Supermatrix models, loop equations, and duality

Abstract

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.