Supermatrix models, loop equations, and duality

TL;DR

This paper analyzes supermatrix integrals with external fields, deriving their topological expansion, loop equations, spectral curve, and a duality relating models with interchanged roles of parameters, using symplectic invariants.

Contribution

It introduces a duality for supermatrix models based on symplectic invariants, extending the understanding of their topological expansion and loop equations.

Findings

Derived a topological expansion in powers of 1/(p-q)

Obtained loop equations and spectral curves for supermatrix integrals

Proved a duality relating models with interchanged external fields

Abstract

We study integrals over Hermitian supermatrices of arbitrary size $p+q$, that are parametrized by an external field $X$ and a source $Y$, of respective size $m+n$ and $p+q$. We show that these integrals exhibit a simple topological expansion in powers of a formal parameter $\hbar$, 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 in arXiv:math-ph/0702045. 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.