Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions

TL;DR

This paper introduces a fermionic representation for two-matrix model integrals, enabling perturbative expansions in terms of Schur functions and establishing connections with integrable hierarchies.

Contribution

It provides a novel fermionic framework for two-matrix models and generalizes Schur function expansions, linking them to integrable systems.

Findings

Derived perturbation series as Schur function expansions

Established links with KP and Toda lattice hierarchies

Extended previous results to more general two-matrix models

Abstract

A new representation of the 2N fold integrals appearing in various two-matrix models that admit reductions to integrals over their eigenvalues is given in terms of vacuum state expectation values of operator products formed from two-component free fermions. This is used to derive the perturbation series for these integrals under deformations induced by exponential weight factors in the measure, expressed as double and quadruple Schur function expansions, generalizing results obtained earlier for certain two-matrix models. Links with the coupled two-component KP hierarchy and the two-component Toda lattice hierarchy are also derived.