BKP and projective Hurwitz numbers

TL;DR

This paper establishes a connection between hypergeometric tau functions of the BKP hierarchy and generating functions for weighted Hurwitz numbers on the real projective plane, extending known results to new geometries and polynomial weights.

Contribution

It introduces the use of BKP hierarchy tau functions as generating functions for projective Hurwitz numbers, including new weighted cases with Hall-Littlewood and Macdonald polynomials.

Findings

Tau functions generate weighted Hurwitz numbers for $ ext{RP}^2$

Extension of $ ext{CP}^1$ results to Klein surfaces

Inclusion of Hall-Littlewood and Macdonald polynomial weights

Abstract

We consider $d$-fold branched coverings of the projective plane $\mathbb{RP}^2$ and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz numbers. In particular we get the $\mathbb{RP}^2$ analogues of the $\mathbb{CP}^1$ generating functions proposed by Okounkov and by Goulden and Jackson. Other examples are Hurwitz numbers weighted by the Hall-Littlewood and by the Macdonald polynomials. We also consider integrals of tau functions which generate projective Hurwitz numbers and Hurwitz numbers related to different Euler characteristics of the base Klein surfaces.