2009/11/09 by Patrick Desrosiers, Bertrand Eynard · 5 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Duality (order theory) #Hermitian matrix #Homotopy and Cohomology in Algebraic Topology #Iterated function #Loop (graph theory) #Quantum Mechanics and Non-Hermitian Physics #Simple (philosophy) #Supermatrix #Symmetry (geometry) #Symplectic geometry #hep-th #math-ph #math.MP #msc:05C30 #msc:14H70 #msc:15A52
paper · pdf · doi:10.1063/1.3430564
published in Journal of Mathematical Physics 51(12) (American Institute of Physics) · 27 pages, 9 figures, ams latex
arxiv created 2009/11/09 · openalex publication_date 2010/12/01 · arxiv updated 2012/08/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 ℏ, 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 Eynard and Orantin [Commun. Number Theory Phys. 1, 347 (2007)]. 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.