Generalized Br\'ezin-Gross-Witten tau-function as a hypergeometric solution of the BKP hierarchy

TL;DR

This paper demonstrates that the generalized Brézin-Gross-Witten tau-function is a hypergeometric solution to the BKP hierarchy, linking it to spin Hurwitz numbers and providing explicit constructions and descriptions.

Contribution

It establishes the hypergeometric nature of the tau-function within the BKP hierarchy and connects it to spin Hurwitz numbers, including explicit operators and Grassmannian points.

Findings

Identifies the tau-function as a hypergeometric solution of BKP hierarchy.

Constructs cut-and-join operators for the family of tau-functions.

Provides explicit descriptions of BKP Sato Grassmannian points.

Abstract

In this paper, we prove that the generalized Br\'ezin-Gross-Witten tau-function is a hypergeometric solution of the BKP hierarchy with simple weight generating function. We claim that it describes a spin version of the strictly monotone Hurwitz numbers. A family of the hypergeometric tau-functions of the BKP hierarchy, corresponding to the rational weight generating functions, is investigated. In particular, the cut-and-join operators are constructed, and the explicit description of the BKP Sato Grassmannian points is derived. Representatives of this family can be associated with interesting families of spin Hurwitz numbers including a spin version of the monotone Hurwitz numbers.