On higher Br\'ezin-Gross-Witten tau-functions

TL;DR

This paper explores the mathematical structures of higher Brézin-Gross-Witten tau-functions, including their operators, spectral curves, and constraints, and introduces a generalization with potential implications for topological recursion.

Contribution

It constructs canonical operators, spectral curves, and constraints for these tau-functions, and introduces a one-parameter generalization.

Findings

Construction of Kac--Schwarz operators and quantum spectral curves

Development of $W^{(3)}$-constraints for tau-functions

Introduction of a one-parametric generalization

Abstract

In this paper, we consider the higher Br\'ezin--Gross--Witten tau-functions, given by the matrix integrals. For these tau-functions we construct the canonical Kac--Schwarz operators, quantum spectral curves, and $W^{(3)}$-constraints. For the simplest representative we construct the cut-and-join operators, which describe the algebraic version of the topological recursion. We also investigate a one-parametric generalization of the higher Br\'ezin--Gross--Witten tau-functions.