Unitary matrix integrals in the framework of Generalized Kontsevich Model. I. Brezin-Gross-Witten Model

TL;DR

This paper introduces a novel approach linking unitary matrix models, especially the Brezin-Gross-Witten Model, to Generalized Kontsevich Models with non-polynomial potentials, revealing new phase and classification insights.

Contribution

It establishes a connection between unitary matrix models and GKM with non-polynomial potentials, providing new classifications and methods for analyzing matrix models in external fields.

Findings

Partition function of BGWM as a GKM with potential 1/X

Identification of strong/weak coupling phases with GKM phases

New method for fixing Ward identities in matrix models

Abstract

We advocate a new approach to the study of unitary matrix models in external fields which emphasizes their relationship to Generalized Kontsevich Models (GKM) with non-polynomial potentials. For example, we show that the partition function of the Brezin-Gross-Witten Model (BGWM), which is defined as an integral over unitary $N\times N$ matrices, $\int [dU] e^{\rm{Tr}(J^\dagger U + JU^\dagger)}$, can also be considered as a GKM with potential ${\cal V}(X) = \frac{1}{X}$. Moreover, it can be interpreted as the generating functional for correlators in the Penner model. The strong and weak coupling phases of the BGWM are identified with the "character" (weak coupling) and "Kontsevich" (strong coupling) phases of the GKM, respectively. This sort of GKM deserves classification as $p=-2$ one (i.e. $c=28$ or $c=-2$) when in the Kontsevich phase. This approach allows us to further identify the Harish-Chandra-Itzykson-Zuber (IZ) integral with a peculiar GKM, which arises in the study of $c=1$ theory and, further, with a conventional 2-matrix model which is rewritten in Miwa coordinates. Inspired by the considered unitary matrix models, some further extensions of the GKM treatment which are inspired by the unitary matrix models which we have considered are also developed. In particular, as a by-product, a new simple method of fixing the Ward identities for matrix models in an external field is presented.