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Spin Hurwitz numbers and topological quantum field theory

Sam Gunningham

Geometry & Topology 20 (2016) 1859–1907

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.

spin Hurwitz numbers, topological quantum field theory
Mathematical Subject Classification 2010
Primary: 81T45
Received: 8 February 2014
Revised: 11 August 2015
Accepted: 24 September 2015
Published: 15 September 2016
Proposed: Jim Bryan
Seconded: Richard Thomas, Peter Teichner
Sam Gunningham
Department of Mathematics
University of Texas at Austin
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