New matrix model solutions to the Kac-Schwarz problem

TL;DR

This paper explores specific solutions to the Kac-Schwarz problem involving matrix models and differential operators, revealing new classes of KP tau-functions expressed as advanced matrix integrals.

Contribution

It introduces novel matrix model solutions to the Kac-Schwarz problem under different operator constraints, expanding the understanding of associated KP tau-functions.

Findings

Pair of constraints with [K_1,W]=1 lead to Kontsevich models

Constraints with [K_1,W]=W produce more complex matrix integrals

New solutions enrich the classification of matrix models in integrable systems

Abstract

We examine the Kac-Schwarz problem of specification of point in Grassmannian in the restricted case of gap-one first-order differential Kac-Schwarz operators. While the pair of constraints satisfying $[{\cal K}_1,W] = 1$ always leads to Kontsevich type models, in the case of $[{\cal K}_1,W] = W$ the corresponding KP $\tau$-functions are represented as more sophisticated matrix integrals.