Spin Hurwitz numbers and topological quantum field theory

Gunningham, Sam

doi:10.2140/gt.2016.20.1859

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Spin Hurwitz numbers and topological quantum field theory

Sam Gunningham

Geometry & Topology 20 (2016) 1859–1907

DOI: 10.2140/gt.2016.20.1859

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 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

Mathematical Subject Classification 2010

Primary: 81T45

References

Publication

Received: 8 February 2014

Revised: 11 August 2015

Accepted: 24 September 2015

Published: 15 September 2016

Proposed: Jim Bryan

Seconded: Richard Thomas, Peter Teichner

Authors

Sam Gunningham
Department of Mathematics University of Texas at Austin 2515 Speedway Stop C1200 Austin, TX 78712 United States

Volume 20, issue 4 (2016)