A connection between the Kontsevich-Witten and Brezin-Gross-Witten tau-functions

TL;DR

This paper establishes a mathematical connection between the Brezin-Gross-Witten and Kontsevich-Witten tau-functions through a representation involving $W_{1+ abla}$ operators, linking two fundamental matrix model partition functions.

Contribution

It introduces a representation of the generalized BGW tau-function using $W_{1+ abla}$ operators that maintains KP integrability, connecting it to the Kontsevich-Witten tau-function via $GL( abla)$ operators.

Findings

Established a $W_{1+ abla}$ operator representation of $ au_N$

Connected BGW and Kontsevich-Witten tau-functions via $GL( abla)$ operators

Preserved KP integrability in the new representation

Abstract

The Brezin-Gross-Witten (BGW) model is one of the basic examples in the class of non-eigenvalue unitary matrix models. The generalized BGW tau-function $\tau_N$ was constructed from a one parametric deformation of the original BGW model using the generalized Kontsevich model representation. It is a tau-function of the KdV hierarchy for any value of $N\in{\mathbb C}$, where the case $N=0$ reduces to the original BGW tau-function. In this paper, we present a representation of $\tau_N$ in terms of the $W_{1+\infty}$ operators that preserves the KP integrability. This naturally establishes a connection between the (generalized) BGW and Kontsevich-Witten tau-functions using $GL(\infty)$ operators, both considered as the basic building blocks in the theory of matrix models and partition functions.