Desrosiers, Patrick; Eynard, Bertrand Supermatrix models, loop equations, and duality. (English) Zbl 1314.82031 J. Math. Phys. 51, No. 12, 123304, 24 p. (2010). Summary: We study integrals over Hermitian supermatrices of arbitrary size \(p + q\), which are parametrized by an external field \(X\) and a source \(Y\) of respective sizes \(m + n\) and \(p + q\). We show that these integrals exhibit a simple topological expansion in powers of a formal parameter \(\hslash\), which can be identified with \(1/(p - q)\). The loop equation and the associated spectral curve are also obtained. The solutions to the loop equation are given in terms of the symplectic invariants introduced by the second author and N: Orantin [Commun. Number Theory Phys. 1, No. 2, 347–452 (2007; Zbl 1161.14026)]. The symmetry property of the latter objects allows us to prove a duality that relates supermatrix models in which the role of \(X\) and \(Y\) are interchanged.{©2010 American Institute of Physics} Cited in 3 Documents MSC: 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics 60B20 Random matrices (probabilistic aspects) 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.) 15A16 Matrix exponential and similar functions of matrices 15B52 Random matrices (algebraic aspects) 15B33 Matrices over special rings (quaternions, finite fields, etc.) 15A75 Exterior algebra, Grassmann algebras 57N35 Embeddings and immersions in topological manifolds Citations:Zbl 1161.14026 PDFBibTeX XMLCite \textit{P. Desrosiers} and \textit{B. Eynard}, J. Math. Phys. 51, No. 12, 123304, 24 p. (2010; Zbl 1314.82031) Full Text: DOI arXiv Link References: [1] 1.Alfaro, J., Medina, R., and Urrutia, L. 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[22] The function ϕ can be used only when the matrix y and the supermatrix Y, as well as their respective sizes, are considered as parameters.We cannot, for instance, set N = 2 and y = diag(1,1) and then apply ϕ. In fact, ϕ is a homomorphism that maps the algebra of symmetric polynomials in the eigenvalues of y to the algebra of polynomials that are symmetric in the two sets of eigenvalues of Y, namely \documentclass[12pt]{minimal}\( \begin{document}y_1,\ldots ,y_p\end{document}\) and \documentclass[12pt]{minimal}\( \begin{document}y_{p+1},\ldots ,y_{p+q}\end{document} \), that become independent of \documentclass[12pt]{minimal}\( \begin{document}y_p\end{document}\) when \documentclass[12pt]{minimal}\( \begin{document}y_p=y_{p+q}\end{document} \). In general, ϕ is not invertible. [23] Cumulants can be defined via \documentclass[12pt]{minimal}\( \begin{document} \langle G_1(M)\ldots G_n(M)\rangle =\sum_{\pi \lbrace 1,\ldots ,n\rbrace }\prod_{J\in \pi }\langle G_{j_1}G_{j_2}\ldots \rangle^c,\end{document} where the sum is over all partitions π of the set {1, …, n}, while \documentclass[12pt]{minimal} \begin{document}J=\lbrace j_1,j_2,\ldots \rbrace\end{document} is an element of π. For instance, \documentclass[12pt]{minimal} \begin{document} \langle AB\rangle =\langle AB \rangle^c+ \langle A \rangle^c\langle B\rangle^c\end{document} and \documentclass[12pt]{minimal} \begin{document} \langle ABC\rangle =\langle ABC\rangle^c+ \langle AB \rangle^c\langle C\rangle^c+\langle AC \rangle^c\langle B \rangle^c+\langle BC \rangle^c\langle A\rangle^c+\langle A\rangle^c\langle B \rangle^c\langle C\rangle^c.\end{document} \) [24] Other nonequivalent definitions for the transpose and the adjoint are possible. They lead, for instance, to a distinct unitary supergroup, namely sU(pq) (See Refs. 3 and 13). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.