# Brezin-Gross-Witten tau function and isomonodromic deformations

## TL;DR

This paper links the Brezin-Gross-Witten tau function to isomonodromic systems, providing new formulas and insights into its structure, hierarchy relations, and geometric interpretation within integrable systems and matrix models.

## Contribution

It proves that a generalized Brezin-Gross-Witten tau function is an isomonodromic tau function, connecting it to monodromy-preserving deformations and deriving explicit formulas.

## Key findings

- Identifies the tau function as an isomonodromic tau function of a 2x2 system.
- Derives effective formulas for correlator generating functions.
- Discusses relations with the Painlevé XXXIV hierarchy.

## Abstract

The Brezin-Gross-Witten tau function is a tau function of the KdV hierarchy which arises in the weak coupling phase of the Brezin-Gross-Witten model. It falls within the family of generalized Kontsevich matrix integrals, and its algebro--geometric interpretation has been unveiled in recent works of Norbury. We prove that a suitably generalized Brezin-Gross-Witten tau function is the isomonodromic tau function of a $2\times 2$ isomonodromic system and consequently present a study of this tau function purely by means of this isomonodromic interpretation. Within this approach we derive effective formul\ae\ for the generating functions of the correlators in terms of simple generating series, the Virasoro constraints, and discuss the relation with the Painlev\'{e} XXXIV hierarchy.