#### Volume 20, issue 4 (2016)

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Spin Hurwitz numbers and topological quantum field theory

### Sam Gunningham

Geometry & Topology 20 (2016) 1859–1907
##### 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 $±1$ according to the parity of the covering surface. These numbers were first defined by Eskin, Okounkov and Pandharipande in order to study the moduli of holomorphic differentials on a Riemann surface. They have also been related to Gromov–Witten invariants of complex $2$–folds by work of Lee and Parker and work of Maulik and 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 and 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 and Teleman.

##### Keywords
spin Hurwitz numbers, topological quantum field theory
Primary: 81T45