Integrable Discrete Linear Systems and One-Matrix Model

TL;DR

This paper explores the connection between one-matrix models and discrete linear systems, revealing how gauge invariances and the continuum limit relate to integrability, Virasoro constraints, and the KdV hierarchy.

Contribution

It demonstrates that the integrability of one-matrix models is governed by gauge invariances in the associated discrete linear systems and links the continuum limit to the KdV hierarchy.

Findings

Virasoro constraints arise from the discrete linear system consistency conditions.

Gauge invariance under time-independent transformations ensures model integrability.

Partition function is identified as the τ-function of the KdV hierarchy.

Abstract

In this paper we analyze one-matrix models by means of the associated discrete linear systems. We see that the consistency conditions of the discrete linear system lead to the Virasoro constraints. The linear system is endowed with gauge invariances. We show that invariance under time-independent gauge transformations entails the integrability of the model, while the double scaling limit is connected with a time-dependent gauge transformation. We derive the continuum version of the discrete linear system, we prove that the partition function is actually the $\tau$-function of the KdV hierarchy and that the linear system completely determines the Virasoro constraints.