Around spin Hurwitz numbers

TL;DR

This paper reviews spin Hurwitz numbers, their connection to $Q$ Schur functions and integrable hierarchies, and how they can be generated by specific $ au$-functions within the BKP hierarchy.

Contribution

It provides a modern framework linking spin Hurwitz numbers to matrix models, integrable hierarchies, and $Q$ Schur functions, including fermionic realizations and factorization formulas.

Findings

Spin Hurwitz numbers relate to $Q$ Schur functions and Sergeev group characters.

Generating functions are hypergeometric $ au$-functions of the BKP hierarchy.

Construction of $ au$-functions entirely from $Q$ Schur functions is demonstrated.

Abstract

We present a review of the spin Hurwitz numbers, which count the ramified coverings with spin structures. They are related to peculiar $Q$ Schur functions, which are actually related to characters of the Sergeev group. This allows one to put the whole story into the modern context of matrix models and integrable hierarchies. Hurwitz partition functions are actually broader than conventional $\tau$-functions, but reduce to them in particular circumstances. We explain, how a special $d$-soliton $\tau$-functions of KdV and Veselov-Novikov hierarchies generate the spin Hurwitz numbers $H^\pm\left( \Gamma^r_d \right)$ and $H^\pm\left( \Gamma^r_d,\Delta \right)$. The generating functions of the spin Hurwitz numbers are hypergeometric $\tau$-functions of the BKP integrable hierarchy, and we present their fermionic realization. We also explain how one can construct $\tau$-functions of this type entirely in terms of the $Q$ Schur functions. An important role in this approach is played by factorization formulas for the $Q$ Schur functions on special loci.