Spin Hurwitz numbers and topological quantum field theory

TL;DR

This paper constructs a topological quantum field theory that computes spin Hurwitz numbers, linking algebraic combinatorics with geometric topology and extending previous formulas to all genera.

Contribution

It introduces a new spin TQFT for calculating spin Hurwitz numbers and generalizes existing formulas using Sergeev algebra combinatorics.

Findings

Constructed a spin TQFT for Hurwitz numbers

Derived a genus formula via Sergeev algebra

Presented a method to average TQFTs over finite covers

Abstract

Spin Hurwitz numbers count ramified covers of a spin surface, weighted by the size of their automorphism group (like ordinary Hurwitz numbers), but signed $\pm 1$ according to the parity of the covering surface. These numbers were first defined by Eskin-Okounkov-Pandharipande in order to study the moduli of holomorphic differentials on a Riemann surface. They have also been related to Gromov-Witten invariants of of complex 2-folds by work of Lee-Parker and Maulik-Pandharipande. In this paper, we construct a (spin) TQFT which computes these numbers, and deduce a formula for any genus in terms of the combinatorics of the Sergeev algebra, generalizing the formula of Eskin-Okounkov-Pandharipande. During the construction, we describe a procedure for averaging any TQFT over finite covering spaces based on the finite path integrals of Freed-Hopkins-Lurie-Teleman.