Unitary Integrals and Related Matrix Models

TL;DR

This paper reviews the fundamental properties of unitary matrix integrals, exploring their connections with three matrix models and addressing complex issues like anomalies and correlators, highlighting the need for further research in this under-explored area.

Contribution

It provides a comprehensive review of unitary matrix integrals, emphasizing technical tools and unresolved problems, and advocates for increased investigation in this field.

Findings

Analysis of the three matrix models related to unitary integrals

Discussion of anomalies and correlator problems in the field

Identification of under-explored areas promising future research

Abstract

Concise review of the basic properties of unitary matrix integrals. They are studied with the help of the three matrix models: the ordinary unitary model, Brezin-Gross-Witten model and the Harish-Charndra-Itzykson-Zuber model. Especial attention is paid to the tricky sides of the story, from De Wit-t'Hooft anomaly in unitary integrals to the problem of correlators with Itzykson-Zuber measure. Of technical tools emphasized is the method of character expansions. The subject of unitary integrals remains highly under-investigated and a lot of new results are expected in this field when it attracts sufficient attention.