Communications in Number Theory and Physics

Volume 13 (2019)

Number 4

Brezin–Gross–Witten tau function and isomonodromic deformations

Pages: 827 – 883



Marco Bertola (Department of Mathematics and Statistics, Concordia University, Montréal, QC, Canada; Centre de recherches mathématiques, Université de Montréal, QC, Canada; and Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy)

Giulio Ruzza (Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy)


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. This tau function admits a natural extension, called generalized Brezin–Gross–Witten tau function. We prove that the latter 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 formulæ for the generating functions of the correlators in terms of simple generating series, the Virasoro constraints, and discuss the relation with the Painlevé XXXIV hierarchy.


Brezin–Gross–Witten tau function, isomonodromic deformations, Painlevé XXXIV hierarchy, Norbury classes

2010 Mathematics Subject Classification

14D21, 14H70, 35Q15

The work of M.B. is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2016-06660 and by the FQRNT grant “Applications des systèmes intégrables à les surfaces de Riemann et les espaces de modules” (2016-PR-190918). G.R. wishes to thank the Department of Mathematics and Statistics at Concordia University where this work was carried out. This project has received funding from the European Union’s H2020 research and innovation programme under the Marie Sklowdoska-Curie grant No. 778010 IPaDEGAN.

Received 19 December 2018

Accepted 20 August 2019

Published 6 December 2019