An alternative approach to KP hierarchy in matrix models

TL;DR

This paper presents a novel method to derive differential hierarchies like KdV directly from one-matrix models without the need for a continuum limit, using Toda lattice reformulation.

Contribution

It introduces an alternative approach to extract integrable hierarchies from matrix models by reformulating the Toda lattice in operator form.

Findings

Successfully derives KdV hierarchy without continuum limit

Reformulates Toda lattice in operator form for matrix models

Provides a new perspective on matrix model integrability

Abstract

We show that there exists an alternative procedure in order to extract differential hierarchies, such as the KdV hierarchy, from one--matrix models, without taking a continuum limit. To prove this we introduce the Toda lattice and reformulate it in operator form. We then consider the reduction to the systems appropriate for one--matrix model.