes1)(g\\otimes1)$ for all $(f,g) \\in \\mathbf C[M]^2$ and by $1 - 1\\otimes 1$ and commutes with the action of $\\mathbf C[\\partial_1, \\ldots, \\partial_n]$ then it preserves the Leibniz ideal and def\\\/ines a~derivation of $\\mathbf C[M^{[n]}]$. This derivation is called the prolongation of $Y$, and it is denoted by~$Y^{[n]}$.\n\nThe same procedure can be used to def\\\/ine the prolongation of analytic or formal vector f\\\/ields on~$M$ to~$M^{[n]}$.\n\n\\subsubsection{The canonical structure}\\label{canonicalst}\n\nThe jet space $M^{[n]}$ is endowed with a dif\\\/ferential structure on its coordinate ring and with a group action by ``reparameterizations''. The compatibility condition between these two structures is well-known (see \\cite[pp.~11--23]{guillemin-sternberg}) and is easily obtained using the construction above.\n\nThe action of $\\partial_j\\colon \\mathbf C[M^{[n]}] \\to \\mathbf C[M^{[n]}]$ can be written in local coordinates and gives the total derivative operator $\\sum_{i,\\alpha} z_i^{\\alpha + 1_j} \\frac{\\partial}{\\partial z_i^{\\alpha}}$. It is the dif\\\/ferential structure of the jet space. The pro-algebraic group \\begin{gather*}\\Gamma = \\big\\{ \\gamma\\colon (\\mathbf C^n,0) \\overset{\\sim}{\\rightarrow} (\\mathbf C^n,0); \\text{ formal invertible}\\big\\}\\end{gather*}\n acts on $M^{[n]}$.This action is denoted by $S \\gamma (r) = r\\circ \\gamma$.\n\nThese two actions arise from the action of the Lie algebra $\\widehat{\\mathfrak{X}} = \\bigoplus \\mathbf C[[x_1,\\ldots,x_n]]\\partial_i$ on $M^{[n]}$. This action is described on the coordinate ring in the following way. For $\\xi \\in \\widehat{\\mathfrak{X}}$, $f\\in \\mathbf C[M]$ and $P \\in \\mathbf C[\\partial_1,\\ldots,\\partial_n]$, we def\\\/ine $\\xi \\cdot ( f \\otimes P) = f\\otimes (P\\circ \\xi)|_0$ where the composition is evaluated in~$0$ in order to get an element of $\\mathbf C[\\partial_1,\\ldots,\\partial_n]$. The action of $\\bigoplus \\mathbf C \\partial_i$ is the dif\\\/ferential structure. The action of $\\widehat{\\mathfrak{X}}^0 = \\mathfrak{lie}(\\Gamma)$, the Lie subalgebra of vector f\\\/ields vanishing at $0$ is the inf\\\/initesimal part of the action of $\\Gamma$.\n\n \\begin{Theorem}[\\cite{guillemin-sternberg}]\\label{canonique} Let $M^{[n]\\ast}$ be the open subset of submersions. The action above gives a~canonical form $\\sigma\\colon T M^{[n]\\ast} \\to \\widehat{\\mathfrak{X}}$ satisfying:\n\\begin{itemize}\\itemsep=0pt\n\\item for any $r \\in M^{[n]\\ast}$, $\\sigma$ is a isomorphism from $T_r M^{[n]\\ast}$ to $\\widehat{\\mathfrak{X}}$;\n\\item for any $\\gamma \\in \\Gamma$, $(S\\gamma)^\\ast \\sigma = \\gamma^\\ast \\circ \\sigma$;\n\\item $d\\sigma = -\\frac{1}{2}[\\sigma, \\sigma]$.\n\\end{itemize}\n \\end{Theorem}\n These equalities are {\\it not} compatible with the projective systems.\n\n\\subsection{Normal Lie algebras of vectors f\\\/ields}\n\nWithout lost of generality, we should\n\\begin{enumerate}\\itemsep=0pt\n\\item[1)] identify $\\mathfrak g$ with its image in $\\mathfrak X(M)$;\n\\item[2)] replace $M$ by a Zariski open subvariety on which $\\mathfrak g$ is def\\\/ined and of maximal rank at any point.\n\\end{enumerate}\nIf $p\\in M$ one can identify $\\mathfrak g$ with a Lie subalgebra of $\\widehat{\\mathfrak X}(M,p)$, the Lie algebra of formal vector f\\\/ields on~$M$ at~$p$.\n\n\\begin{Definition} For a Lie subalgebra $\\mathfrak g \\subset \\mathfrak X[M]$, its normalizer at $p\\in M$ is \\begin{gather*} \\widehat{N}(\\mathfrak g,p) = \\big\\{ Y \\in \\widehat{\\mathfrak X}(M,p) \\, |\\, Y,\\mathfrak g] \\subset \\mathfrak g\\big\\}.\\end{gather*}\n\\end{Definition}\n\n\\begin{Definition}A Lie subalgebra $\\mathfrak g \\subset \\mathfrak X[M]$ is said to be normal if for generic $p \\in M$ on has $ \\widehat{N}(\\mathfrak g,p) = \\mathfrak g$.\n\\end{Definition}\n\n\\begin{Lemma} If $\\mathfrak g$ is transitive then the Lie algebra $ \\widehat{N}(\\mathfrak g,p) $ is finite-dimensional.\n\\end{Lemma}\n\\begin{proof} Let $k$ be an integer large enough so that the only element of $\\mathfrak g$ vanishing at order~$k$ at~$p$ is~$0$. If $\\widehat{N}(\\mathfrak g,p) $ is not f\\\/inite-dimensional then there exists a non-zero $Y \\in \\widehat{N}(\\mathfrak g,p)$ vanishing at order $k+1$ at $p$. For $X \\in \\mathfrak g$, the Lie bracket $[Y,X]$ is an element of $\\mathfrak g$ vanishing at order~$k$ at~$p$. It is zero meaning that $Y$ is invariant under the f\\\/lows of vector f\\\/ields in~$\\mathfrak g$. The transitivity hypothesis together with $Y(p)=0$ proves the lemma.\n \\end{proof}\n\\begin{Lemma}If there exists a point $p\\in M$ such that $\\mathfrak g$ is maximal among finite-dimensional Lie subalgebra of~$\\widehat{\\mathfrak X}(M,p)$ then $\\mathfrak g$ is normal.\n\\end{Lemma}\n\\begin{proof}\nBecause of the preceding lemma, if such a point exists then $\\mathfrak g = \\widehat{N}(\\mathfrak g,p)$ in $\\widehat{\\mathfrak X}(M,p)$. By transitivity, for any couple of points $(p_1,p_2) \\in M^2$ there is a composition of f\\\/lows of elements of~$\\mathfrak g$ sending~$p_1$ on~$p_2$. These f\\\/lows preserve $\\mathfrak g$ thus the equality holds at any~$p$.\n\\end{proof}\n\n\\begin{Example} Let $M$ be $n$-dimensional and $\\mathfrak g$ be a transitive Lie subalgebra of rational vector f\\\/ields isomorphic to $\\mathfrak{sl}_{n+1}(\\mathbf C)$. Then $\\mathfrak g$ is normal (see~\\cite{cartan}).\n\\end{Example}\n\n\\subsection[Centerless, transitive and normal $\\Rightarrow$ isogenous to a homogeneous space]{Centerless, transitive and normal $\\boldsymbol{\\Rightarrow}$ isogenous to a homogeneous space}\n\n\\begin{Theorem}\\label{homogeneous} Let $M$ be a smooth irreducible algebraic variety over $\\mathbf C$ and $\\mathfrak g$ be a transitive, centerless, normal, finite-dimensional Lie subalgebra of $\\mathfrak X(M)$. Then there exists an algebraic group $G$, an algebraic subgroup $H \\subset G$ and an isogeny between $(M,\\mathfrak g)$ and $(G\/H, \\mathfrak{lie}(G))$. Moreover, if $N_G(\\mathfrak{lie}(H)) = H$ then the isogeny is a dominant rational map~$M \\dasharrow G\/H$.\n\\end{Theorem}\n\nBecause of the f\\\/initeness and the transitivity, there exists an integer~$k$ such that at any $p \\in M$ and for any $Y \\in \\widehat{N}(\\mathfrak g,p)$, $j_k(Y)(p) \\not = 0$, unless $Y=0$.\n\nLet $r_o\\colon (\\mathbf C^n,0) \\to M$ be an invertible formal map with $r_o(0) =p$ a regular point. Let us consider the subspace of $M^{[n]}$ def\\\/ined by\n\\begin{gather*}V = \\{ r \\colon (\\mathbf C^n,0) \\to M \\, |\\, r^\\ast \\mathfrak g = r_o^\\ast \\mathfrak g\\}.\\end{gather*}\n\n\\begin{Lemma} $V$ is finite-dimensional. \\end{Lemma}\n\\begin{proof} If $r_o^{-1}\\circ r$ is tangent to the identity at order $k$ then the induced automorphism of $\\mathfrak g$ is the identity. The map $r_o^{-1}\\circ r$ f\\\/ixes $p$, thus it is the identity. This proves the lemma.\n\\end{proof}\n\nUsing $r_o$ one can identify the Lie algebra $\\widehat{N}(\\mathfrak g,p)$ with a Lie subalgebra of $\\widehat{\\mathfrak{X}}$. The latter acts on $M^{[n]}$ as described in Section~\\ref{canonicalst}. As an application of the Theorem~\\ref{canonique}, one gets:\n\n\\begin{Lemma} The restriction of the canonical structure of $M^{[n]}$ gives an parallelism\n\\begin{gather*} TV = r_o^\\ast(\\widehat{N}(\\mathfrak g,p)) \\times V,\\end{gather*} called the canonical parallelism.\n\\end{Lemma}\n\n\\begin{Lemma}The horizontal sections of the reciprocal Lie connection of the canonical parallelism are~$Y^{[n]}$ for $Y \\in \\widehat{N}(\\mathfrak g,q)$ for $q\\in M$.\n\\end{Lemma}\n\n\\begin{Lemma}Under the hypothesis of normality of $\\mathfrak g$, $V$ has two commuting parallelisms of type~$\\mathfrak g$.\n\\end{Lemma}\n\nUsing Corollary \\ref{th:pair}, $\\mathfrak g$ is the Lie algebra of an algebraic group $G$ isogeneous to $V$. $V$ is foliated by the orbits of the subgroup $K$ of $\\Gamma$ stabilizing $V$. This group is algebraic with Lie algebra $\\mathfrak k = r_o^\\ast (\\mathfrak g) \\cap \\widehat{\\mathfrak{X}}^0$. Let $\\mathfrak h \\subset \\mathfrak{lie}(G)$ be the Lie algebra corresponding to $\\mathfrak k$ by the isogeny. Then the orbits of $\\mathfrak h$ are algebraic. This means that~$\\mathfrak h$ is the Lie algebra of an algebraic subgroup~$H$ of~$G$, and that~$V\/K$ and~$G\/H$ are isogenous.\n\nAssume that $N_{G}(\\mathfrak{lie}(H)) = H$. If $W$ is the isogeny between $V$ and $G$. The push-forward of a~local analytic deck transformation of $W \\to V$ is a transformation of~$G$ preserving each element of~$\\mathfrak g$, it is a right translation. A deck transformation preserves the orbits of the pull-back of~$\\mathfrak k$ on~$W$. Its push-forward preserves the orbits of a group containing~$H$ with the same Lie algebra. By hypothesis the push-forward is in~$H$ and then the isogony obtained by taking the quotient under~$K$ and~$H$ is the graph of a dominant rational map.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe martensitic transition in Heusler alloys is predominantly controlled by the addition of impurities resulting in numerous applications. Some of the fascinating properties which originate as a result of doping include large magnetocaloric effect \\cite{Krenke42005,Moya752007}, giant barocaloric effect \\cite{Manosa92010}, large magnetoresistance \\cite{Koyama892006,Sharma892006,Yu892006} and magnetic field induced strains \\cite{Kainuma4392006}. The transformation behaviour largely depends on the type and concentration of impurities. In Ni-Mn-Z (Z = In, Sn, Sb) Heusler alloys, structural transformation from cubic L$2_1$ austenite to lower symmetry martensite or vice-versa occurs when the dopant concentration exceeds a critical value. This is associated with the alteration of magnetic ground state from ferromagnetic to non-magnetic\/antiferromagntic \\cite{Krenke732006,Priolkar872013} leading to some interesting phenomena like kinetic arrest \\cite{Ito922008,Sharma762007}, exchange bias \\cite{Khan912007}, as well as emergence of non-ergodic states like magnetic cluster glass \\cite{Yadav92019,Nevgi322020}.\n\nRecent studies have also shown the occurrence of strain glass in Heusler alloys with impurity doping \\cite{Wang982012,Nevgi1122018}. Strain glass phase reportedly occurs in all ferroelastic\/martensitic alloys beyond a critical dopant concentration \\cite{Ren2012}. The presence of a defect phase like body centred Ni phase in Ni rich NiTi alloys, impede the long range ordering of the elastic strain vector leading to a frozen ferroelastic phase. On the other hand, no pre-transition phases are reported in Mn-rich Ni-Mn-In Heusler alloys. The ability of the Heusler structure to accommodate the stress caused by atomic size mismatch between Mn and In atoms is believed to be responsible for the absence of non-ergodic phases \\cite{Nevgi322020}. Doping of transition metal like Fe in the magnetic shape memory alloys is known to suppress martensitic transition. In some cases, Fe addition results in an emergence of structural impurity phases \\cite{Chen1012012,Tan72017,Zhang2016481,Lobo2014116}. Site selectivity of the dopant atom appears to play a key role in the ground state of the resultant alloy. It is observed that doping Fe at the expense of Mn to realize Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ leads to suppression of the martensitic state. At a critical concentration of $x = 0.1$, despite the average structure being martensitic, strain glassy phase appears below the glass transition temperature $T_g$ = 350 K \\cite{Nevgi1122018}. On the other hand when Fe is doped to replace Ni, to realize Ni$_{2-y}$Fe$_y$Mn$_{1.5}$In$_{0.5}$, the ferromagnetic interactions are enhanced with a complete suppression of the martensitic state at $y = 0.2$ \\cite{Nevgi7972019}.\n\nTherefore, it is essential to understand the role of the dopant in the occurrence of strain glassy phase, in particular the nature of the structural defect created, that leads to the formation of the strain glass. It is also equally important to understand the apparent site selectivity of the dopant atom that results in a cubic ferromagnetic ground state. To that effect, we have studied, in detail, the structural and magnetic properties of Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ ($0 \\le x \\le 0.2$) and Ni$_{2-y}$Fe$_y$Mn$_{1.5}$In$_{0.5}$ ($0 \\le y \\le 0.2$). The results suggest that in Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ where the X sublattice of X$_2$YZ Heusler is fully occupied, dopant Fe segregates into an impurity phase leading to a non-ergodic ground state. Whereas, when Fe is sought to be doped into the X sublattice to achieve Ni$_{2-y}$Fe$_y$Mn$_{1.5}$In$_{0.5}$, it promotes an A2 type antisite disorder by itself occupying Y\/Z sublattice and facilitates a cubic ferromagnetic ground state.\n\n\n\\section{Experimental}\n\nThe alloys were prepared by arc melting stoichiometric amounts of high purity elements ($>$ 99.9\\%)in an argon atmosphere. The homogeneity of the ingots was ensured by flipping the individual ingot a few times during the preparation process. The ingots were cut, powdered, covered in tantalum foil, and were vacuum sealed in quartz tubes to be annealed at 750 $^\\circ$C for 48 hours and eventually quenched in ice cold water. The alloy compositions were verified using scanning electron microscopy with energy dispersive x-ray (SEM-EDX) analysis and were found to be within $\\pm$2\\% of the stoichiometric values. Room temperature x-ray diffraction measurements were performed on the powdered alloys using Mo $K_\\alpha$ radiation in the angular range of 10$^\\circ$ to 50$^\\circ$. Synchrotron x-ray diffraction measurements were carried out on BL-18B at Photon Factory, KEK, Tsukuba, Japan using incident photons of 16 KeV at 300 K and 500 K to trace the structural changes with temperature. Temperature dependent magnetization measurements M(T) were carried out in the temperature range of 10 K to 380 K using a SQUID magnetometer in the applied magnetic field of 5 mT and 5 T  wherein the samples were first cooled in zero applied magnetic field from room temperature to the lowest temperature and the data was recorded while warming (ZFC), the subsequent cooling (FCC) and warming (FCW) cycles. The resistivity measurements were performed using standard four probe method in the temperature interval of 200 K -- 400 K and frequency dependence of AC storage modulus and loss were carried out using Dynamical Mechanical Analyzer (Q800, TA Instruments) as described in \\cite{Nevgi1122018}. Extended X-ray Absorption Fine structure (EXAFS) were employed to perform local structural studies at Ni K (8333 eV), Mn K (6539 eV) and Fe K (7111 eV) edges at the P65 beamline (PETRA III Synchrotron Source, DESY, Hamburg, Germany). Using gas ionization chambers as detectors, the incident (I0) and the transmitted (I) photon energies were simultaneously recorded. The thickness $t$ of the absorbers were adjusted so as to obtain the absorption edge jump, $\\Delta \\mu t \\leq 1$. For the Ni and Mn K edge EXAFS data was averaged over three scans collected in transmission mode while and at the Fe K edge ten scans were collected in fluorescence mode to average the statistical noise. The EXAFS data was analyzed using well established procedures in Demeter suite \\cite{Raval200512}.\n\n\\section{Results}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{XRD.eps}\n\\caption{The room temperature Lebail refined x-ray diffraction data as the function of two theta and wave vector q for the alloys 0.075 $\\leq x \\leq$ 0.2 in the series Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ (a to c) and for the alloys $y$ = 0.1, 0.2 in the series Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$ (d and e). Inset highlights the presence of minor fcc cubic phase seen the alloy $y$ = 0.2 (c).}\n\\label{fig:XRD}\n\\end{center}\n\\end{figure}\n\nFig. \\ref{fig:XRD} represents the room temperature x-ray diffraction data analyzed by Le Bail method using Jana 2006 software\\cite{Petricek2292014}. In the series Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$, the alloys with lower Fe concentration ($x \\leq 0.1$) exhibit 7M modulated martensitic structure (Fig. \\ref{fig:XRD}a and b) just like the parent Ni$_{2}$Mn$_{1.5}$In$_{0.5}$ \\cite{Nevgi7972019}. With the increase in Fe content ($x = 0.2$), there is a structural transition from the martensitic phase to a major $L2_1$ and a minor impurity phase, as seen in Fig. \\ref{fig:XRD}c. The additional Bragg peaks of the impurity phase can be fitted to a face centred cubic (fcc) phase identified later as $\\gamma -$(Fe,Ni) phase. In the second series Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$, the 7M modulated phase is converted to a cubic Heusler phase with the increase in Fe concentration from $y = 0.1$ to $y = 0.2$ (Fig. \\ref{fig:XRD}d and e). The refinement results are summarized in Table \\ref{table0}.\n\n\\begin{table}[h]\n\\caption{\\label{table0} The refined crystallographic data for the alloy compositions $x$ = 0.075, 0.1 and 0.2 in the series Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$, and $y$ = 0.1, 0.2 in the series Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$.}\n\\setlength{\\tabcolsep}{0.3pc}\n\\vspace{0.3cm}\n\\centering\n\\begin{tabular}{|l|l|l |}\n\\hline\n\\textbf{Chemical formula} & \\textbf{Space group} & \\textbf{Lattice parameters}\\\\[0.5ex]\n\\hline\nNi$_{2}$Mn$_{1.425}$Fe$_{0.075}$In$_{0.5}$   & $I2\/m(\\alpha0\\gamma)00$ & a = 4.450(5) \\AA, b = 5.624(7) \\AA, c = 4.376(4) \\AA \\\\\n                                             &                         & $\\beta$ = 93.93(1)$^\\circ$, $q$ = 0.342(5)$c^*$ \\\\\n\\hline$\\deg$\nNi$_{2}$Mn$_{1.4}$Fe$_{0.1}$In$_{0.5}$       & $I2\/m(\\alpha0\\gamma)00$ & a = 4.438(2) \\AA, b = 5.644(1) \\AA, c = 4.355(1) \\AA\\\\\n                                             &                         & $\\beta$ = 93.32(2)$^\\circ$, $q$ = 0.337(1)$c^*$\\\\\n\\hline\nNi$_{2}$Mn$_{1.3}$Fe$_{0.2}$In$_{0.5}$       & $Fm-3m$ ($L2_1$)          & a = 5.983 (3) \\AA\\\\\n                                             & $Fm-3m$ ($\\gamma$)        & a = 3.616 (2) \\AA\\\\\n\\hline\nNi$_{1.9}$Fe$_{0.1}$Mn$_{1.5}$In$_{0.5}$     & $I2\/m(\\alpha0\\gamma)00$ & a = 4.471(5) \\AA, b = 5.651(7) \\AA, c = 4.381(4) \\AA\\\\\n                                             &                         & $\\beta$ = 92.97(1)$^\\circ$, $q$ = 0.339(4)$c^*$\\\\\n\\hline\nNi$_{1.8}$Fe$_{0.2}$Mn$_{1.5}$In$_{0.5}$     & $Fm-3m$ ($L2_1$)          & a = 6.012 (2) \\AA\\\\[1ex]\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{MT.eps}\n\\caption{Magnetization as a function of temperature for the alloys $x$ = 0.1 and $x$ = 0.2 in the series Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ (a, b) and for the alloys $y$ = 0.1 and $y$ = 0.2 in the series Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$ (c, d) during warming after cooling the alloys in zero field (ZFC) and subsequent warming (FCW) and cooling (FCC) cycles in magnetic field.}\n\\label{fig:MT}\n\\end{center}\n\\end{figure}\n\nThe temperature dependent magnetization measurements M(T) recorded in applied magnetic fields of 5 mT and 5T for the alloys $x$ = 0.075 (a), and $x$ = 0.1 (b) belonging to the series Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ and for the alloys $y$ = 0.1 (c) and $y$ = 0.2 (d) from the second series Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$ are shown in Fig. \\ref{fig:MT}. The alloys $x$ = 0.075, $x$ = 0.1 and $y$ = 0.1 exhibit a first order martensitic transition evident from the hysteresis in the warming and cooling data. The broad peak at the blocking temperature $T_{B}$  in ZFC data recorded in magnetic field of 5 mT along with the large splitting between ZFC and the field cooled (FC) curves signify a non-ergodic behaviour similar to that observed in Ni$_2$Mn$_{1+x}$In$_{1-x}$ \\cite{Nevgi322020}.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{DMARESmagn.eps}\n\\caption{Imaginary part of the storage modulus $\\tan\\delta$ at different frequencies (a), resistivity during warming and cooling cycles (b) and magnetization during ZFC, FCC and FCW cycles recorded in H = 5 mT (c) as a function of temperature for Ni$_2$Mn$_{1.4}$Fe$_{0.1}$In$_{0.5}$}\n\\label{fig:DMARESMagn}\n\\end{center}\n\\end{figure}\n\nIn Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$, as the Fe concentration increases from 0.075 to 0.1, the martensitic transition temperature $T_M$ decreases from 360 K to about 302 K and the $T_{B}$ shifts towards higher temperature, indicating a strengthening of ferromagnetic interactions at the expense of martensitic interactions. Interestingly in $x = 0.1$, the ZFC curve approaches a negative magnetization value below $T$ = 100 K (see Fig. \\ref{fig:MT}(b)) signifying a magnetic compensation. It should be noted that the alloy $x = 0.1$ was reported to have a strain glassy ground state below $T_g =$ 350 K \\cite{Nevgi1122018}. Frequency dependence of the imaginary part of the storage modulus ($\\tan\\delta$) seen in Fig. \\ref{fig:DMARESMagn}(a) follows Vogel-Fulcher law indicating glassy nature of the ground state. However, as the crystal structure of this alloy ($x = 0.1$) was 7M modulated at room temperature, the strain glass state was classified as unusual one consisting of large undoped martensitic grains separated by an impurity phase. The unconventional nature of the glassy state is also evident from the thermal hysteresis in the resistivity, as shown in Fig. \\ref{fig:DMARESMagn}(b). As the hysteresis in magnetization measurement recorded in 5 mT Fig. \\ref{fig:DMARESMagn}(c) and resistivity coincide, the observation of the first--order transition in magnetization could be the signature of the martensitic transformation of the large undoped grains. Further, the martensitic transformation of $x = 0.1$ alloy shifts to a higher temperature with an increase in applied field from 5 mT to 5 T (see Fig.\\ref{fig:MT}(b)). Such a behaviour of increasing the martensitic transformation  with magnetic field is not seen in the other two transforming alloys, $x = 0.075$ and $y = 0.1$. Contrary to this, the magnetization of Ni$_{1.8}$Fe$_{0.2}$Mn$_{1.5}$In$_{0.5}$ increases sharply displaying a paramagnetic to ferromagnetic transition at $T_C$ = 314 K (Fig. \\ref{fig:MT}(d)). However, the presence of hysteresis in the warming and the cooling magnetization curves in the temperature range of 150 K to 350 K need a deeper understanding of the structure.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{SYNCXRD.eps}\n\\caption{X-ray diffraction data in the limited two theta and wave vector q range highlighting the evolution of phases with temperature and Fe concentration in the series Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ (a, b) and Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$ (c, d). The presence of minor impurity phases arising due to Fe substitution are marked as (*).}\n\\label{fig:SYNCXRD}\n\\end{center}\n\\end{figure}\n\nTo detect the presence of possible impurity phases as a result of Fe doping in the alloys belonging to Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ and Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$, synchrotron x-ray diffraction measurements were performed at 300 K and at 500 K, which is well above the martensitic transition temperature of the undoped alloy and the phases were identified using the Le Bail refinement. Fig. \\ref{fig:SYNCXRD} highlights the diffraction patterns in the limited two theta and wave vector q region. In the series, Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$,  the alloy $x$ = 0.1 (Fig. \\ref{fig:SYNCXRD} a) exhibits 7M modulated martensitic structure at 300 K, which is transformed to a cubic Heusler phase accompanied by a minor fcc impurity phase (marked as $*$) at 500 K. The same impurity phase appears to grow in intensity in $x$ = 0.2 alloy (Fig. \\ref{fig:SYNCXRD} b) and is present at both the temperatures along with the major Heusler phase. It appears that the modulated monoclinic structure of the alloy $x$ = 0.1 masked the impurity phase at 300 K, and hence the alloy appeared as a single phase.  The presence of the impurity phase in $x = 0.1$ lends weight to the earlier proposition that the strain glass phase in Ni$_{2}$Mn$_{1.4}$Fe$_{0.1}$In$_{0.5}$ occurs due to large martensitic grains separated by an impurity phase. The martensitic grains transform into austenitic phase above 300K. Since the phase fraction of the impurity phase increases with Fe doping, it either consists of Fe or the impurity phase itself is induced by Fe doping.\n\nIn the second series, Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$ the alloy $y$ = 0.1 (Fig. \\ref{fig:SYNCXRD}c) undergoes a transition from the modulated phase at 300 K to a cubic Heusler phase at 500 K. The alloy $y$ = 0.2 (Fig. \\ref{fig:SYNCXRD}d) on the other hand exhibits a cubic Heusler phase with a minor percentage of monoclinic phase (I2\/m) at 300 K. This monoclinic phase disappears at 500 K, perhaps due to its transformation to the cubic austenitic phase. Same was not seen in the diffraction pattern recorded using a laboratory source, and hence its phase fraction should be meager. The presence of this martensitic impurity phase could be the reason for observed hysteresis in the warming and cooling magnetization curves of $y = 0.2$ alloy. Despite the presence of the impurity phase, these alloys do not exhibit a strain glassy ground state; instead, the martensitic state is completely suppressed by a ferromagnetic cubic phase. Therefore, it appears that the ground state of Fe doped Ni-Mn-In alloys depends on the doping site. In these Mn rich Ni-Mn-In alloys, Ni atoms are expected to occupy the X sites while the Mn atoms are present on the Y sublattice and along with In atoms on the Z sublattices. To determine the site occupancy of the Fe atoms and to study the structural interactions that play a role in the determination of the ground state, we have studied the local environment of Ni, Mn and Fe atoms via EXAFS experiments performed at 300 K and 50 K.\n\nThe Ni K and Mn K EXAFS spectra in all alloy compositions have been fitted using  a common structural model based on the crystal structure and similar to that described earlier\\cite{Nevgi322020}. It employs 14 independent parameters, including correction to bond length ($\\Delta R$) and mean square variation in bond length $\\sigma^{2}$ for every scattering path used. The coordination number of the structural correlations were fixed to those obtained for Ni$_{2}$Mn$_{1.5}$In$_{0.5}$ composition. The amplitude reduction factor, $S_0{^2}$ for the two edges were obtained from the analysis of the standard metal spectra and were also kept fixed throughout the analysis.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{EXAFS1.eps}\n\\caption{Fourier transform magnitude of EXAFS spectra recorded at 50 K at the Ni K and Mn K edges for the alloys $x$ = 0.1 and $x$ = 0.2 in the series Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$.}\n\\label{fig:EXAFS1}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{EXAFS2.eps}\n\\caption{Magnitude of Fourier transform of Ni K and Mn K EXAFS spectra for the alloys, Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$ with $y$ = 0.1 and $y$ = 0.2 at 50 K.}\n\\label{fig:EXAFS2}\n\\end{center}\n\\end{figure}\n\nThe best fits for  Ni K and Mn K EXAFS spectra obtained in both the cases, when Fe is doped for Mn and Ni are presented in Fig. \\ref{fig:EXAFS1} and Fig. \\ref{fig:EXAFS2} respectively and the best fit values of bond length ($R$) and mean square radial displacement ($\\sigma^{2}$) at 300 K and 50 K are listed in Table \\ref{table1}. The results are in good agreement with previously reported EXAFS studies on such Mn rich Ni-Mn-In alloys \\cite{Nevgi322020,Priolkar20113}.  As expected, the nearest neighbour,  Ni--Mn distance is shorter than the  Ni--In bond distance. Similarly, the Mn$_Y$--Mn$_Y$ distance (Mn$_Y$ describes Mn atoms in the Y sublattice of X$_2$YZ Heusler structure) decreases from  $\\sim$ 4.4 \\AA~ in alloys undergoing a martensitic transformation in the paramagnetic state to $\\sim$ 4.2 \\AA~ in alloys with dominant ferromagnetic interactions. This is very clearly evident as the Fe content is increased from $y = 0.1$ to $y = 0.2$ in Ni$_{2-y}$Fe$_y$Mn$_{1.5}$In$_{0.5}$, where in the crystal structure at 300 K changes from 7M monoclinic to cubic Heusler along with the strengthening of ferromagnetic interactions. The cubic Heusler structure demands the bond distances Mn--In and Mn$_Y$--Mn$_Z$ (Mn$_Z$ represents Mn atoms occupying Z sublattice) to be equal. This change is also clearly seen as the structure of Fe doped alloys converts from martensitic to cubic Heusler structure.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{EXAFS3.eps}\n\\caption{Fe K edge EXAFS spectra in $R-$space (Fourier transform magnitude) obtained at 300 K for the alloys Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ with (a) $x = 0.1$ and (b) $x= 0.2$ and Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$ with (c) $y = 0.1$ and (d) $y= 0.2$.}\n\\label{fig:EXAFS3}\n\\end{center}\n\\end{figure}\n\n\n\\begin{table}[h]\n\\caption{\\label{table1} The results of the best fits obtained at 300 K and 50 K at Ni K and Mn K edges in the  $k-$range 3--12 \\AA$^{-1}$, and in $R-$range 1--3 \\AA. The parameter $R$ gives the bond length while the mean square disorder in the bond length is described by $\\sigma^{2}$. Figures in parenthesis indicate uncertainty in the last digit.}\n\\setlength{\\tabcolsep}{1pc}\n\\vspace{0.3cm}\n\\begin{adjustbox}{width=\\columnwidth,center}\n\\begin{tabular}{cccccccc}\n\\hline\n\\textbf{Alloys}                         &\\textbf{Temperature}  &              &\\textbf{Bonds}        \\\\[0.5ex]\n                                        &                      &              & Ni - Mn     & Ni - In    &Mn - In    &Mn$_Y$ - Mn$_Y$    &Mn$_Y$ - Mn$_Z$\\\\\n\\hline\\hline\nNi$_{2}$Mn$_{1.5}$In$_{0.5}$            &{300 K}              & {R (\\AA)}            & 2.571(8)\t& 2.620(7)\t &2.88(4)    & 4.40(5)   & 3.05(4)  \\\\\n                                        &                     &{$\\sigma^{2}$ (\\AA$^2$)}       & 0.0057(4)\t& 0.0099(8)\t & 0.050(5)  & 0.025(8)  &0.04(3)    \\\\\n\\hline\n                                        &{50 K}               & {R (\\AA)}             &2.58(1)    &2.618(5)    &2.88(4)    &4.45(1)    &3.01(4)   \\\\\n                                        &                     &{$\\sigma^{2}$ (\\AA$^2$)}        &0.0018(5)  & 0.0068(5)  & 0.008(5)  & 0.02(1)   &0.016(7) \\\\\n\\hline\nNi$_{2}$Mn$_{1.4}$Fe$_{0.1}$In$_{0.5}$ &{300 K}              & {R (\\AA)}            & 2.57(1)\t& 2.62(1)\t &2.86(4)    & 4.39(6)   & 3.05(4)  \\\\\n                                       &                     &{$\\sigma^{2}$ (\\AA$^2$)}       &0.0105(7)\t& 0.005(1)\t & 0.050(5)  & 0.025(8)  &0.04(3)    \\\\\n\\hline\n                                       &{50 K}               & {R (\\AA)}             &2.572(8)   &2.61(1)    &2.89(4)    &4.45(1)    &3.01(4)   \\\\\n                                       &                     &{$\\sigma^{2}$ (\\AA$^2$)}        &0.001(1)  & 0.0062(5)  & 0.008(5)  & 0.02(1)   &0.016(7) \\\\\n\\hline\nNi$_{2}$Mn$_{1.3}$Fe$_{0.2}$In$_{0.5}$ &{300 K}              & {R (\\AA)}            & 2.54(1)\t& 2.63(1)\t &2.93(3)    & 4.25(7)   & 2.98(4)  \\\\\n                                       &                     &{$\\sigma^{2}$ (\\AA$^2$)}       & 0.011(1)\t& 0.004(1)\t &0.003(4)  & 0.02(1)    &0.010(9)    \\\\\n\\hline\n                                       &{50 K}               & {R (\\AA)}             &2.542(9)    &2.64(2)    &2.94(3)    &4.24(4)    &2.96(2)   \\\\\n                                       &                     &{$\\sigma^{2}$ (\\AA$^2$)}        &0.002(1)  & 0.0049(6)  & 0.002(2)  & 0.016(4)   &0.003(3) \\\\\n\\hline\nNi$_{1.9}$Fe$_{0.1}$Mn$_{1.5}$In$_{0.5}$ &{300 K}              & {R (\\AA)}          & 2.577(9)\t& 2.633(8)\t &2.88(4)    & 4.39(6)   & 3.05(4)  \\\\\n                                       &                     &{$\\sigma^{2}$ (\\AA$^2$)}       & 0.0048(8)\t& 0.0116(8)\t & 0.050(5)  & 0.025(8)  &0.04(3)    \\\\\n\\hline\n                                       &{50 K}               & {R (\\AA)}             &2.588(7)    &2.619(8)    &2.89(4)    &4.45(1)    &3.02(4)   \\\\\n                                       &                     &{$\\sigma^{2}$ (\\AA$^2$)}        &0.0015(8)   &0.0072(4)  & 0.008(5)  & 0.02(1)   &0.016(7) \\\\\n\\hline\nNi$_{1.8}$Fe$_{0.2}$Mn$_{1.5}$In$_{0.5}$ &{300 K}              & {R (\\AA)}           & 2.54(2)\t& 2.62(1)\t &2.93(6)    & 4.21(6)   & 2.95(5)  \\\\\n                                       &                     &{$\\sigma^{2}$ (\\AA$^2$)}       & 0.013(2)\t& 0.003(1)\t & 0.026(5)  & 0.02(1)   &0.005(7)    \\\\\n\\hline\n                                       &{50 K}               & {R (\\AA)}             &2.576(7)    &2.625(6)    &2.93(2)    &4.25(5)    &2.95(1)   \\\\\n                                       &                     &{$\\sigma^{2}$ (\\AA$^2$)}        &0.0080(7)  & 0.0009(5)  & 0.001(1)  & 0.04(4)   &0.08(1)\\\\ [1ex]\n\\hline\n\\end{tabular}\n\\end{adjustbox}\n\\end{table}\n\n\n\\begin{table}[h]\n\\caption{\\label{table2} The results of the best fits obtained at 300 K at Fe K edge in the $k-$range 3--12 \\AA$^{-1}$, and in $R-$range 1--3 \\AA. The parameter $R$ gives the  bond length while the thermal variation in bond length is described by $\\sigma^{2}$. Figures in parenthesis indicate uncertainty in the last digit.}\n\\setlength{\\tabcolsep}{1pc}\n\\vspace{0.3cm}\n\\centering\n\\begin{tabular}{ccccc}\n\\hline\n\\textbf{Alloys}                           &                  &\\textbf{Bonds}        \\\\[0.5ex]\n                                          &                  & Fe - Ni     & Fe - Fe    &Fe - In \\\\\n\\hline\\hline\nNi$_{2}$Mn$_{1.4}$Fe$_{0.1}$In$_{0.5}$    & {R (\\AA)}         & 2.508(8)  \t& 3.546(8)\t & \\\\\n                                          &{$\\sigma^{2}$ (\\AA$^2$)}    & 0.006(3)\t    & 0.02(2)\t &  \\\\\n\\hline\nNi$_{2}$Mn$_{1.3}$Fe$_{0.2}$In$_{0.5}$    & {R (\\AA)}         & 2.525(4)  \t& 3.571(4)\t & \\\\\n                                          &{$\\sigma^{2}$ (\\AA$^2$)}    & 0.009(1)\t    & 0.015(6)\t &  \\\\\n\\hline\nNi$_{1.9}$Fe$_{0.1}$Mn$_{1.5}$In$_{0.5}$  & {R (\\AA)}         & 2.46(5)  \t& 2.8(1)\t &2.97(8) \\\\\n                                          &{$\\sigma^{2}$ (\\AA$^2$)}    & 0.01(1)\t    & 0.02(3)\t &0.01(1) \\\\\n\\hline\nNi$_{1.8}$Fe$_{0.2}$Mn$_{1.4}$In$_{0.5}$  & {R (\\AA)}         & 2.41(2)  \t& 2.93(4)\t &3.1(1)\\\\\n                                          &{$\\sigma^{2}$ (\\AA$^2$)}    & 0.004(3)\t    & 0.008(6)\t &0.01(1)\\\\[1ex]\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe Fe K EXAFS spectra recorded at 300 K appears to differ from that of Ni K and Mn K edge spectra in Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ indicating a different local environment around Fe  compared to Ni and Mn atoms. Since $x = 0.1$ and $x = 0.2$ compositions reveal presence of an fcc impurity phase, we tried to fit the Fe K edge EXAFS to structural correlations obtained for the fcc structure. A good fit is obtained by considering 12 Ni atoms at $\\sim$ 2.5 \\AA~ and 6 Fe atoms at $\\sim$ 3.5 \\AA~ and the same can be seen in Fig. \\ref{fig:EXAFS3}(a and b). All attempts to include In atoms as scatterers either in the first shell ($\\sim$ 2.5 \\AA) or the second shell ($\\sim$ 3.5 \\AA) did not result in physically acceptable parameters. Therefore, the impurity phase segregated in Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$ consists of Fe and Ni and from the Fe-Ni binary phase diagram was identified as $\\gamma -$(Fe,Ni) phase.\n\nIn the second series Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$, though Fe is doped for Ni, the local structure of Fe is similar to that of Mn at the Y\/Z site rather than Ni at the X site. It may be mentioned here that in the Heusler structure, X (Ni) atoms have Y (Mn) and Z (In) atoms in the first coordination shell and X (Ni) atoms in the second coordination shell while the Y (Mn) atoms have only X (Ni) in their first coordination shell and Z (In\/Mn) atoms in the second coordination shell. Therefore, if Fe replaces Ni at the X site, then it should have Mn and In atoms as nearest neighbours and Ni atoms as the second nearest neighbours. However, EXAFS signal can be fitted with only Ni atoms in the first coordination shell and In\/Mn atoms as second neighbours. The best fit to the experimental data is shown in Fig. \\ref{fig:EXAFS3}(c and d), and the parameters obtained from fitting are presented in Table \\ref{table2}.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{XRD1.eps}\n\\caption{A comparison of the x-ray diffraction patterns of the alloys Ni$_{2}$Mn$_{1.3}$Fe$_{0.2}$In$_{0.5}$ and Ni$_{1.8}$Fe$_{0.2}$Mn$_{1.5}$In$_{0.5}$ featuring the absence of 200 peak and the presence of $A2$ disorder in the later.}\n\\label{fig:XRD1}\n\\end{center}\n\\end{figure}\n\nIn the X$_2$YZ Heusler structure, the intensities of the super lattice reflections, (111) and (200), are sensitive to the antisite disorder. A disorder in occupancy of Y and Z sublattices of the Heusler structure is identified as $B2$ disorder and results in lowering of intensities of both (111) and (200) reflections. While a disorder involving X and Y sublattices affect the intensity of (200) reflection and is known as $A2$ disorder \\cite{Webster101969,Takamura1052009}. Substitution of Fe in the Y\/Z sublattice in Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$ should result in the occupation of the vacant X sites by Mn atoms and therefore result in an $A2$ type disorder. Fig. \\ref{fig:XRD1} compares the x-ray diffraction patterns of Ni$_{2}$Mn$_{1.3}$Fe$_{0.2}$In$_{0.5}$ and Ni$_{1.8}$Fe$_{0.2}$Mn$_{1.5}$In$_{0.5}$, highlighting the absence of (200) reflection in Ni$_{1.8}$Fe$_{0.2}$Mn$_{1.5}$In$_{0.5}$ and confirming the presence of $A2$ type disorder.\n\n\\section{Discussion}\n\nThe above studies illustrate that Fe doping in the martensitic Ni$_{2}$Mn$_{1.5}$In$_{0.5}$ alloy results in the suppression of the martensitic state. The ground state is decided based on whether the dopant atom occupies one of the crystallographic sites in the Heusler structure or segregates in an impurity phase.  These two scenarios are confirmed from both XRD as well as EXAFS studies. In the series Ni$_{2}$Mn$_{1.5-x}$Fe$_{x}$In$_{0.5}$, with the increase in Fe content, an impurity phase, $\\gamma -$(Fe,Ni) is segregated in addition to the martensitic Heusler phase. Even though the large Heusler grains undergo martensitic transformation just above room temperature, the ordering of the elastic strain vector is spatially disturbed by the presence of the impurity phase leading to a strain glass phase. This is clear from the fact that the glass transition temperature $T_g = 350$ K is higher than the martensitic transition temperature $T_M =$ 304 K. Furthermore; there is an increase in the $T_M$ of Ni$_{2}$Mn$_{1.4}$Fe$_{0.1}$In$_{0.5}$ alloy from 304 K to 340 K when the applied magnetic field is increased from 5 mT to 5T. In Mn--excess Ni-Mn-In alloys, the martensitic transition temperature generally decreases with magnetic field. This contravening behaviour of increasing $T_M$ under magnetic field can be understood to be due to a strong coupling between magnetic and elastic degrees of freedom. Increased magnetic field lowers the entropy of the system, thereby promoting an increased order of the elastic strain vector and higher transformation temperature.\n\nOn the other hand, when Fe is doped for Ni in Ni$_{2-y}$Fe$_{y}$Mn$_{1.5}$In$_{0.5}$, Fe promotes antisite disorder by occupying Y or Z sublattices and forcing Mn to the vacancies in the X sublattice of the Heusler alloy. This $A2$ disorder suppresses martensitic transformation and strengthens ferromagnetic interactions leading to a cubic ferromagnetic ground state. It appears that when the X sublattice in X$_2$YZ Heusler structure is fully occupied, Fe doping results in segregation of an impurity phase, and in the case of Fe being doped into the X sublattice, it promotes antisite disorder and accommodates itself in the Y\/Z sublattice of the Heusler structure.\n\n\\section{Conclusions}\n\nIn conclusion, when Mn is sought to be replaced by Fe in the martensitic Ni$_{2}$Mn$_{1.5}$In$_{0.5}$, the alloy phase separates into a major Heusler phase and a minor, $\\gamma -$(Fe,Ni) phase. This $\\gamma$ phase serves as an impediment for the long range ordering of the elastic strain vector, promoting a strain glassy ground state. On the other hand, when Fe is added at the expense of Ni, it replaces Mn in the Y\/Z Heusler sublattices forcing Mn to occupy the X sublattice along with Ni and thus creating an $A2$  disorder. The presence of antisite disorder suppresses martensitic transition by promoting a ferromagnetic ground state. This study implies that the suppression of martensite via strain glass occurs if the Fe addition facilitates the segregation of an impurity phase. However, if the dopant Fe is accommodated in the Heusler phase even via an antisite disorder; the resultant stronger ferromagnetic interactions forbid martensitic transformation of the resultant alloy.\n\n\\section*{Acknowledgements}\n\nKRP and RN acknowledge the Science and Engineering Research Board, Govt. of India under the project SB\/S2\/CMP-0096\/2013 for financial assistance and Department of Science and Technology, Govt. of  India for the travel support within the framework of India\\@ DESY collaboration. RN thanks the Council of Scientific and Industrial Research, Govt. of India for Senior Research fellowship. Edmund Welter and Ruidy Nemausat are thanked for experimental assistance at P65 beamline, PETRA III, DESY Hamburg.\n\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Acknowledgement}\nThis work is\nsupported by Shanghai NSF Grant (No.~18ZR1425100) and NSFC\nGrant (No.~61703195).\n\\section{Task Inference Strategy}\\label{sec:app}\n\n\n\nWe now introduce our task reasoning strategy for meta-RL with limited data. Our goal is to use the limited exploration to generate experiences with sufficient information for task inference and to effectively infer the task posterior from the given task experiences. In Sec.~\\ref{sec:exp}, we explain how we learn an explorer that pursues task-informative experiences with randomized behavior and mitigates train-test mismatch. In Sec.~\\ref{sec:inf}, we elaborate on the network design of the task encoder, which enables relational modeling between task experiences.\n\n\\subsection{Learning the Exploration Policy}\\label{sec:exp}\n\nOur exploration policy aims to enrich the task-related information of experiences within limited episodes. To this end, below we introduce two reward shaping methods: first, we increase the coverage of task experiences to obtain task-informative experiences; second, we improve the quality of each sample to have as much information gain as possible. \n\n\\subsubsection{Increasing Coverage}\n\nThe first idea of improving exploration is to increase its coverage of task experiences by adding stochasity into the agent's behavior, which is motivated by the empirical observations of the performance\nobtain with replay buffer (see the off-policy curve in Fig.~\\ref{fig:exp} ). Off-policy data have more coverage of experiences as the samples are uniformly drawn from \nmultiple different trajectories, compared to samples in online rollouts with smaller variations.\n\nTo this end, we encourage the randomized behavior of the explorer by leveraging SAC~\\cite{haarnoja2018soft}. SAC derives from the entropy-regularized RL objective~\\cite{levine2018reinforcement}, which essentially adds entropy to the reward (and value) functions. Note that we optimize both $\\pi$ and $\\xi$ with SAC, but for different purposes. SAC favors exploration in policy learning for the actor $\\pi$, which will run deterministically at meta-test time. In contrast, here SAC is used to guide the explorer towards randomized task-exploration, which is a stochastic policy during deployment. \n\nIt is worth noting that training an exploration policy with randomized behavior also helps mitigate train-test mismatch, which is caused by the different data distributions for policy adaptation (buffer data v.s. online rollouts). Our task-exploration explicitly learns the ability for randomized exploration that (empirically) brings the online rollout data distribution closer to the off-policy buffer data.\n\n\n\\subsubsection{Improving Quality}\nTo improve the quality of exploration, we also design a reward shaping that \nguides the explorer towards highly informative experiences. However, it is non-trivial \nto quantify the informativeness in a sample. In our case, the explorer collects experiences given a task hypothesis, so we are more interested in evaluating the mutual information as the reward for the explorer, \n\\begin{align}\n  \\tilde{r}_i = \\mathcal{I}(z|\\tilde{x}; x_i) = \\mathcal{H}(q(z|\\tilde{x}))-\\mathcal{H}(q(\\hat{z}|\\tilde{x},x_i))\n\\end{align}\nwhere we differentiate the new task hypothesis $\\hat{z}$ against previously hypothesized task $z\\sim q(z|\\tilde{x})$, $\\tilde{x}=\\{x_j\\}_{1:K}$ denotes the repository of collected experiences and $x_i \\notin \\tilde{x}$ is the sample for assessment. Intuitively, we expect the credit $\\tilde{r}_i$ to reflect the information gain from the sample $x_i$ given the current hypothesis of task $z\\sim q(z|\\tilde{x})$.\n\nHowever, directly computing the mutual information is impractical, as it involves evaluating the new posterior after incorporating each new sample $x_i$. Instead, we adopt the following proxy for the mutual information:\n\\begin{align}\n  \\mathcal{I}(z|\\tilde{x};x_i) \\propto \\log \\frac{1}{p(\\hat{z}=z|x_i,\\tilde{x})}\n\\end{align}\nThis proxy implies that we believe a sample $x_i$ brings larger information gain if the new hypothesis $\\hat{z}$ is less likely to be the same as the prior belief $z$. To compute $p(\\hat{z}=z|x_i, \\tilde{x})$, we apply the Bayes Rule:\n\\begin{align}\n   p(\\hat{z}=z|x_i, \\tilde{x}) = \\frac{p(R=y(\\tilde{x}), \\hat{z}=z|x_i,\\tilde{x})}{P(R=y(\\tilde{x})|\\hat{z}=z,x_i,\\tilde{x})}\n\\end{align}\nAs the joint distribution of $(\\hat{z},R)$ is constant given $(x_i,\\tilde{x})$, the posterior of $\\hat{z}$ is proportional to the inverse of the likelihood.\nSimilar to Sec.~\\ref{sec:ps}, we use the state-action value function to compute the likelihood $P(R=y(\\tilde{x})|x_i, \\tilde{x})$, but we instead assume a laplace distribution as we empirically find it is more stable to use the L1-norm than the L2-norm induced by a Gaussian. As a result, we have the following score function that gives the shaped reward $\\tilde{r}_i$: \n\\begin{align}\n  &\\tilde{r}_i = \\lambda  \\log P\\left(R=y(\\tilde{x})|\\hat{z}=z,x_i,\\tilde{x}\\right) \\\\\n  &= \\lambda \\left\\|Q\\left(s_i,a_i,z\\right)-\\left(r_i+\\gamma \\max_{a_i^\\prime} Q\\left(s_i^\\prime,a_i^\\prime\\right)\\right)\\right\\|_{L_1}\\label{eq:td}\n\\end{align}\nwhere the hyperparameter $\\lambda$ is the reward scale, and the greedy policy $\\pi(a|s)$ is learned to compute $\\max_a Q(s,a)$~\\cite{dpg,ddpg,haarnoja2018soft}.\n\\subsection{Context-aware Task Encoder}\\label{sec:inf}\n\nOur task inference network computes the posterior of latent variable $z$ given a set of experience data, aiming to extract task information from experiences. To this end, we design a network module with the following properties:\n\\begin{enumerate}[leftmargin=4mm]\n  \\item \\textit{Permutation-invariant}, as the output should not vary with the order of the inputs\n  \\item \\textit{Input size-agnostic}, as the network would encounter variable size of inputs within the arbitrary number of rollouts.\n  \\item \\textit{Context-aware}, as extracting cues from a single sample should incorporate the context formed by other samples\\footnote{Imagine in a 2d-navigation task where the agent aims to navigate to a goal location, a sample may indicate the possible location of the goal due to the high rewards, and can further eliminate possibilities by another sample that shows what locations are not possible by its low rewards.}. \n\\end{enumerate}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{figs\/encoder.jpg}\n  \\caption{\\small The task encoder.The first aggregation constructs a bipartite graph with full connections from the $n$ nodes in $V^i$ to the $c$ latent nodes in $V^h$. Self-attention operates on $V^h$, which are assembled to one latent node in the second aggregation.}\n  \n  \\label{fig:inference}\n\\end{figure}\n\nSpecifically, we adopt a latent Graph Neural Network architecture~\\cite{zhang2019latentgnn}, which integrates self-attention with learned weighted-sum aggregation layers. \nFormally, we introduce a set of latent node features $H=[h_1,\\cdots,h_d]^T$ \nwhere $h_i\\in \\mathbb{R}^c$ is a c-channel feature vector same as the \nset of input node features $X=[x_i,\\cdots,x_n]^T$. Note that $n$ can be \narbitrary number while $d$ should be a fixed hyperparameter. We construct a graph \n$\\mathcal{G}=(\\mathcal{V},\\mathcal{E})$ with the $n+d$ nodes and full connections \nfrom all of the input nodes to each of the latent nodes, i.e., \n$(v_i,v_j)\n\\in \\mathcal{E}, v_i\\in\\mathcal{V}^i, v_j\\in\\mathcal{V}^h$ where \n$\\mathcal{V}^i,\\mathcal{V}^h$ refers to the set of input nodes and latent nodes respectively. The graph network module is illustrated in Fig.~\\ref{fig:inference}.\n\nThe output of the aggregation layer $f_{\\text{AGG}}$ is the latent node features $H$, which are computed as follows:\n\\begin{align*}\n  h_k = \\sum_{j=1}^n \\psi(x_j,\\phi_k)x_j, 1\\leq k\\leq d\n\\end{align*}\nwhere $\\psi(x,\\phi_k)$ is a learned affinity function parameterized by $\\phi_k$\nthat encodes the affinity between any input node $x$ and the $k^{th}$ latent node.\nIn practice, we instantiate this function as the dot product followed with normalization, i.e.,\n$\\text{softmax}_j (\\phi_k x_j^T)$.\n\nWe then combine the above aggregation layer \nwith the following self-attention layer $f_\\text{ATN}$:\n\\begin{align*}\n  \\tilde{x}_j = \\sum_{i=1}^n f(x_i,x_j)x_j, 1\\leq i\\leq n\n\\end{align*}\nwhere we use the the scaled dot product attention \\cite{vaswani2017attention}.\n\nFollowing \\cite{zhang2019latentgnn}, we propagate messages through a shared space \nwith full-connections between latent nodes. We first pass the input nodes through an aggregation layer \nwith $d$ latent nodes, where $d<<n$, then perform self-attention on the latent nodes (for multiple iterations), and finally \npass the latent nodes through another aggregation layer with $1$ final latent node to obtain \nthe final output, i.e., $f_\\text{AGG}\\circ f_\\text{ATN}\\circ f_\\text{AGG}:\n\\mathbb{R}^{n\\times c}\\mapsto \\mathbb{R}^{d\\times c}\\mapsto \\mathbb{R}^{d\\times c}\\mapsto \\mathbb{R}^{1\\times c}$. The final output provides the parameters for $q(z|x)$ which is a Gaussian distribution. \nThe network can be viewed as a multi-stage summarization process, in which we group the inputs \ninto several summaries, and operate on these summaries to compute the relationships between entities, \nand produce a final summary of the entities and relationships.\n\n\\section{Conclusion}\nWe divide the problem of meta-RL into \nthree sub-tasks, and take on probabilistic inference via variational \nEM learning. We thus present CASTER, a novel meta-RL method that learns dual agents with a task encoder for task inference. \nCASTER \nperforms efficient task-exploration via a curiosity-driven \nexploration policy, from which the collected experiences are exploited by \na context-aware task encoder. The encoder is equipped with the capacity for relational reasoning, with which the action policy adapts \nto complete the current task. Through extensive experiments, we show the superiority \nof CASTER over prior methods in sample efficiency, and empirically reveal that \nthe learned exploration strategy efficiently acquires task-informative experiences with randomized behavior, which effectively helps mitigate meta-overfitting.\n\\section{Experiments}\nIn this section, we demonstrate the efficacy of our design and \nthe behavior of our method, termed CASTER (shorthand for \\textit{Context-Aware task encoder with Self-supervised Task ExploRation for efficient meta-reinforcement learning}), through a series of experiments.\nWe first introduce our experimental setup in \nSec.~\\ref{exp:setup}. In Sec.~\\ref{exp:sota}, we evaluate our CASTER against three meta-RL algorithms in terms of sample efficiency. We then compare the behavior of CASTER with PEARL \nregarding overfitting and exploration in Sec.~\\ref{exp:und}. Finally, in Sec.~\\ref{exp:abl}, we conduct ablation study on our design of the exploration strategy and the task encoder.\n\n\\subsection{Experiemtal Setup}\\label{exp:setup}\nWe evaluate our method on four benchmarks proposed \nin \\cite{rothfuss2018promp}\\footnote{Two experiments ``cheetah-forward-backward'' and ``ant-forward-backward'' are not used because they only include two tasks (the goal of going forward and backward), and do not match the meta-learning assumption that there is a task distribution from which we sample the meta-train task set and a held-out meta-test task set. Such benchmark does not provide convincing evidence for the efficacy of meta-learning algorithms.} and an environment introduced in \\cite{rakelly2019efficient} for ablation. They are implemented on the OpenAI Gym \\cite{brockman2016openai} with \nthe MuJoCo simulator \\cite{todorov2012mujoco}. All of the experiments characterize locomotion tasks, \nwhich may vary either in the reward function or the transition function. We briefly describe the environments as follows.\n\\begin{itemize}[leftmargin=4mm]\n  \\item \\textbf{Half-cheetah-velocity.} Each task requires the agent to reach a different target velocity.\n  \\item \\textbf{Ant-goal.} Each task requires the agent to reach a goal location on a 2D plane .\n  \\item \\textbf{Humanoid-direction.} Each task requires the agent to keep high velocity without falling off in a\n  specified direction.\n  \\item \\textbf{Walker-random-params.} Each task requires the agent to keep high velocity without falling\n  off in different system configurations.\n  \\item \\textbf{Point-robot.} Each task requires a point-mass robot to navigate to a different goal location on a 2D plane.\n\\end{itemize}\nWe adopt the following evaluation protocol throughout this section: first, the estimated per-episode performance on each task is averaged over (at least) three trials with different random seeds; second, the test-task performance is evaluated on the held-out meta-test tasks and is an average of the estimated\nper-episode performance over all tasks with at least three trials. More details about the experiments can be found at \\href{https:\/\/github.com\/HazekiahWon\/CASTER}{our Github repository}.\n\n\n\n\\subsection{Performance}\\label{exp:sota}\nIn this section, we compare with three meta-RL algorithms that are the representatives \nof the three lines of works mentioned in Sec.~\\ref{sec:meta-rl}: 1) inference-based method, PEARL~\\cite{rakelly2019efficient}; 2) optimization-based method, ProMP~\\cite{rothfuss2018promp}; 3) black-box method, $\\text{RL}^2$~\\cite{rl-squared}.\n\nFor those baselines, we reproduce the experimental results via their officially released code, following their proposed meta-training \nand testing pipelines.\nWe note that prior works~\\cite{rl-squared,rothfuss2018promp} are not designed to optimize for sample efficiency but we keep their default hyperparameter settings in order to reproduce their results. \n\n\n\\begin{figure*}[th]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figs\/overf\/overfit_curve_final.png}\n\t\\caption{\\small\\textbf{Overfitting in off-policy meta-RL.} Each column in the plot corresponds to a different environment. We pick three environments most prone to meta-overfitting, i.e., `cheetah-vel', `ant-goal' and `point-robot' from left to right. }\\vspace{-3mm}\n\t\\label{fig:meta-over}\n\\end{figure*}\n\n\nTo demonstrate the training efficiency and testing efficiency, we plot the test-task performance as a function of the number of samples. Fig.~\\ref{fig:sota} shows the comparison results on the training efficiency. Here the x-axis indicates the number of interactions \nwith the environment used to collect buffer data. At each x-tick, we evaluate the test-task performance with \\textbf{2} episodes for all methods. \nWhile for testing efficiency, the x-axis refers to the number of adaptation \nepisodes used by different methods to perform policy adaptation (or task inference). \n\nWe can see CASTER outperforms other methods by a sizable margin: CASTER achieves the same status of performance with much fewer environmental interactions (up to 400\\%) while being able to reach higher performance (up to $300\\%$). \nPEARL and CASTER incorporate off-policy learning and naturally enjoy an advantage in training sample efficiency.\nOur CASTER achieves better performances due to two reasons. On the one hand, efficient exploration potentially offers CASTER richer information within limited episodes (two in Fig.~\\ref{fig:sota}), and pushes higher the upper bound of accurate task inference. On the other hand, context-aware relational reasoning extracts information effectively from the task experiences, and thus improves the ability of task inference.\n\nTesting efficiency is shown in Fig.~\\ref{fig:test-eff}, and CASTER still stands out against other methods. \nNotably, CASTER achieves large performance boost with the first several episodes, indicating that the learned exploration policy is critical in task inference. By contrast, other models either fail to improve from zero-shot performance (e.g. flat lines of $\\text{RL}^2$, PEARL in \\textit{humanoid-dir}, ProMP in \\textit{walker-rand-params}) via exploration or have unstable performance that drops after improvement (e.g., PEARL in \\textit{cheetah-vel} and \\textit{ant-goal}). \n\n\n\\subsection{Understanding CASTER's Behavior}\\label{exp:und}\n\n\\subsubsection{Meta-overfitting}\nAs mentioned in \\ref{sec:exp}, meta-overfitting arises due to train-test mismatch, an issue particular for \nmeta-RL methods that incorporate off-policy learning. \nWe compare the test-task performance obtained with different adaptation data distribution, i.e., the off-policy buffer data v.s. the on-policy exploration data, to see the performance gap when the data distribution shifts. We pick three environments that we find most prone to train-test mismatch, i.e., \\textit{cheetah-vel}, \\textit{ant-goal}, \\textit{point-robot}. Note that in the experiments, the number of transitions in the off-policy data equals the number of transitions in two episodes of the on-policy data.\n\nFig.~\\ref{fig:meta-over} shows the results on the three environments\\footnote{The \\textit{ant-goal} curve reproduced with PEARL's public repo has a discrepancy from the result reported in~\\cite{rakelly2019efficient}, which we suspect to be the train-task performance with off-policy data. We recommend the reader to check with the publicly available code.}. CASTER takes a large leap \ntowards bridging the gap between train and test sample distribution, reducing up to $75\\%$ performance \ndrop as compared to PEARL. This can be credited to better stochasity inherent in the explorer's episodes (Sec.~\\ref{sec:vis-exp}) and better task inference of CASTER. The stochasity in the online rollouts brings its distribution closer to the off-policy data distribution, since at meta-train time data is randomly sampled from the buffer. Task inference that is aware of relations between samples better extracts the information in the few trajectories, which also narrow the gap by improving task inference.\n\n\\subsubsection{The Learned Exploration Strategy}\\label{sec:vis-exp}\nWe investigate the exploration process by visualizing the histogram \nof the rewards in the collected trajectories. \nWe take three consecutive trajectories rolled out by the explorer on \ntwo benchmarks : 1) the reward-varying environment \\textit{ant-goal}, 2) the transition-varying environment \\textit{walker-rand-params}. \nWe compare with PEARL \nsince it also resorts to posterior sampling for exploration~\\cite{strens2000bayesian,osband2013more,osband2016deep}.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{figs\/vis\/ant2.png}\n  \\caption{\\small\\textbf{Learned exploration behavior in \\textit{ant-goal}.} The reward histogram consists of 3 consecutive trajectories. \n  Left: CASTER and Right: PEARL. Different color denotes different trajectories.}\\vspace{-3mm}\n  \\label{fig:vis-rew}\n\\end{figure}\n\nIn Fig.~\\ref{fig:vis-rew}, it is shown that CASTER's exploration spans a large reward region of $[-25,0]$, while PEARL ranges from $-7$ to $-3$. This is consistent with our goal to increase the coverage of task experiences within few episodes, which potentially increases the chances to reach the goal and enables the task encoder to reason with both high rewards and low rewards, i.e., what region of state space might be the goal and might not. By contrast, PEARL tends to exploit the higher extreme values at the risk of following a wrong direction (e.g., in the second plot, PEARL proceeds to explore in a narrow region of relatively lower reward than the preceding episode).\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{figs\/vis\/walker2.png}\n  \\caption{\\small\\textbf{Learned exploration behavior in \\textit{walker-rand-params}.} }\n  \\label{fig:vis-trans}\n\\end{figure}\n\nIn Fig.~\\ref{fig:vis-trans}, the two methods are evaluated in the transition-varying environment \\textit{ant-goal} in which higher rewards do not necessarily carry the information of the system parameters. In this case, PEARL clings to higher reward regions as expected, while CASTER favors lower reward region. Supported by the results in Fig.~\\ref{fig:sota},\\ref{fig:test-eff}, PEARL's exploration is sub-optimal. CASTER performs better because the exploration is powered by information gain, and the explorer discovers task-discriminative experiences that happen to be around lower reward region. \n\n\n\\subsection{Ablations}\\label{exp:abl}\nIn this section, we investigate our design choices of the exploration strategy and the task encoder via a set of ablative experiments on the \\textit{point-robot} environment.\nWe also provide the test-task performance \nwith off-policy data (adaptation data from a buffer collected beforehand) to eliminate the effect of insufficient data source for task inference that induces train-test mismatch.\n\\subsubsection{The Task Encoder}\\label{exp:inf}\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{figs\/abl\/abl_inf_final.png}\n  \\caption{\\small\\textbf{Ablation study of the task encoder. We show the test-task performance with on-policy data (left) and off-policy data (right).}}\n  \\label{fig:inf}\n  \\vspace{-3mm}\n\\end{figure}\n\nPEARL~\\cite{rakelly2019efficient} builds its task encoder by stacking a Gaussian product (Gp) aggregator \non top of an MLP, while we propose to combine self-attention \nwith a learned weighted-sum aggregator for better relational modeling. \nWe examine the following models: 1) \\textit{Enc(Gp)-Exp(None)} that uses the Gp task encoder, 3) \\textit{Enc(WS)-Exp(None)} that uses a learned weighted-sum aggregator without self-attention for graph message-passing,  3) \\textit{Enc(GNN)-Exp(None)} that uses the proposed GNN encoder, 4) and \\textit{Enc(GNN)-Exp(RS)} the proposed model. Note that for the first three baselines, the explorer is disabled to eliminate the impact of the exploration policy.\n\nIn Fig.~\\ref{fig:inf}, all models tend to converge to similar asymptotic performance with off-policy data. However \\textit{Enc(WS)-Exp(None)} learns much slower than its counterparts, while \\textit{Enc(Gp)-Exp(None)} seems a reasonable design w.r.t. the off-policy performance. \nWe conjecture that it can be hard for \\textit{Enc(Gp)-Exp(None)} to learn a weighting strategy as Gp, in which the weights are determined by the relative importance of a sample w.r.t. the whole pool of samples (${\\sigma_i^2}\/{\\sum\\sigma_j^2}$), and \\textit{Enc(Gp)-Exp(None)} has no access to other samples when computing the weight of each sample. By contrast, \\textit{Enc(GNN)-Exp(None)} provides more flexibility for the final weighted average pooling, by incorporating the interactions between samples. \nSuch relational modeling enables it to extract more task statistics within the few episodes, superior to Gp in the on-policy performance.\n\n\\subsubsection{The Exploration Strategy}\\label{exp:exp}\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{figs\/abl\/abl_exp_final.png}\n  \\caption{\\small\\textbf{Ablation study of exploration. We show the test-task performance with on-policy data (left) and off-policy data (right).}}\n  \\label{fig:exp}\n  \\vspace{-5mm}\n\\end{figure}\n\n\nWe aim to show the efficacy of the proposed reward shaping for task-exploration. The baseline models are: 1) \\textit{Enc(Gp)-Exp(None)} that uses no exploration policy, \n2) \\textit{Enc(Gp)-Exp(Rand)} that uses a (uniformly) random explorer, 3) \\textit{Enc(Gp)-Exp(RS)} that uses an explorer guided by the proposed reward shaping. Here the Gp task encoder is used by default.\n\nAs shown in Fig.~\\ref{fig:exp}, all models perform equally in terms of \nasymptotic performance with off-policy data, \nsince sufficient coverage of experiences are inherent in the data randomly sampled from buffers. This demonstrates the significance of improving coverage of task experiences for task inference.\n\nFor the on-policy performance, we can see \\textit{Enc(Gp)-Exp(Rand)} suffers from significant overfitting, \nas the coverage brought by the randomness doesn't suffice to benefit it within few episodes. \\textit{Enc(Gp)-Exp(RS)} performs much better than \n\\textit{Enc(Gp)-Exp(Rand)} because high rewards is an informative guidance in a reward-varying environment.\nOur approach combines the merits of both broad coverage over task experiences and informative guidance regarding task relevance, and hence achieves the best performance.\n\n\\section{Introduction}\nModern reinforcement learning has achieved great successes in solving certain complex tasks by utilizing deep neural networks, which can even be trained from scratch~\\cite{alphazero, poker, alphastar}. Such successes, \nhowever, require a large amount of training experiences for new tasks. In contrast, human \nlearners are able to exploit past experiences when facing a novel problem and quickly learn \nskills for a related task~\\cite{lake2017building}. To achieve such fast adaptation is a critical step \ntowards building a general AI agent capable of solving multiple tasks in real-world environments. \n\nA principled way to tackling the problem of efficient adaptation is the meta \nlearning framework~\\cite{finn2017model}, which aims to capture shared knowledge across \ntasks and hence enables an agent to learn a similar task using only a few experiences. \nIn the reinforcement learning setting, however, as the learning agent also needs to explore \nin each novel task, it is particularly challenging to design an efficient meta-RL algorithm.       \nA majority of prior works adopt on-policy RL algorithms~\\cite{rl-squared,finn2017model,gupta2018meta,rothfuss2018promp}, \nwhich are data-inefficient during \nmeta-training~\\cite{rakelly2019efficient}.\nTo remedy this, \\citet{rakelly2019efficient} propose an alternative strategy, PEARL, that \nrelies on off-policy RL methods to achieve sample efficiency in meta-training. \nBy introducing a latent variable to represent a task, their method decomposes the problem into online task inference and task-conditioned policy learning that uses experiences from a replay buffer (i.e., off-policy learning). \n\nNevertheless, such an off-policy strategy has several limitations during meta-test stage, particularly for the few-shot setting. First, it ignores the role of exploration in the task inference (cf. meta-episodes in \\cite{rl-squared}), which is critical in efficient adaptation as the exploration is responsible for collecting informative experiences within the few episodes to infer tasks. \nIn addition, the agent has to explore in an online fashion for \ntask inference during meta-test, which typically has a different sample distribution from the replay buffer that provides adaptation data at meta-train stage. PEARL partially alleviates this problem of train-test mismatch~\\cite{vinyals2016matching} by adopting a replay buffer of recently collected data. Such an in-between strategy~\\cite{rakelly2019efficient}, however, still leads to severe  ``meta-overfitting'', particularly for online task inference (cf. Sec~\\ref{sec:exp}\\&\\ref{exp:und}).    \nFurthermore, the adaptation data acquired by exploration are simply aggregated by a weighted average pooling for task inference. As the experience samples are not iid, such aggregation fails to capture their dependency relations, which is informative for task inference.\n\nIn this work, we propose a context-aware task reasoning strategy for meta-reinforcement learning to address the aforementioned limitations. \nAdopting a latent representation for tasks, we formulate the meta-learning as a POMDP, which learns an approximate posterior distribution of the latent task variable jointly with a task-dependent action policy that maximizes the expected return for each task. \nOur main focus is to develop an adaptive task inference strategy that is able to effectively map from few-shot experiences of a task to its representation without suffering from the meta-overfitting.  \n\nTo achieve this, we design a novel task inference network that consists of an exploration policy network and a structured task encoder shared by all tasks. Learning an exploration policy allows us to explore a task environment more efficiently and to introduce regularization to bridge the gap between meta-train and meta-test, leading to better generalization. The structured task encoder is built on a context-aware graph network, which is input permutation-invariant and size-agnostic in order to cope with variable number of exploration episodes. Our graph network encodes task-related dependency in experience data samples, capable of capturing complex task statistics from exploration to achieve more data-efficient task inference. \n\nTo train our meta-learner, we develop a variational EM formulation that alternately optimizes the exploration \npolicy network that collects experiences for a task, the context-aware graph network that performs \ntask inference based on the collected task data, and an action policy network that aims to complete \ntasks towards maximum rewards given the inferred task information. \nOur meta-learning objective, formulated under the POMDP framework, is composed of two state-action value functions for the two respective policies. The state-action value for the exploration policy is guided by a reward shaping strategy that encourages the policy to collect task-informative experiences, and is learned with Soft Actor Critic (SAC)~\\cite{haarnoja2018soft} for randomized behavior to narrow the gap between online rollouts and offline buffers.\nIn essence, our method decomposes the meta-RL problem into three sub-tasks, task-exploration, task-inference and task-fulfillment, in which we learn two separate policies for different purposes and a task encoder for task inference.\n\nWe evaluate our meta-reinforcement learning framework on four benchmarks with a diverse task \ndistributions, in which our approach outperforms prior works by improving training efficiency (up to 400x) and testing efficiency with better asymptotic performance (up to 300\\%) while effectively mitigating the meta-overfitting by a large margin (up to 75\\%). \nOur contributions can be summarized as follows:\n\\begin{itemize}[leftmargin=4mm]\n \n \n  \\item We propose a sample-efficient meta-RL algorithm that achieves the state-of-the-art performances on multiple benchmarks with diverse task distributions. \n  \\item We present a dual-agents design with a reward shaping strategy that explicitly optimizes for the ability for task exploration and mitigates meta-overfitting.\n  \\item We develop a context-aware graph network for task inference that models \n  the dependency relations between experience data in order to efficiently infer task posterior.\n\\end{itemize}\n\\section{Variational Meta-Reinforcement Learning}\\label{sec:ps}\nWe aim to address the fast adaptation problem in reinforcement learning, which allows the learned agent to explore a new task (up to a budget) and adapts its policy to the new task. \nWe adopt the meta-learning framework to enable fast learning, incorporating a prior of past experiences in structurally similar tasks. \n\nFormally, we assume a task distribution $p(\\mathcal{T})$ from \nwhich multiple tasks are i.i.d sampled for meta-training and testing. \nAny task $\\mathcal{T} \\sim p(\\mathcal{T})$ can naturally be defined by its MDP $(S,A,\\rho(s_0),P, \\gamma, r)$ in RL, where $S$ is the state space, $A$ is the action space, $\\rho(s_0)$ is the distribution of the initial states, $P(s^\\prime|s,a)$ is the transition distribution, $\\gamma$ is the discount in rewards which we will omit for clarity, and $r(s,a)$ is the reward function. \nHere the task distribution is typically defined by the distribution of the \nreward function and the transition function, which can vary due to some underlying parameters. \nWe introduce a latent variable $z\\sim p(z)=\\mathcal{N}(0,1)$ to represent the cause of task variations, and formulate the meta RL problem as a POMDP in which $z$ serves as the unobserved part of the state space. \n\n\n\n\nWe first introduce two policies, including an action policy $\\pi$ that aims to maximize expected reward under a given task $z$ and an exploration policy $\\xi$ for generating task-informative samples by interacting with the environment. We then adopt the off-policy Actor-Critic framework~\\cite{dpg}, and learn two Q-functions $Q^\\pi,Q^\\xi$ for the action and exploration policy respectively. To achieve this, we define the following learning objective, which maximizes the expected log likelihood of returns under two policies (we omit the expectation over tasks for clarity): \n\\begin{align}\n\\underset{\n\t\\substack{\\beta(s,a)\\\\\n\t\tE(s^\\prime,r|\\cdot)}\n}{\\mathbb{E}}\\left[\\log P(R=y^\\pi|s,a)\\right]\n+\\underset{\n\t\\substack{\\rho(s,a)\\\\\n\t\tE(s^\\prime|\\cdot),r^\\xi\n\t}\n}{\\mathbb{E}}\\left[\\log P(R=y^\\xi|s,a)\\right]\\label{eq:obj}\n\\end{align}\nwhere $\\beta,\\rho$ denotes the $(s,a)$ distribution visited by the two behavior policies for $\\pi,\\xi$, $E$ denotes the environment, and $r^\\xi$ is the reshaped reward for $\\xi$ (Sec.~\\ref{sec:exp}). Typically, $\\beta,\\rho$ is approximated by the experience replay buffers of $\\pi,\\xi$, denoted as $\\mathcal{B},\\mathcal{X}$ respectively~\\cite{mnih2013playing,ddpg,haarnoja2018soft}. We will slightly abuse the notation $\\beta,\\rho$ to denote the joint distribution of $(s,a,s^\\prime,r )$ for brievity. \n\nHere $P(R|s,a)$ denotes the likelihood of return $R$ given a state-action pair, and $y=r+\\gamma\\max_{a^\\prime}Q(s^\\prime, a^\\prime)$ is the target value estimated using the optimal Bellman Equation. Note that by assuming $P(R|s,a)=\\mathcal{N}(R;\\mu,\\sigma)$ where $\\mu=Q(s,a)$ and $\\sigma$ a constant, then maximizing the likelihood $P(R=y|s,a)$ is equivalent to the Temporal Difference learning objective~\\cite{tesauro1995temporal,konda2000actor,mnih2013playing}:\n\\begin{align}\n  &\\max_Q \\log P(R=y|s,a) =\\max_Q \\log \\left[\\alpha \\exp \\left(-\\beta \\left(Q(s,a)-y\\right)^2\\right)\\right]\\nonumber\\\\\n  &\\Leftrightarrow \\min_Q \\left(Q(s,a)-y\\right)^2\\label{eq:likelihood}\n\\end{align}\nwhere $\\sigma$ leads to the constants $\\alpha,\\beta>0$.\n\nWe utilize the variational EM learning strategy~\\cite{bishop2006pattern} to maximize the objective. Denoting the sampled experiences as $x$, we introduce a variational distribution $q(z|x)$ to approximate the intractable posterior $p(z|x)$, in which we alternate between optimizing for $q(z|x)$ and for other parameterized functions $Q^\\pi, Q^\\xi, \\pi, \\xi$.\nWe refer to $q(z|x)$ as the task encoder in our model.\n\nFor E-step, we maximize a variational lower bound derived from \\eqref{eq:obj} to find the optimal $q(z|x)$. In order to decouple the action and exploration policy, we also introduce an auxiliary experience distribution $p(x)$ and minimize the following free energy:  \n\\begin{align}\n  &\\underset{\n    \\substack{p(x),\\beta(s,a,s^\\prime,r)\\\\\n              z\\sim q(z|x)\n            }\n  }{\\mathbb{E}}\n \n  \\left[\\left(Q^\\pi(s,a,z)-y^\\pi\\right)^2+\\text{KL}(q(z|x)\\|p(z))\\right]+c                        \n\\end{align}\nwhere in the first term the TD Error derives from $\\log \\int_Z P(R=y^\\pi|s,a,z)p(z)dz$, and the constant term $c$ comes from the second term in \\eqref{eq:obj} which is irrelevant to $z$.   \nFor the auxiliary distribution $p(x)$, we adopt the exploration policy's sample distribution $\\rho(\\cdot)$, as $\\xi$ aims to sample task-informative experiences for task inference. Hence the E-step objective $\\mathcal{J}_e(q)$ has the following form:\n\\begin{align}\n \\underset{\n    \\substack{x\\in \\mathcal{X},z\\sim q(z|x)\\\\\n              (s,a,s^\\prime,r)\\in \\mathcal{B}\\\\\n             }\n  }{\\mathbb{E}} \\left[\\left(Q^\\pi(s,a,z)-y^\\pi\\right)^2+\\text{KL}(q(z|x)\\|p(z))\\right]\\label{eq:approx2}\n \n\\end{align}\n\nFor M-step, given $q(z|x)$, we minimize the following free-energy objective $\\mathcal{J}_m(\\pi, \\xi, Q^\\pi, Q^\\xi)$ derived from \\eqref{eq:obj} :\n\\begin{align}\n  \\underset{\\substack{\n    x\\in\\mathcal{X}\\\\\n    z\\sim q(z|x)\\\\\n    (s,a,s^\\prime)\\sim\\mathcal{B}\n  }}{\\mathbb{E}}\\left[\\left(Q^\\pi(s,a,z)-y^\\pi\\right)^2\\right]\n  +\n  \\underset{\\substack{\n    x\\in\\mathcal{X}\\\\\n    z\\sim q(z|x)\\\\\n    (s,a,s^\\prime)\\sim\\mathcal{X}\n  }}{\\mathbb{E}}\n  \\left[\\left(Q^\\xi(s,a,z)-y^\\xi\\right)^2\\right]\n\\end{align}\nBased on this variational EM algorithm, our approach learns dual agents for task-exploration and task-fulfillment respectively. The meta-training process interleaves the data collection process with the alternating EM optimization process, as shown in Alg.~\\ref{algo}. \n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{figs\/overview4.png}\n\t\\caption{\\small \\textbf{Context-aware task reasoning for RL adaptation}. We separate the task into \n\t\ttask-exploration, task inference and task-fulfillment. The explorer interacts with the environment to collect experiences for the task encoder to update the belif of task. After an iterative process of $K-1$ rounds, the task encoder takes all the collections and gives the final task hypothesis to adapt the actor.}\\label{fig:overview}\n\\end{figure}\n\\IncMargin{1em}\n\\begin{algorithm}[t]\n\t\\SetAlgoLined\n\n\n\n\n\t\\SetKwInOut{Input}{input}\\SetKwInOut{Output}{output}\n\t\\Input{Meta-train tasks $T=\\{\\mathcal{T}_i\\}_{1:M},\\sim p(\\mathcal{T})$, training steps $N$, E-step $e$, M-step $m$, meta-batch size $B$, task sample size $C$ for buffer update, learning rate $\\lambda$.}\n\t\\Output{ $q(z|x),\\pi,\\xi,Q^\\pi, Q^\\xi$.}\n\tInitialize $q,\\pi,\\xi,Q^\\pi, Q^\\xi$ with network parameters $\\theta,\\phi$, buffer $\\mathcal{B}$ for $\\pi$, buffer $\\mathcal{X}$ for $\\xi$.\\\\\n\t\\For {each task $\\mathcal{T}_i\\in T$}{\n\t\tcollect episodes into buffers $\\mathcal{B},\\mathcal{X}$ with the policies $\\pi,\\xi$ respectively.\n\t\n\t}\n\t\\While {not converged}{\n\t\trandomly sample $C$ tasks from $T$ to form a set $T_C$.\\\\\n\t\t\\For {task $\\mathcal{T}_i \\in T_C$}{\n\t\t\tadd episodes into buffers $\\mathcal{B},\\mathcal{X}$ with the policies.\n\t\t\n\t\t}\n\t\t\\For {each step in $N$ training steps}{\n\t\t\trandomly sample $B$ tasks from $T$ to form a set $T_B$.\\\\\n\n\t\t\t\\eIf{\n\t\t\t\t$\\textrm{step} \\mod (e+m)<e$\n\t\t\t}{\n\t\t\t\t\n\t\t\t\tget the inputs (E-step):\\\\ \n\t\t\t\t$\\left\\{x_{q}\\right\\}_{1:B}\\sim\\mathcal{X}, \\left\\{(s,a,s^\\prime,r)_\\pi\\right\\}_{1:B}\\sim\\mathcal{B}$.\\\\\n\t\t\t\tcompute the loss: $\\mathcal{L}(q;\\theta)\\leftarrow\\mathcal{J}_e(q)$.\\\\\n\t\t\t\tupdate the model: $\\theta\\leftarrow \\theta-\\lambda \\nabla_\\theta\\mathcal{L}(q;\\theta)$\n\t\t\t}{\n\t\t\t\tget the inputs (M-step):\n\t\t\t\t$\\left\\{x_{q}\\right\\}_{1:B}\\sim\\mathcal{X}$, $\\left\\{(s,a,s^\\prime,r)_\\pi\\right\\}_{1:B}\\sim\\mathcal{B}$, $\\left\\{(s,a,s^\\prime,r)_\\pi\\right\\}_{1:B}\\sim\\mathcal{X}$.\\\\\n\t\t\t\tcompute the loss: $\\mathcal{L}(\\pi,\\xi,Q^\\pi, Q^\\xi;\\phi)\\leftarrow\\mathcal{J}_m(\\pi,\\xi,Q^\\pi, Q^\\xi)$.\\\\\n\t\t\t\tupdate the model: $\\phi\\leftarrow \\phi-\\lambda \\nabla_\\phi\\mathcal{L}(\\pi,\\xi,Q^\\pi, Q^\\xi;\\phi)$\n\t\t\t}\n\t\t\t\n\t\t}\n\t}\n\t\\caption{The Meta-training algorithm}\\label{algo}\n\\end{algorithm} \nIn contrast to \\cite{taskinf,rakelly2019efficient}, our variational EM formulation derives from a unified learning objective and enables fast learning of dual agents with different roles. The explorer learns the ability of \\textbf{task-exploration} that targets at efficient exploration to (actively) collect sufficient task information for task inference,\nwhile the actor learns for \\textbf{task-fulfillment} that accomplishes the task towards high \nrewards. Both the explorer and the actor are task-conditioned, so that they are able to perform temporally-coherent exploration for different goals, given the current task \nhypothesis \\cite{gupta2018meta}. \n\nWith such a formulation, we need to answer the following questions to achieve\nefficient and effective task inference to solve the meta-RL problem:\n\\begin{itemize}[leftmargin=4mm]\n  \\item How to guide the explorer towards active exploration for sufficient task information that is crucial in task inference with few experiences?\n  \\item How to achieve effective task inference that captures the relationships between experiences and reduce task uncertainty?\n  \\item Since we incorporate off-policy learning algorithms, how to mitigate the train-test mismatch issue due to the use of replay buffer \n   v.s. online rollouts during testing? \n\\end{itemize}\nWe will address the above three questions in Sec.~\\ref{sec:app}, which introduces our task inference strategy in detail.\n\n\\paragraph{Meta-test} At the meta-test time, we apply the same data collection procedure as in the meta-training stage.\nFor each task, the explorer $\\xi(a|s,z)$ first samples a hypothesis $z$ from the posterior (initialized as $\\mathcal{N}(0,1)$), and then explores optimally according to the hypothesis. The collected experiences $x$ during the exploration are used by the task encoder $q(z|x)$ to update the posterior of $z$. This process iterates until the explorer uses up the maximum number of rollouts. All the experiences are fed to the task encoder to produce a final posterior representing the belief of MDPs. We then sample from the posterior to generate a task-conditioned action policy $\\pi(a|s,z)$, which interacts with the environment in an attempt to fulfill \nthe task. This process is shown in Fig.~\\ref{fig:overview}.\n\n\\section{Related work}\n\\subsection{Meta-reinforcement Learning}\\label{sec:meta-rl}\nPrior meta reinforcement learning methods can be categorized into the following three  \nlines of work. \n\nThe first line of work adopts a learning-to-learn strategy~\\cite{finn2017model, stadie2018some,gupta2018meta,rothfuss2018promp,xu2018learning}. In particular, MAML~\\cite{finn2017model} meta-learns a model initialization from which to adapt the parameters \nwith policy gradient methods. To tackle issues in computing second-order derivatives for MAML, ProMP~\\cite{rothfuss2018promp} \nfurther proposes a low-variance curvature estimator to perform \nproximal policy optimization that bounds the policy chang\nRecently, MAESN~\\cite{gupta2018meta} improves MAML with more temporally coherent exploration by injecting structured noise\nvia a latent variable conditioned policy, and enables fast learning of exploration strategies.\n\nThe second category of approaches uses recurrent or recursive models to directly meta-learn a \nfunction that takes experiences as input and generates a policy for an agent~\\cite{mishra2017simple,reiforcelearn,rl-squared}. \nAmong them, $\\text{RL}^2$~\\cite{rl-squared} trains a recurrent agent with on-policy \nmeta-episodes that comprise a sequence of exploration and task-fulfillment episodes, \nwhich aims to maximize the expected return of the task-fulfillment episode.\nTheir method essentially learns to extract task information from the first few rounds of exploration, encoded in the latent states of its RNN, and complete the task in the final episode based on the known task information (latent states). \n\nThe first two groups of work often learn a single policy to perform exploration for policy adaptation and task-fulfillment, which overloads the agent with two distinct objectives. By contrast, we learn two separate policies, one focusing on exploration for policy adaptation and the other for task-fulfillment.\n\n\nIn the third line of work, \\citet{rakelly2019efficient} and \\citet{taskinf} propose to first infer task with task experiences and adapts the agent according to the task knowledge. Framing meta-RL as a POMDP problem \nwith probabilistic inference, \\citet{taskinf} formulate task inference as belief \nstate in POMDP, and learn the inference procedure in a supervised manner with privileged information \nas ground truth. \nPEARL \\cite{rakelly2019efficient} learns to infer tasks in an unsupervised manner by introducing an extra  reconstruction term and incorporates off-policy learning to improve sample efficiency. \nHowever, \nboth approaches ignore the role of exploration in task inference, and PEARL suffers from train-test mismatch in the data distribution for task inference.\n\nBy contrast, we propose to further disentangle meta-RL into task-exploration, task-inference and task-fulfillment, and introduce an exploration policy within a variational EM formulation. Our method enhances the inference procedure via active task exploration and effective task inference, achieves data-efficiency both in meta-train and meta-test, and mitigates meta-overfitting.\n\n\\subsection{Relational Modeling on Sets}\n\nThe problem of inference on a set of samples has been widely explored in literature~\\cite{battaglia2018relational,gilmer2017neural,li2015gated, bruna2013spectral,kipf2016semi,hamilton2017inductive}, and here we focus on those approaches that learn a permutation-invariant and input size-agnostic model.\n \nDeepSets~\\cite{zaheer2017deep} proposes a general design principles for permutation invariant \nfunctions on sets. As an instantiation, Sets2Sets~\\cite{vinyals2015order}\nencodes a set using an attention mechanism, which retrieves vectors in a manner immune to the shuffle\nof memory and accepts variable size of inputs. \nHowever, they typically do not consider relations among input elements, which is critical for modeling short experiences for task inference.\n \nSelf-attention module~\\cite{internet,vain,santoro2017simple,wang2018non} is commonly used to model pairwise relationships between entities of a set.\n\nAmong them, Nonlocal Neural Networks~\\cite{wang2018non} are capable of capturing global context using the scaled dot product attention \\cite{vaswani2017attention}. Along this direction, \ngraph networks~\\cite{battaglia2018relational,gilmer2017neural} provide a flexible framework that can model arbitrary relationships among entities and is invariant to graph isomorphism. Self-attention on sets, \na special case of graphs, can naturally be assimilated into the graph network framework~\\cite{zhang2019latentgnn,battaglia2018relational,velivckovic2017graph}.\nRecently, LatentGNN~\\cite{zhang2019latentgnn} develops a novel \nundirected graph model to encode non-local relationships through a shared latent space \nand admits efficient message passing due to its sparse connection. In this work, the design of our graph-based inference network is inspired by LatentGNN and DeepSets, aiming to efficiently model the dependency relationship\nin experience data.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Results}\n\nThe holographic correspondence has revolutionized the way we think about both quantum field theory and string theory. Moreover, it has indicated that the two are intimately related, and that string theory can be viewed as the dynamics of sources of quantum field theory (QFT), \\cite{LV,string,sslee}, namely  a form of quantum renormalization group.\n\nA particular item of the QFT tool box, renormalization group (RG) flows have an elegant description in the string theory\/gravity language. They correspond to bulk solutions of the equations of motion with appropriate boundary conditions. There is a way of writing the bulk equations using the Hamilton-Jacobi formalism, so that they look formally similar to RG equations, \\cite{deboer,papa,bianchi}, a fact that has been exploited in numerous situations. There have been various takes on the form of the RG\/coarse graining procedure in the holographic case, \\cite{holorg}, including the Wilsonian approach to IR physics \\cite{cgkkm}.\n\nIt is not yet known how to generalize the Hamilton-Jacobi formalism to cases where we are discussing non-trivial states in QFT, namely states with non-trivial temperature and finite density.\nHowever, even in that case there are other techniques that allow the calculation of effective potentials, \\cite{eff}, and effective actions, \\cite{nitti}, along with  the related RG equations.\n\nAn important set of flows in QFT are generated by nearly marginal (marginally relevant) operators. This is the example of QCD, as well as other asymptotically free QFTs at zero or finite density (like the Kondo problem).\nThere have been essentially very few investigations of such flows in the literature, \\cite{gkn,hartnoll}, because they are not generic in top-down models, and they are explicitly avoided (because of their computational difficulty) in bottom up models. The only examples that have been studied do not seem to be exhaustive. They include, asymptotically-soft AdS minima of the potential, relevant for holographic models of YM, \\cite{gkn}, that were argued recently to also be relevant in the cosmology of primordial inflation, \\cite{cosmo}, and marginally relevant asymptotics of holographic Lifshitz fixed points, \\cite{hartnoll}.\n\nThe purpose of this note is to analyze marginally relevant holographic flows in their full generality as they can arise in different contexts, under distinct realizations. They will also be compared to their field theory counterparts.\n\nMoreover, the holographic approach allows the incorporation of non-perturbative corrections to the $\\beta$-functions as well as the flow, due to non-trivial vacuum expectation values. We will calculate their form in full generality.\n\n\\bigskip\nOur results are as follows:\n\n\\begin{enumerate}\n\n\\item We review the case of relevant operators, and the superpotential formalism in order to establish notations.\n\n\\item We show that the geometric singularities of a solution correspond to places where the potential $V$ and\/or the superpotential $W$ diverge.\n\n\\item We define the holographic $\\beta$-function, and derive a non-linear first-order equation for it, valid to leading order at strong coupling and large$-N$.\n\n\\item We analyze the case of classically marginal operators whose potential has zero mass term and the first non-trivial term is $\\phi^n$, $n\\geq 3$ in the regime $\\phi\\to 0$.\n    The solutions around the UV fixed point ($\\phi=0$) are derived and indicate that such operators have a perturbative $\\beta$-function that starts at $n-2$ loop order.\n\n\\item We find the leading non-perturbative corrections, that behave exponentially in the inverse of the coupling constant.\n\n\\item We find however that the scalar diverges as it approaches the boundary, and this case seems inconsistent with the original assumptions ($\\phi\\ll 1$).\n\n\\item We also analyze classical marginal operators near a UV fixed point at $\\phi\\to \\pm\\infty$. This case corresponds to asymptotically-soft fixed points advocated for IHQCD,\\cite{gkn}, and appears also in cosmology in Starobinsky-like models, \\cite{staro,cosmo}. We find that they have always a non-trivial one-loop $\\beta$-function, in close analogy to YM in four dimensions. Moreover, the scalar coupling asymptotes properly to the UV fixed point at the boundary of the geometry.\n\n\\item The leading non-perturbative corrections to the $\\beta$-function are evaluated and shown to be similar to those expected in YM.\n\n\\item The multiscalar case is analyzed in a similar spirit. Here there is also a Zamolodchikov metric that enters the bulk description apart from the scalar potential. The perturbative RG flows are constructed near a UV fixed point, and it is shown that the contributions of the Zamolodchikov metric to the $\\beta$-functions appear first at two-loops.\n\n\n\n\\end{enumerate}\n\nWe conclude  with an important open question to which we know the answer only in very special cases : which concrete cases of marginal operators arise from top-down string theory constructions at string tree level?\n\n\n\n\n\n\\section{Generic Perturbations Near the AdS Boundary}\n\nWe will start our study by setting up the general problem of RG flows of a scalar operator. To do this we will have to confine ourselves to Einstein-Dilaton (ED) theory which at the two derivative level\\footnote{Generically this is justified at the strong coupling and large N limit, however adding higher derivatives of the scalar and the metric, does not affect the structure of the $\\beta$ functions in the scaling region.}  after field redefinitions, takes the form:\n\\begin{equation}\n\\label{1}\nS_{2d} =M^{d-1}\\int d^{d+1}x \\sqrt{-g}\n\\biggl\\{R -{1\\over 2} \\partial_{\\mu}\\phi\\partial^{\\mu}\\phi+V(\\phi)\\biggl\\}\n\\end{equation}\nHere $d$ is the space-time dimension of the QFT (living on the boundary), while the space-time dimension of the bulk ED gravity is $d+1$.\n\nThe metric as usual is dual to the stress tensor in the QFT and is always included, as it is always sourced even in the vacuum saddle-point (gravitational) solution\\footnote{The basic rules of the holographic correspondence are detailed in several papers and textbooks, see eg. \\cite{thebook}.}. We also have included a bulk scalar field $\\phi$ dual to a scalar operator in the QFT.\n\nIn this section, there are no restrictions on the form of the potential, apart from the requirement of at least one extremum at $\\phi=\\phi_*$ which yields an $AdS_{d+1}$ geometry as the vacuum solution, with curvature scale $\\ell$ and signals the existence of a CFT:\n\\begin{equation}\nV(\\phi_*)={d(d-1)\\over \\ell^2} + \\cdots\n\\label{2}\n\\end{equation}\nThe general form of the equations of motion obtained from \\eqref{1} is then:\n\\begin{align}\n&\n \\nabla^{\\mu} \\nabla_{\\mu} \\phi + V'(\\phi) = 0, \\label{eomphi}  \\\\\n&\\label{efe}\nR_{\\mu \\nu} - {1 \\over 2} g_{\\mu \\nu } \\left( R - {1 \\over 2 }  \\left( \\partial \\phi \\right) ^2 + V(\\phi) \\right) - {1 \\over 2} \\partial_\\mu \\phi \\partial_\\nu \\phi  = 0.\n\\end{align}\n\n\\bigskip\n\\subsection{The Standard Perturbative Method}\n\\label{stdperturb}\n\\bigskip\n\n\nWe will proceed now to study RG flows in the neighborhood of the $AdS$ fixed point, (\\ref{2}), also known in QFT as the scaling region.\n\nIn the \\emph{conformal coordinate frame} for the metric,  the ansatz that we will be studying for the Poincar\\'e-invariant field theory metric and the dilaton is:\n\\begin{equation}\nds^2=e^{2A(r)}\\left[dr^2+dx_{\\mu}dx^{\\mu}\\right]\\;\\;\\;,\\;\\;\\; \\phi(r)\\,.\n\\label{3}\n\\end{equation}\nwere the radial direction is the holographic direction and will be related to the RG scale as is standard in holography.\n\nThe $AdS_{d+1}$ solution:\n\\begin{equation}\ne^{2A}={\\ell^2\\over r^2}\\;\\;\\;,\\;\\;\\; \\phi=0 \\;\\;\\;,\\;\\;\\;\n\\label{4}\\end{equation}\nthen corresponds to the (UV) CFT, with the boundary at $r=0$. Substituting the ansatz \\eqref{3} into \\eqref{eomphi} and \\eqref{efe}, we obtain the equations for the scale factor and the scalar:\n\\begin{align}\n&(d-1) \\left[ A'(r)^2 - A''(r) \\right] - {1 \\over 2} \\phi'(r)^2 = 0, \\label{eq1} \\\\\n&(d-1) A ''(r) + (d-1)^2 A'(r)^2 - e^{2A(r)} V(\\phi) = 0, \\label{eq2} \\\\\n&e^{-2A(r)} \\left[ \\phi '' (r) + (d-1)A'(r) \\phi'(r) \\right] + {d V(\\phi)  \\over d \\phi }= 0, \\label{eq3}\n\\end{align}\nwhere we have used the convention\n$' \\equiv {d \\over dr}$.\nThese equations capture the solutions with Poincar\\'e invariance in $d$ dimensions.\n\n\nEquations (\\ref{eq1})-(\\ref{eq3}) can be solved perturbatively near the boundary. We choose the UV minimum to be at $\\phi_*=0$ without loss of generality\\footnote{It can be made to vanish by a shift in $\\phi$ that does not affect the other terms in the action.}. We then expand the potential near the minimum to quadratic order\\footnote{The convention chosen here for the mass term of the potential differs from \\cite{aharony}. There it appears with a minus sign.}:\n\\begin{equation}\nV(\\phi) = {d(d-1) \\over \\ell^2} +\\frac12 m^2 \\phi^2+\\cdots\\;\\;\\;,\\;\\;\\; \\phi\\to 0.\n\\label{pot}\\end{equation}\nWorking at next to leading order, we obtain:\n\\begin{align}\n\\label{fi}\\phi(r) = & c_1 r^{\\Delta_+} + c_2 r^{\\Delta_-}+\\cdots\\, , \\, \\, \\Delta_{\\pm} = {d \\pm \\sqrt{d^2 - 4 m^2 \\ell^2} \\over 2}\\, , \\,\\\\\n A(r) =& \\log {\\ell \\over r}  - {c_2^2 \\Delta_- r^{2\\Delta_-} \\over 4(d-1)(1+ 2 \\Delta_-)} - {c_1^2 \\Delta_+ r^{2\\Delta_+} \\over 4(d-1)(1+ 2 \\Delta_+)} - {c_1 c_2 \\Delta_+ \\Delta_- r^d \\over d(d^2-1)}+\\cdots\n \\label{nexto}\n\\end{align}\nwith $c_{1,2}$ real.\n\nWe therefore derive the basic rule of the  $AdS\/CFT$ dictionary, namely that\n\\begin{equation}\nm^2 \\ell^2 = \\Delta_+\\Delta_-=\\Delta \\left( d - \\Delta \\right)\n\\end{equation}\nwhere $\\Delta$ is the scaling dimension of the operator ${\\cal O}$ dual to $\\phi$ in the dual QFT.\n\nThere are several cases to be distinguished depending on the values of the mass:\n\\begin{enumerate}\n\n\\item ${d^2\\over 4}>m^2\\ell^2>0$ corresponds to relevant operators. If $m^2\\ell^2>    {d^2\\over 4} $,  the dimensions $\\Delta_{\\pm}$ are complex (and therefore this case is unacceptable). The scalar violates the BF bound.\n\n    There are two important subcases:\n\\begin{itemize}\n\n\\item     $0<m^2\\ell^2<{d^2\\over 4}-1$ in which only $\\Delta=\\Delta_+$ and $c_1$ is the vev while $c_2$ is the source in (\\ref{fi}).\n\n\\item     ${d^2\\over 4}>m^2\\ell^2>{d^2\\over 4}-1$, where there are two quantizations of the scalar field as both linearly independent solutions are normalizable near the boundary. In one, $\\Delta=\\Delta_+$ and $c_{1,2}$ are as above, while in the other, $\\Delta=\\Delta_-$ and the role of $c_{1,2}$ is reversed.\n\n\\end{itemize}\n\n\\item $m^2<0$ corresponds to irrelevant operators.\n\n\\item $m=0$ corresponds to operators that to this order are marginal.\n\n\\end{enumerate}\n\n\\bigskip\n\\subsection{The Superpotential Method}\n\\label{supmethod}\n\\bigskip\n\nThere is however a more convenient formalism for our purposes, \\cite{gkn,eff}, which uses a different set of coordinates, the \\emph{domain wall coordinates}.\nThis is convenient because we will write the radial evolution equations for the scalar as first order RG equations in terms of a $\\beta$-function.\n\n  We use a new bulk coordinate $u$, related to the previous one through the relation $r = \\ell e^{-u\/\\ell }$ in the $AdS$ case. Hence the boundary of an asymptotically $AdS$ solution now lies at $u = + \\infty$. Using these coordinates, the ansatz (\\ref{3}) for the metric takes the form:\n\\begin{equation}\nds^2 = du^2 + e^{2A(u)} dx_{\\mu} dx^{\\mu}.\n\\end{equation}\nThe $AdS_{d+1}$ solution is now:\n\\begin{equation}\nA(u) = { u \\over \\ell} \\; , \\; \\phi(u) = 0.\n\\end{equation}\nSubstituting the new form of the ansatz into the equations of motion \\eqref{eomphi} and \\eqref{efe}, we obtain:\n\\begin{align}\n& d(d-1)\\dot{A}(u)^2-{1\\over 2}\\dot\\phi^2 - V(\\phi) = 0, \\label{eq11} \\\\\n& 2(d-1) \\ddot{A}(u) + \\dot{\\phi}(u)^2 = 0, \\label{eq22} \\\\\n& \\ddot{\\phi}(u) + d \\dot{A}(u) \\dot{\\phi}(u) + { dV(\\phi) \\over d \\phi}=0. \\label{eq33}\n\\end{align}\nWe then introduce a function $W(\\phi)$ that we  call the \\emph{superpotential}\\footnote{In supergravity actions it is indeed the superpotential, but it is a more general concept, and as shown in \\cite{eff} it gives the boundary QFT effective potential for the scalar.}. It is a function of the scalar only but does not depend explicitly on the radial coordinate. The superpotential satisfies:\n\\begin{equation}\n\\label{weq}\n{dW^2\\over 2(d-1)}-W'^2=2V(\\phi).\n\\end{equation}\nwhere the prime is derivative with respect to $\\phi$.\nIf (\\ref{weq}) is valid then the first order equations:\n\\begin{equation}\n\\label{1oeq}\n\\dot A = -{W\\over 2(d-1)} \\;\\;\\;,\\;\\;\\; \\dot\\phi= {dW\\over d\\phi}\\equiv W' \\;,\n\\end{equation}\nare equivalent to the equations of motion (\\ref{eq11}), (\\ref{eq22}) and (\\ref{eq33}).\n\nThe equations (\\ref{1oeq}) are first order equations. Hence we only need one initial condition for each function $A$ and $\\phi$. However, $W$ is determined through (\\ref{weq}) which is also a a first order differential equation, and we therefore need one initial condition for $W$ to solve the equation uniquely.\n In total we have 3 initial conditions for this system of equations.\n\n This is the correct number of the initial conditions of the original set of equations (\\ref{eq11})-(\\ref{eq33}) as the $\\phi$ equation is second order and has two arbitrary initial conditions while the equation for $A$ is first order, and has another integration constant.\n\nThe non-trivial integration constant of equation (\\ref{weq}) controls  the (non-perturbative) expectation value of the operator dual to $\\phi$.\nTypically, as the equation (\\ref{weq}) is non-linear, the space of solutions consists of two generic families that lead to singular solutions for the scalar, and a unique solution in between that gives a regular evolution for the scalar, \\cite{thermo, hr}.\n\n\\subsection{Perturbative Calculation of the Superpotential}\n\n\n\nThe equation (\\ref{weq}) can be solved perturbatively near the boundary ($\\phi\\to 0$) as a regular expansion in $\\phi$:\n\\begin{equation}\nW(\\phi) = W_0 + W_1 \\phi +  W_2 \\phi^2 +  \\cdots.\n\\end{equation}\nExpanding to second order and using the potential (\\ref{pot}), we obtain the following set of equations:\n\\begin{equation}\n{d W_0^2 \\over 2 (d-1)} - W_1^2  = {2d(d-1) \\over \\ell^2} \\;\\;\\;,\\;\\;\\; {d W_1 W_0 \\over (d-1) } - 4 W_1 W_2 = 0 \\;\\;\\;,\\;\\;\\;\n\\end{equation}\n\\begin{equation}\n{dW_1^2\\over 2(d-1)}+{dW_0W_2\\over d-1}-4W_2^2-6W_1W_3=m^2.\n\\end{equation}\nwhich are solved in terms of $W_0$:\n\\begin{equation}\nW_1 = \\pm  \\sqrt{ {2d \\over \\ell^2 } - {dW_0^2 \\over 2 (d-1)^2}} \\;\\;\\;,\\;\\;\\; W_2 = {dW_0 \\over 4(d-1)}.\n\\label{e1}\\end{equation}\nHence, at quadratic order the expansion is:\n\\begin{equation}\nW(\\phi) =  W_0 \\pm  \\sqrt{ {2d \\over \\ell^2 } - {dW_0^2 \\over 2 (d-1)^2}} \\phi +  {dW_0 \\over 4(d-1)} \\phi^2  +  O(\\phi^3).\n\\label{ww}\\end{equation}\nHowever, this solution is too general for our purposes and does not satisfy important holographic requirements. We need that after solving the first order equations using this superpotential, that we obtain the neighborhood of the boundary.\n We solve (\\ref{1oeq}) using (\\ref{ww}) to linear order,  $W(\\phi) = W_0 + W_1 \\phi$. The equation $\\dot \\phi = W_1$ gives:\n\\begin{equation}\n\\phi(u) = W_1 (u - u_0).\n\\end{equation}\nwhere we have adjusted the boundary at $u=u_0$ so that $\\phi$ vanishes there as it should.\nFor the metric, we are led to solve $\\dot A = - W \/ 2(d-1)$ which implies\n\\begin{equation}\nds^2=du^2+e^{A_0-{\\left(W_1^2(u-u_0)+W_0\\right)^2\\over 2(d-1)W_1^2}}dx^{\\mu}dx_{\\mu}.\n\\end{equation}\nIt is clear, that $u=u_0$ is not a boundary for this metric (where the scale factor must diverge) unless $W_1=0$.\n Hence we must impose $W_1=0$ and the expansion (\\ref{ww}) is no longer valid. Instead we  obtain\n\\begin{equation}\nW(\\phi) = -{2 (d-1) \\over \\ell} -2{\\Delta_{\\pm}\\over \\ell}\\phi^2+ O(\\phi^{3}).\n\\label{e}\\end{equation}\nBoth overall signs for $W$ solve the equations. The minus sign corresponds to leaving a UV Fixed point. This can be seen from (\\ref{1oeq}), and the fact that $u$ decreases away from the boundary at $u=+\\infty$ and the fact that $A$ should decrease.\n\n\n\n\n\\subsection{The Holographic $\\beta$-function}\n\nIn holography with a two-derivative bulk action, the scale factor $e^{A}$ plays the role of an RG scale. Apart from the fact that near the AdS boundary $r\\to 0$ (or $u\\to+\\infty$) it coincides with the radial coordinate, it can be shown from the equations (\\ref{eq11})-(\\ref{eq33}) that it is a monotonic function, that diverges in the UV and vanishes in the IR, \\cite{zafa}.\nWe may therefore associate the exponent of the scale factor $A\\leftrightarrow \\log \\mu$ to the logarithm of the RG scale.\n\nIn view of this we may define the holographic analogue of the $\\beta$ function as\n\n\\begin{equation}\n{d\\phi\\over dA}\\equiv \\beta_{\\phi}=-{2(d-1) \\over W} {d\\over d\\phi}W(\\phi)\\;,\n\\end{equation}\nwhere we have used the first order equations of motion (\\ref{1oeq}) in the righthand side above.\n\n\nThe nonlinear equation for the superpotential (\\ref{weq}) implies a non-linear equation for the $\\beta$-function.\n Differentiating \\eqref{weq} and using \\eqref{1oeq}, we obtain a first order non-linear  differential equation for the holographic $\\beta$-function:\n\\begin{equation}\n\\beta = (d-1)  {d \\over d \\phi} \\log \\left( {\\beta^2 - 2d(d-1)  \\over V }\\right).\n\\label{nonl}\\end{equation}\n\n\\subsection{Regularity of the Solutions}\n\nTaking $\\phi$ as a coordinate instead of $u$, the metric becomes:\n\\begin{equation}\nds^2={d\\phi^2\\over W'^2}+e^{2A}(dx_{\\mu}dx^{\\mu}),\n\\end{equation}\nwhere we have used the equations of motion \\eqref{1oeq}. The Ricci scalar can be computed to be\n\\begin{align}\nR =&{d\\over 4(d-1)}\\left[W^2-8V\\right].\n\\label{r1}\n\\end{align}\nwhere we have once again used the equations of motion and where all derivatives are with respect to $\\phi$. On the other hand, taking the trace of the Einstein equations (\\ref{efe}) yields:\n\\begin{equation}\nR = {1\\over 2}(\\partial\\phi)^2-{d+1\\over d-1}V.\n\\end{equation}\nSubstituting this back into the Einstein equations allows us to obtain a simple expression for the Ricci tensor:\n\\begin{equation}\nR_{\\mu\\nu}={1\\over 2}\\partial_{\\mu}\\phi\\partial_{\\nu}\\phi-{V\\over d-1}g_{\\mu\\nu}.\n\\end{equation}\nFrom this we can readily deduce the following invariant:\n\\begin{align}\nR_{\\mu\\nu}R^{\\mu\\nu} =&{d\\over 16(d-1)^2}\\left[16(d+1)V^2-8d VW^2+dW^4\\right].  \\label{r2}\n\\end{align}\nFinally, we can calculate directly the Riemann invariant:\n\\begin{align}\nR_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma}= &{d\\over 8(d-1)^3}\\left[32(d-1)V^2-16(d-1)VW^2+(2d-1)W^4\\right].\n\\label{r3}\\end{align}\n\nFrom these three formulae (\\ref{r1}), (\\ref{r2}) and (\\ref{r3}) we can conclude the following concerning the regularity of the metric (finiteness of the curvature invariants):\n\\begin{enumerate}\n\n\\item If at a point $\\phi$ both V and W are finite, then the geometry is regular.\n\n\\item If at a point $\\phi$, $V$ diverges but $W$ remains bounded, or vice versa, this is a curvature singularity.\n\n\\item If $V\\to \\infty$ as $\\phi\\to \\phi_*$, then in order for the scalar curvature to be regular , $W$ should also diverge as\n$W^2\\sim 8V$. However in that case the Ricci square is always divergent.\n\n\\end{enumerate}\nWe conclude that the geometric singularities correspond to places where $V$ and\/or $W$ diverge.\n\nIf we now assume that we have an Einstein space, i.e. that $R_{\\mu\\nu}=\\Lambda(\\phi)g_{\\mu\\nu}$, then from (\\ref{r1}) and (\\ref{r2}) we obtain:\n\\begin{align}\n& \\Lambda^2(\\phi) (d+1) = {d\\over 16(d-1)^2}\\left[16(d+1)V^2-8d VW^2+dW^4\\right]\\\\\n&  \\Lambda(\\phi) (d+1) = {d\\over 4(d-1)}\\left[W^2-8V\\right].\n\\end{align}\nEliminating $W$ out from these equations gives:\n\\begin{equation}\nV(\\phi)=-(d-1)\\Lambda.\n\\end{equation}\nIn particular in the case of AdS space $\\Lambda$ is constant and then the potential must be constant.\n\n\n\n\\section{Marginal Operators: The Cubic and Higher Orders Cases}\n\\label{cubic}\n\n\nIn this section we will assume that  the extremal point of the potential is degenerate, $m=0$ and therefore the operator is marginal to leading order.\nWe will also assume that the first non-trivial derivative comes in at order $n\\geq 3$.\n\nWe use the superpotential method of section \\ref{supmethod} to study the gravitational solutions with the following potential:\n\\begin{equation}\nV(\\phi) = {d(d-1) \\over \\ell^2} + {g_n \\over \\ell^2} \\phi^n+\\cdots ,\n\\end{equation}\nnear $\\phi\\to 0$ where we have chosen the critical point of this potential to be.\n\n\n\n\\label{super}\n\nThe equation (\\ref{weq}) can be solved perturbatively near the boundary by expanding $W$ around the $AdS$ solution $\\phi=0$:\n\\begin{equation}\nW(\\phi) = \\sum_{n=0}^{\\infty} W_n \\phi^n .\n\\end{equation}\nWe obtain\\footnote{We assume here the presence only of the $\\phi^n$ term in the potential in order to track which terms in the $\\beta$-function are affected.}:\n\\begin{equation}\n  \\label{wsol}\n W(\\phi) = -{1\\over \\ell} \\left( {2(d-1)} + {g_n \\over d } \\phi^{n} + {g_n^2 n^2 \\over 2  d^3} \\phi^{2n-2} +\\delta_{n,3} {54g_n^3 \\over d^5}\\phi^{2n-1} \\right)+O(\\phi^{2n}).\n \\end{equation}\n\nWe may now use the superpotential found,  to compute the (perturbative) $\\beta$-function as\n\\begin{equation}\n \\beta_{\\phi}\\equiv {d\\phi\\over dA}=-{2(d-1) \\over W} {d\\over d\\phi}W(\\phi).\n\\end{equation}\nUsing the perturbative solution (\\ref{wsol}) for $W$, we find to next to leading order:\n\\begin{equation}\n \\beta_\\phi = - {g_n n \\over d} \\phi^{n-1} - {g_n^2 n^2 (n-1) \\over d^3}\\phi^{2n-3} + O(\\phi^{2n-1}).\n\\label{50}\\end{equation}\nFrom (\\ref{50}) we can interpret this $\\beta$ function as follows.\n If $n=2$, at lowest order, we obtain a linear term in $\\phi$, proportional to $d- \\Delta$. This the standard classical contribution to the $\\beta$-function of non-marginal operators.\n\nMarginal operators have no quadratic term around the fixed point. If the first non-trivial term is cubic ($n=3$), then the leading $\\beta$ function is quadratic and this is similar to the QFT case where  a one-loop contribution is the leading contribution to the $\\beta$-function.\n\nWhen $n>3$, then this is similar to the field theory case, where the first non-trivial contribution arises at $n-2$ loops.\nWe also observe, that a single bulk $\\phi^n$ interaction, generates an infinite number of terms in the perturbative $\\beta$ functions as seen in (\\ref{50}).\nFor $n=3$,  all higher terms appear, while for higher $n$ there are gaps at low number of loops. This is explicitly seen in appendix \\ref{wsoll} where the superpotential and the $\\beta$ function are computed to higher orders.\n\n\nThe standard perturbative method can be used to obtain explicit expressions for the metric\\footnote{Here $A$ denotes the logarithm of the scale factor.} and scalar field. For a dual operator $\\mathcal{O}$ with $\\langle \\mathcal{O} \\rangle =0$, we obtain:\n\\begin{equation}\n\\phi(A) =\\phi_0- {g_n n\\over d} A \\phi_0^{n-1} - {g_n^2 n^2(n-1) \\over d^2} A \\left( {1 \\over d} - \\frac12 A \\right) \\phi_0^{2n-3} + O(\\phi_0^{2n-1}),\n\\label{a222}\\end{equation}\nwhere $\\phi_0$ is the infinitesimal source for $\\phi$. We can also express the scale factor in terms of the radial coordinate $r$:\n\\begin{equation}\nA(r) = \\log {\\ell \\over r} - {g_n \\over 2 d(d-1)} \\phi_0^n + {g_n^2 n^2 \\over 4(d-1) d^3} \\left[ 1- 2 d \\left( 1- \\log {r \\over \\ell} \\right) \\right] \\phi_0^{2n-2} + O(\\phi_0^{2n-1}) .\n\\end{equation}\n\n\n\\subsubsection*{Non-perturbative Corrections}\nWe will now try to extend this perturbative solution by including non-perturbative terms that originate in non-trivial vevs for the bulk scalar. In the case of a massive scalar field, it is known that the full non-perturbative solution is expressed as follows, \\cite{eff} :\n\\begin{equation}\nW(\\phi) = \\sum_{k = 0}^{\\infty} W_k(\\phi) \\;\\;\\;,\\;\\;\\; W_k(\\phi) = \\mathcal C^k \\phi^{k \\delta} \\sum_{j = 0}^{\\infty} A_{k,j} \\phi^{j} \\;\\;\\;,\\;\\;\\;\n\\label{d1}\\end{equation}\nwhere $\\delta$ is equal to $d\/\\Delta_-$, and where $\\mathcal C$ is related to the vev $\\langle \\mathcal O \\rangle$. $W_0(\\phi)$ denotes the perturbative power series solution, valid in the $\\phi\\to 0$ regime.\n\nFor the marginal case, the non-perturbative contributions do not take the form of non-integer powers of $\\phi$. To obtain the form of the possible non-perturbative terms, we will expand the superpotential again as in (\\ref{d1}) and we will treat $W_{k>1}$ as small. This is equivalent to a small vev expansion, and we will formalise it by writing\n\\begin{equation}\nW = \\sum_{k =0}^\\infty x^k W_k.\n\\label{d2}\\end{equation}\nand do perturbation theory in $x$ and then set it to one at the end.\n\nSubstituting (\\ref{d2}) into \\eqref{weq}, and separating powers of $x$, we obtain for any integer $k\\geq 0$:\n\\begin{equation}\n{d \\over 2 (d-1)} \\sum_{i + j = k} W_i W_j - \\sum_{i+ j = k } W_i' W_j' =  \\delta_{k,0} 2V.\n\\end{equation}\n\nAt each order, we obtain a first-order linear equation for $W_k$ of the form:\n\\begin{equation}\n{d \\over (d-1)}W_0 W_k - 2W_0' W_k' = f_k(\\phi),\n\\end{equation}\nwhere $f_k(\\phi)$ is a function depending on the $W_m$ and $W_m'$ for $0 \\leq m \\leq k-1$. The general solution to this equation is:\n\\begin{equation}\nW_k = \\exp\\left[-{d}\\int {d\\phi\\over \\beta_0(\\phi)}\\right] \\left( \\mathcal C _k - \\int { f_k(\\phi) \\over 2 W_0'} \\exp\\left[{d}\\int {d\\tilde \\phi \\over \\beta_0(\\tilde \\phi)}\\right] d\\phi \\right).\n \\label{nonpsol}\n\\end{equation}\nFor the leading correction $W_1$, we obtain \\eqref{weq}:\n\\begin{equation}\n{d \\over d-1}W_0 W_1 - 2 W_0' W_1' = 0.\n\\label{nonpert1}\n\\end{equation}\nand therefore\n\\begin{equation}\nW_1 = \\mathcal C _1\\exp\\left[-{d}\\int_{0}^{\\phi} {d\\chi\\over \\beta_0(\\chi)}\\right],\n\\end{equation}\nwhere we have used the definition of the perturbative holographic $\\beta$-function:\n\\begin{equation}\n\\beta_0 = -2 (d-1) {W_0' \\over W_0}.\n\\end{equation}\nThe constant $\\mathcal C _1$ is non-perturbative and can be fixed only if a global and regular flow solution for the scalar and the metric has been found. Similarly, for $k=2$, we obtain:\n\\begin{equation}\n{d \\over (d-1)} W_0 W_2 - 2W_0' W_2' = W_1'^2 - {d \\over 2(d-1)} W_1^2 \\equiv f_2.\n\\end{equation}\nwith solution\n\\begin{align}\nW_2 =&\\exp\\left[-{d}\\int_{0}^{\\phi} {d\\phi\\over \\beta_0(\\phi)}\\right]   \\left( \\mathcal C _2 + \\int_{0}^{\\phi} \\left( {d \\over \\beta_0^2} - {1 \\over 2(d-1)} \\right) {d(d-1) \\over  \\beta_0} \\mathcal C _1^2e^{ - \\int_{0}^{\\phi} \\beta_0 \\left( {d \\over \\beta_0^2} - {1 \\over 2(d-1)}\\right) }d \\chi  \\right)\n\\end{align}\nwhere we have used:\n\\begin{equation}\nW_0=e^{-{1\\over 2(d-1)}\\int d\\phi \\beta_0(\\phi)}\\;\\;\\;,\\;\\;\\;  W_1=\\mathcal C _1 e^{-d\\int {d\\phi\\over \\beta_0}} \\;\\;\\;,\\;\\;\\; W_0' = - {\\beta_0 \\over 2(d-1)} e^{- {1 \\over 2(d-1)} \\int d\\phi \\beta_0}.\n\\end{equation}\nNote that the integration constant $\\mathcal C _2$ can be absorbed into $\\mathcal C _1$ in $W_1$. This is as it should, as the full solution should contain a single undetermined constant as the equation for the superpotential is of the first order.\n\nBy induction it is straightforward  to see that all the higher terms $W_i$, $i \\geq 3$ can also be expressed in terms of integrals of $\\beta_0$.\nWe will now compute explicitly $W_1$, using the perturbative form \\eqref{wsol} obtained for $W_0$. From \\eqref{nonpert1}, we have:\n\\begin{equation}\n{d \\log W_1 \\over d \\phi } = {d \\over 2(d-1)} {W_0 \\over W_0'} = {d^2 \\over g_n n \\phi^{n-1}} - {n-1 \\over \\phi} - \\delta_{n,3} {18 g_n \\over d^2} + O(\\phi),\n\\end{equation}\nwhich is solved by:\n\\begin{align}\nW_1(\\phi) &= \\mathcal C _1 e^{- {d^2 \\over g_n n(n-2) \\phi^{n-2} }-(n-1) \\log \\phi -  \\delta_{n,3}{18g_n\\over d^2}\\phi+\\cdots }.\n\\label{W1+}\n\\end{align}\n\nIt should be noted here that the non-perturbative solution is captured completely in the non-linear equation for the $\\beta$ function (\\ref{nonl}) we derived earlier. Also the arbitrary integration constant that appears in the non-perturbative part, is eventually fixed by demanding that the superpotential gives rise to a globally regular solution. This constraints the integration constant to take a restricted set of values, typically only one, \\cite{eff}.\n\nWe can also compute the contribution $\\beta_1$ of the non-perturbative part of the superpotential to the $\\beta$-function:\n\\begin{align}\n\\beta =& -2(d-1) {d \\over d \\phi} \\log W = \\underbrace{-2(d-1) {d \\over d \\phi} \\log W_0}_{\\beta_0} \\underbrace{- 2(d-1) {d \\over d \\phi} {W_1 \\over W_0}}_{\\beta_1}+ \\cdots\n\\end{align}\nwith\n\\begin{equation}\n\\beta_1=\n-2(d-1)\\left({W_1'\\over W_0}-{W_0'\\over W_0}{W_1\\over W_0}\\right)=\\left({2d(d-1)\\over \\beta_0}-\\beta_0\\right){\\mathcal C_1\\over W_0}e^{-d\\int {d\\phi\\over \\beta_0}}.\n\\end{equation}\n\nWe obtain:\n\\begin{align}\n\\beta_1 =& \\ell \\mathcal C _1 e^{- {d^2 \\over g_n n (n-2) \\phi^{n-2}} - (n-1) \\log \\phi - \\delta_{n,3} {18 g_n \\over d^2}\\phi} \\nonumber \\\\\n& \\times \\Big( {d^2 \\over g_n n \\phi^{n-1}}  -{n-1 \\over \\phi} - \\delta_{n,3} {18 g_n \\over d^2} + O(\\phi) \\Big).\n\\end{align}\n\nTherefore for $n=3$ corresponding to a one-loop $\\beta$-function we obtain\n\\begin{equation}\n\\beta_1\\simeq \\mathcal C {d^2\\over 3g_3\\phi^2}e^{- {d^2 \\over 3g_3 \\phi}}.\n\\end{equation}\nwhich has the typical non-perturbative form we are familiar with from QCD.\n\n\n\\subsubsection*{Asymptotic Behavior of the Coupling }\n\n\nWe can now use the expression for $\\beta_\\phi$ to discuss the asymptotic behaviors depending on the coupling constant $g_n$. For this, we integrate (\\ref{50}) in the following way:\n\\begin{align}\n& \\int_{\\phi(u_0)}^{\\phi(u)} {d \\phi \\over \\beta_\\phi} = \\int_{A(u_0)}^{A(u)} dA\n\\end{align}\nwhere $u_0$ is some arbitrary scale, and which then gives:\n\\begin{align}\n&  - {d \\over g_n n}\\int_{\\phi(u_0)}^{\\phi(u)} {1 \\over \\phi^{n-1}} \\left( 1 - {g_n n (n-1) \\over d} \\phi^{n-2} + O(\\phi^n) \\right) d\\phi = A(u) - A(u_0).\n\\end{align}\nFor $n>2$, the result is:\n\\begin{align}\n{d \\over g_n n (n-2)} \\left( {1 \\over \\phi^{n-2}(u) } - {1 \\over \\phi^{n-2}(u_0) } \\right) + (n-1) \\log \\left( {\\phi(u) \\over \\phi(u_0)} \\right) + O(\\phi^2)= A(u) - A(u_0).\n\\end{align}\nSince the UV (or near-boundary limit) limit is at $u \\rightarrow + \\infty$ and $A(u) \\sim  u\/\\ell$, we observe that the theory is asymptotically free for $g_n >0$. The IR limit is at $u \\rightarrow - \\infty$ and this time the theory will be IR free for $g_n < 0$.\n\nInverting the above formula, one obtains:\n\\begin{equation}\n\\phi\n\\simeq  \\phi_0 - {g_n n \\over d} \\phi_0^{n-1} A  + {g_n^2 n^2 (n-1) \\phi_0^{2n-3} \\over 2 d^2}  (A^2+{\\cal O}(A)) + {\\cal O}(\\phi_0^{2n-2}),\n\\label{a223}\\end{equation}\nin agreement with (\\ref{a222}).\n\n\nWe summarize the solution as follows: we have set our UV fixed point to $\\phi=0$ without loss of generality, and expanded the potential near this fixed point, with $\\phi\\ll 1$. We have found that near the boundary, $A\\to\\infty$,  the solution for $\\phi$ is given by (\\ref{a222}) or (\\ref{a223}) and therefore, $\\phi\\to\\infty$ as we approach the boundary, $A\\to \\infty$. This contradicts our assumptions and therefore this case is inconsistent.\n\n\n\n\\subsubsection*{Single-term $\\beta$-function}\n\nWe have seen that a single non-trivial term in the bulk scalar potential produces an infinite number of terms in the $\\beta$ function.\nIn this part we will ask the opposite question: are there bulk potentials that give a (perturbative)  $\\beta$ function containing as single term?\n\nThis is equivalent to imposing:\n\\begin{equation}\n- 2(d-1) {W' \\over W} = - k \\phi^{n-1}\n\\end{equation}\nwhich can be solved by :\n\\begin{equation}\nW(\\phi) = \\mathcal{C} e^{{ k \\over 2n(d-1)}\\phi^n},\n\\end{equation}\nwhere $\\mathcal{C}$ is an integration constant. Substituting this result back into (\\ref{weq}), we get:\n\\begin{align}\nV(\\phi) &= \\frac12 \\left( {d \\over 2(d-1)} \\left( \\mathcal{C} e^{{ k \\over 2n(d-1)}\\phi^n}\\right)^2 - \\left(\\mathcal{C} { k \\over 2(d-1)}\\phi^{n-1} e^{{ k \\over 2n(d-1)}\\phi^n}  \\right)^2 \\right) \\\\\n& = \\frac12  \\mathcal{C}^2 e^{{ k \\over n(d-1)}\\phi^n}   \\left( {d \\over 2 (d-1)} -{k^2 \\over 4(d-1)^2} \\phi^{2n-2} \\right)\n\\end{align}\nThe potential therefore has Liouville-type interactions.\n\n\n\n\\bigskip\n\\section{Marginal Operators: Asymptotically Soft Fixed Points}\n\\bigskip\n\n\nIn our previous discussion we have assumed that the UV fixed point is a finite value of the scalar field. This is not the only option, and in this section we will describe the only alternative, namely asymptotically soft nearly marginal fixed points. This is the case utilized in holographic models of YM like Improved Holographic QCD (IHQCD), \\cite{gkn,rev}, and is also relevant in cosmology (see \\cite{cosmo}).\n\nThe critical point now will be at $\\phi\\to\\pm \\infty$. Without loss of generality we consider  the following potential:\n\\begin{equation}\nV(\\phi)={d(d-1)\\over \\ell^2}\\left[1+\\sum_{n=1}^{\\infty} V_n~\\lambda^n\\right]\\;\\;\\;,\\;\\;\\; \\lambda\\equiv e^{-a\\phi}\\;\\;\\;,\\;\\;\\;  a>0 \\;\\;\\;,\\;\\;\\;\n\\end{equation}\nand we will be working in the region $\\phi\\to \\infty$, so that $\\lambda\\to 0$.\n$\\lambda=0$ is an AdS fixed point. This potential has the property that around the AdS fixed point, all derivatives of the potential vanish justifying the term asymptotically soft fixed-point.\n\nSolving perturbatively \\eqref{weq}, we obtain for the first few coefficients:\n\\begin{equation}\nW=\\sum_{n=0}^{\\infty}W_n \\lambda^n\\;\\;\\;,\\;\\;\\; W_0 =- {2(d-1) \\over \\ell}\\;\\;\\;,\\;\\;\\;  W_1 = W_0 {V_1 \\over 2 } \\;\\;\\;,\\;\\;\\;\n\\end{equation}\n\\begin{equation}\nW_2 = W_0\\left({V_2 \\over 2}  - \\left[1 -   { 2  (d-1)a^2 \\over d} \\right]  {V_1^2  \\over 8}\\right),\n\\end{equation}\nwhere once again the sign of $W_0$ has been chosen so that the solution leads to an $AdS$ boundary.\n\nThe $\\beta$-function is calculated as follows:\n\\begin{equation}\n\\beta_\\lambda \\equiv  {d \\lambda \\over dA} = -a \\lambda {d\\phi \\over dA} = 2(d-1) a \\lambda {W' \\over W}.\n\\end{equation}\nSubstituting the previous results found for the superpotential, we obtain:\n\\begin{align}\n\\beta_\\lambda =& - (d-1) a^2V_1\\lambda^2 - 2(d-1) a^2  \\left[ V_2  - \\left[1 -   { 2  (d-1)a^2 \\over d} \\right]  {V_1^2  \\over 2}  \\right] \\lambda^3 + \\dots\n\\end{align}\nThis is again a $\\beta$-function corresponding to a marginal coupling, with the difference that here there is no option on what is the leading term in the $\\beta$ function. It always is the ``one-loop\" term.\n\nFrom here, we can compute the corrections to the metric and to the dilaton by solving \\eqref{1oeq}. For the dilaton we obtain\n\\begin{equation}\n\\dot \\phi = - {1 \\over a \\lambda  }{d\\lambda \\over d u  } = -a W_1 \\lambda -2a W_2 \\lambda^2 + \\dots,\n\\end{equation}\nwhich is solved by:\n\\begin{align}\n\\lambda(u) = {\\ell \\over a^2 (d-1) V_1 u} + { \\ell^2 \\over a^4  (d-1)^2 V_1  } \\left(   \\left[1 -   { 2  (d-1)a^2 \\over d} \\right]  {1  \\over 2} - {2V_2 \\over  V_1^2}\\right){\\log u \\over u^2}.\n\\end{align}\n\nWe can then turn our attention to the metric, for which the equation to solve is:\n\\begin{align}\n\\dot A =& -{1 \\over 2(d-1)} \\left( W_0 + W_1 \\lambda + W_2\\lambda^2 + \\dots \\right) \\nonumber \\\\\n=&  {1 \\over \\ell} + {V_1 \\over 2 \\ell}  \\lambda + {1 \\over \\ell }\\left({V_2 \\over 2}  - \\left[1 -   { 2  (d-1)a^2 \\over d} \\right]  {V_1^2  \\over 8}\\right)\\lambda^2 + \\dots\n\\end{align}\nThis is solved by:\n\\begin{align}\nA(u) =&  {u \\over \\ell } + {1 \\over a^2 2  (d-1) V_1 } \\log u \\nonumber \\\\\n& - { \\ell \\over a^4  2(d-1)^2   } \\left(   \\left[1 -   { 2  (d-1)a^2 \\over d} \\right]  {1  \\over 2} - {2V_2 \\over  V_1^2}\\right){\\log u \\over u } \\nonumber \\\\\n&-  {\\ell \\over a^4 (d-1)^2} \\left( -  {V_2 \\over 2 V_1^2} + \\left[ 1 - {2(d-1) a^2 \\over d} \\right]{1 \\over 8}    \\right) {1 \\over u }.\n\\end{align}\n\nWorking again at next to leading order, we can also express $\\lambda$ as a function of $A$:\n\\begin{align}\n\\lambda =  {1 \\over a^2 (d-1) V_1  A} +  {1 \\over a^4 (d-1)^2 V_1} \\left( \\left[ 1 - {2(d-1) a^2 \\over d}\\right] \\frac12 - {1 \\over 2 V_1}- {2V_2 \\over V_1^2} \\right)  {\\log  A \\over A ^2 } \\nonumber \\\\\n\\end{align}\n\nNote that in this case, and unlike that of the previous section, the expansion of the potential, and the RG flow are compatible. $\\lambda$ remains small during the flow.\n\n\nUsing the setup in section \\ref{super}, where $W_{np}$ denotes the leading non-perturbative correction to the superpotential, we are led to consider the following equation:\n\\begin{equation}\nW_0 W_{np} - {2(d-1) \\over d} W_0' W_{np}' = 0,\n\\end{equation}\nWe obtain\n\\begin{align}\n{d \\log W_{np} \\over d \\phi} = - {d \\over (d-1) a V_1 \\lambda}  - {d \\over (d-1) a} \\left( 1 - {2 V_2 \\over V_1^2 } - {(d-1) a^2 \\over d } \\right)+\\cdots,\n\\end{align}\nwhich is integrated to\n\\begin{equation}\nW_{np}(\\lambda) = \\mathcal C~ e^{- {d \\over (d-1) a^2 V_1 \\lambda} }~~ \\lambda^{{d \\over (d-1) a^2} \\left( 1 - {2 V_2 \\over V_1^2 } - {(d-1) a^2 \\over d } \\right)}+\\cdots.\n\\label{nonpertasy}\n\\end{equation}\n\n$W_{np}$ affects the expression for the scalar field. The equation to solve is:\n\\begin{align}\n-{\\dot \\lambda \\over a \\lambda} = -a \\left( W_{1} \\lambda + 2 W_{2} \\lambda^2 \\right) - {d \\over (d-1) a  V_1 \\lambda}  W_{np}+\\cdots,\n\\end{align}\nwhere we have only kept the leading non-perturbative term in \\eqref{nonpertasy}. This gives:\n\\begin{align}\n\\lambda(u) =& {\\ell \\over a^2 (d-1) V_1 u }  + { \\ell^2 \\over a^4  (d-1)^2 V_1  } \\left(   \\left[1 -   { 2  (d-1)a^2 \\over d} \\right]  {1  \\over 2} - {2V_2 \\over  V_1^2}\\right){\\log u \\over u^2} \\nonumber \\\\\n& - {\\ell \\over (d-1)} \\left( 1 + {\\ell \\over  d V_1 ^2 u} + {2 \\ell^2  \\over d^2  u^2 }  \\right) \\mathcal C _1 e^{-{d u \\over \\ell }}+\\cdots.\n\\end{align}\n\nFinally, we compute the contribution of the non-perturbative term to the $\\beta$-function. We have this time:\n\\begin{align}\n\\beta_\\lambda \\equiv & {d \\lambda \\over dA } = - a \\; {d\\phi \\over d A} = 2(d-1) a \\lambda {d \\over d \\phi} \\log W \\nonumber \\\\\n=& 2(d-1) a \\lambda {d \\over d \\phi} \\log \\left( W_{pert} + W_{np} \\right) =  2(d-1) a \\lambda {d \\over d \\phi} \\left( \\log W_{pert}  + \\log \\left[ 1 + {W_{np} \\over W_{pert}}\\right] \\right)\\;.\n\\end{align}\nWe are interested in computing the second term, namely:\n\\begin{equation}\n\\beta_{\\lambda,np} \\equiv   2(d-1) a \\lambda {d \\over d \\phi} {W_1 \\over W_0 }\n=  {\\ell d \\lambda  \\over (d-1)} W_{np} \\left[  {1 \\over V_1 \\lambda} +  \\frac12 - {2V_2 \\over V_1^2 } - {(d-1)a^2 \\over d}  \\right]+\\cdots.\n\\end{equation}\n$$\n=\\mathcal C~ {\\ell d \\lambda  \\over (d-1)} ~ \\left[  {1 \\over V_1 \\lambda} +  \\frac12 - {2V_2 \\over V_1^2 } - {(d-1)a^2 \\over d}  \\right]~ e^{- {d \\over (d-1) a^2 V_1 \\lambda} }~~ \\lambda^{{d \\over (d-1) a^2} \\left( 1 - {2 V_2 \\over V_1^2 } - {(d-1) a^2 \\over d } \\right)}+\\cdots\n$$\n\nThis is in agreement with general dimensional expectations.\n\nWe summarize, that this second case of marginal operators seems self-consistent and described flows similar to those of asymptotically-free gauge theories in the weakly coupled regime.\n\nIts realization in string theory so far can be found in higher derivative contexts, several possibilities of which have been analyzed in \\cite{dis}, (see also \\cite{ke} for a discussion of supergravity embeddings).\n\n\\section{The Multiscalar Case}\n\nIn the generic case, a single scalar operator, even if it is the only one sourced in the UV, mixes as it runs with other scalar operators. This is captured by a multi-scalar general action that we will investigate in this section.\n\nThe important difference is that apart from a bulk potential, there is a scalar-field metric now that also controls the holographic RG flow.\nOne of our goals will be to see at what order in the scaling region the metric affects the flow.\n\n\n\nWe consider therefore the case of $N$ scalars. We start from a modified version of \\eqref{1}:\n\\begin{equation}\nS_{2d} = M^{d-1} \\int d^{d+1}x \\sqrt{-g} \\left\\{ R - \\frac12 G_{ij}(\\phi) \\partial_\\mu \\phi^i \\partial^\\mu \\phi^j + V(\\phi) \\right\\}.\n\\end{equation}\nThe bulk equations can be easily deduced from previous calculations:\n\\begin{equation}\nR_{\\mu\\nu} - \\frac12 g_{\\mu\\nu} \\left( R - \\frac12 G_{ij}(\\phi) \\partial_\\mu \\phi^i \\partial^\\mu \\phi^j + V(\\phi) \\right) - \\frac12 G_{ij}(\\phi) \\partial_\\mu \\phi^i \\partial^\\mu \\phi^j = 0 \\;\\;\\;,\\;\\;\\;\n\\end{equation}\n\\begin{equation}\n\\nabla^\\mu \\left( G_{ij} \\nabla_\\mu \\phi^j \\right) + {\\partial V \\over \\partial \\phi^i} - \\frac12 {\\partial G_{kl} \\over \\partial \\phi^i} \\partial_\\mu \\phi^k \\partial^\\mu \\phi^l = 0.\n\\end{equation}\n\nFrom this, using the ansatz in the domain wall frame, we obtain:\n\\begin{equation}\nd(d-1) \\dot A ^2 + (d-1) \\ddot A - V = 0 \\;\\;\\;,\\;\\;\\;  \\label{eomM1}\n\\end{equation}\n\\begin{equation}\n 2(d-1) \\ddot A + G_{ij} \\dot \\phi ^i \\dot \\phi^ j = 0 \\;\\;\\;,\\;\\;\\;\n\\label{eomM2}\n\\end{equation}\n\\begin{equation}\n {\\Gamma'}^i_{jk} \\dot \\phi^j \\dot \\phi^k + \\ddot \\phi^i + d \\dot A  \\dot \\phi ^i + G^{ij}{\\partial V \\over \\partial \\phi^j} = 0 \\;\\;\\;,\\;\\;\\;\n \\label{EinsteinM}\n\\end{equation}\n where $\\Gamma'$ are the  Christoffel symbols for the field metric.\n\nA generalization of the first order equations \\eqref{weq} and \\eqref{1oeq} to the case of multiple scalars is:\n\\begin{align}\n& \\dot A = - {W \\over 2 (d-1)} \\;\\;\\;,\\;\\;\\; \\dot \\phi ^i = G^{ij} {d W \\over d \\phi^j} \\;\\;\\;,\\;\\;\\; \\label{1oeqM}\\\\\n&{d W^2 \\over 2  (d-1)} - G^{kl} {d W \\over d \\phi^k} {d W \\over d \\phi ^l } = 2 V. \\label{weqM}\n\\end{align}\nThese reproduce the equations of motion \\eqref{eomM1}, \\eqref{eomM2} and \\eqref{EinsteinM}.\n\n\nIn order to compute the superpotential and the corresponding $\\beta$-function, we will choose Riemann normal coordinates (RNC) around an $AdS$ extremum. With this choice, the potential takes the following form:\n\\begin{equation}\nV(\\phi) = {d(d-1) \\over \\ell^2}  + \\frac12{M^2}_{ij} \\phi^i \\phi^j +g_{ijk}\\phi^i\\phi^j\\phi^k\\cdots \\;.\n\\label{normalp}\n\\end{equation} Using the properties of the RNC, we can expand the metric around the $AdS$ extremum and obtain:\n\\begin{equation}\nG_{ij} = \\delta_{ij} + \\frac12 \\phi^k \\phi^l  \\left. {d^2 G_{ij} \\over d \\phi^k d \\phi^l} \\right|_{\\phi = 0}.\n\\label{metricexpand}\n\\end{equation}\nExpressing the second derivatives of the metric in terms of the Riemann tensor, we obtain:\n\\begin{equation}\nG^{ij} = \\delta^{ij} - {1\\over 3} \\delta^{im} \\delta^{jn} R_{ mk  n l}\\phi^{k}\\phi^l + O(\\phi^3).\n\\label{normalinverse}\n\\end{equation}\n\nUsing the set up of the previous section, we can solve perturbatively \\eqref{weqM}. The superpotential $W$ takes the general form:\n\\begin{equation}\nW(\\phi) = W_0 + \\sum_{i = 1}^N W_i \\phi^i + \\sum_{i,j=1}^N W_{ij}\\phi^i \\phi^j + \\sum_{i,j,k=1}^N W_{ijk} \\phi^i \\phi^j \\phi^k + O(\\phi^4),\n\\end{equation}\nwhere the convention is that:\n\\begin{equation}\nW_{i_1 \\dots i_k  } = \\left. {d^k W \\over d\\phi^{i_1} \\cdots d\\phi^{i_k}} \\right|_{\\phi=0}.\n\\end{equation}\n\nAs in the single scalar case, we should put the first derivatives to zero, in order for the vanishing of the scalars to happen at the extremum.\nSolving perturbatively the equation \\eqref{weqM} for the superpotential, we obtain the following set of equations:\n\\begin{align}\n&{d \\over 2(d-1)} W_0^2  = {2 d(d-1) \\over \\ell^2} \\\\\n& {d \\over (d-1)} W_0 W_{ij}  - \\delta^{kl} 4W_{jk}W_{il} = M_{ij}^2 \\\\\n& {d \\over (d-1)} W_0\\sum_{i,j,p=1}^N W_{ijp} \\phi^i \\phi^j \\phi^p - 12  \\delta^{kl}  \\sum_{p=1}^N W_{pk} \\phi^p  \\sum_{i,j=1}^N W_{ijl} \\phi^i \\phi^j = 2 g_{i,j,k}\\phi^{i}\\phi^{j}\\phi^{k}\n\\end{align}\nwhich can be simplified to:\n\\begin{align}\n& W_0 = - {2(d-1) \\over \\ell} \\label{M1}\\\\\n& -{2d  W_{ij} \\over \\ell}  - \\delta^{kl} 4W_{il}W_{kj} = M_{ij}^2 \\label{M2} \\\\\n& -{2d \\over \\ell} W_{ijp} - 12 \\delta^{kl} W_{k(p} W_{ij)l} = 2\\delta_{n,3} g_{ijp} \\label{M3}\n\\end{align}\nwhere indices are raised and lowered using the flat Euclidean metric since all quantities are evaluated at the center of the RNC, and where the sign of $W_0$ is chosen in order to recover an $AdS$ solution when all the scalar fields vanish.\n\nWithout loss of generality, we can take $M^2_{ij}$ and $W_{ij}$ to be real and symmetric. It is then straightforward to show using \\eqref{M2} that $\\left[ M^2, W \\right] = 0$. Hence there exists a coordinate system in the scalar fields space in which both these matrices can be simultaneously diagonalized by a constant $O(N)$ rotation. With this choice, we can write:\n\\begin{equation}\nM^2 = \\text{diag} \\left( m^2_1, m^2_2, \\dots, m^2_N \\right).\n\\end{equation}\nThe eigenvalues are real, and some of them can be equal to zero, in which case this would correspond to a marginal operator. Denoting $w_i$ the eigenvalues of $W_{ij}$, the equation  $\\eqref{M2}$ can be written as:\n\\begin{equation}\n 4w_i^2 + {2d \\over \\ell}w_i +  m^2_i=0,\n\\end{equation}\nwhich is solved by:\n\\begin{equation}\nw_i = {- d\\pm \\sqrt{d^2 - 4 \\ell^2 m_i^2} \\over 4\\ell }.\n\\end{equation}\n\nWe now can define the holographic version of the $\\beta$-function for the case of multiple scalars:\n\\begin{equation}\n\\beta_\\phi ^i \\equiv  {d \\phi^i \\over d A} = - 2(d-1) G^{ij}  {d \\over d \\phi^j} \\log W.\n\\end{equation}\nUsing \\eqref{normalinverse}, we obtain:\n\\begin{align}\n\\beta^i_\\phi =&2 \\ell w_i \\phi^i + 3\\ell  \\sum_{p,q=1}^N W_{ipq} \\phi^p \\phi^q \\nonumber \\\\\n&+ \\sum_{p,q,r=1}^N \\left( 4 \\ell  W_{ipqr} + { \\ell^2 \\over (d-1)}  w_pw_i\\delta^{pq}\\delta^{ir} - {2 \\ell \\over 3} w_rR_{iprq} \\right) \\phi^p \\phi^q \\phi^r + O(\\phi^4).\n\\end{align}\nFrom this last equation and \\eqref{M1}, \\eqref{M2} and \\eqref{M3}, we conclude the following\n\n\\begin{itemize}\n\n\\item  If a subset of operators are marginal, then their leading $\\beta$-function starts generically at one-loop.\n\n\\item The mixing with relevant operators is also happening first at the one-loop. This persists even when all operators are marginal.\n\n\\item The field space metric (Zamolodchikov metric), equivalent to the curvature of the field space first affects the $\\beta$-functions at two-loop level.\n\n\n\\end{itemize}\n\n\n\n\\section{Acknowledgements}\\label{ACKNOWL}\n\n\nWe would like to thank J. Gauntlet, N. Warner and K. Pilch  for correspondence.\n\n\nThis work was supported in part by European Union's Seventh Framework Programme under grant agreements (FP7-REGPOT-2012-2013-1) no 316165,\nPIF-GA-2011-300984, the EU program ``Thales'' MIS 375734, by the European Commission under the ERC Advanced Grant BSMOXFORD 228169and was also co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program ``Education and Lifelong Learning'' of the National Strategic Reference Framework (NSRF) under ``Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes''. The research leading to these results has also received funding from the People Programme  (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7\/2007-2013\/ under REA Grant Agreement No 317089 (GATIS).\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}