Tau functions of the UC hierarchy as partition functions of matrix models

TL;DR

This paper links tau functions of the UC hierarchy to matrix model partition functions, introducing new models related to sphere products and extending to multi-matrix cases for the multi-component hierarchy.

Contribution

It establishes a novel connection between UC hierarchy tau functions and matrix models, including new models for sphere products and multi-matrix generalizations.

Findings

Partition functions of certain matrix models are tau functions of the UC hierarchy.

New matrix models associated with products of two spheres and embedded graphs.

Extension to multi-matrix models corresponding to multi-component UC hierarchy.

Abstract

We present a family of matrix models such that their partition functions are tau functions of the universal character (UC) hierarchy. This develops one of the topics of our previous paper arXiv:2410.14823. We found new matrix models associated with the product of two spheres with embedded graphs via a gluing matrix. We also generalize these studies to multi-matrix models case, which corresponds to the multi-component UC hierarchy.