Virasoro versus superintegrability. Gaussian Hermitian model

TL;DR

This paper explores how superintegrability enhances the understanding of Virasoro constraints in Gaussian Hermitian matrix models, leading to elegant recursive formulas and clearer separation of matrix size dependence.

Contribution

It demonstrates that superintegrability simplifies Virasoro constraints into recursive identities for Schur functions, clarifying their relation in Gaussian Hermitian models.

Findings

Virasoro constraints can be reformulated as identities for Schur functions.

Superintegrability allows explicit formulas for Gaussian averages of characters.

The approach separates matrix size dependence from the constraints.

Abstract

Relation between the Virasoro constraints and KP integrability (determinant formulas) for matrix models is a lasting mystery. We elaborate on the claim that the situation is improved when integrability is enhanced to super-integrability, i.e. to explicit formulas for Gaussian averages of characters. In this case, the Virasoro constraints are equivalent to simple recursive formulas, which have appropriate combinations of characters as their solutions. Moreover, one can easily separate dependence on the size of matrix, and deduce superintegrability from the Virasoro constraints. We describe one of the ways to do so for the Gaussian Hermitian matrix model. The result is a spectacularly elegant reformulation of Virasoro constraints as identities for the Schur functions evaluated at appropriate loci in the space of time-variables.