Mathematics – Quantum Algebra
Scientific paper
2012-01-05
Mathematics
Quantum Algebra
23 pages. Introduction rewritten to include motivation from ordinary Hurwitz numbers, and references to related work
Scientific paper
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 Atiyah invariant (parity) of the covering surface. These are related to Gromov-Witten invariants of complex 2-folds by the work of Lee-Parker and Maulik-Pandharipande. A formula was given in the genus 1 case by Eskin-Okounkov-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 (or equivalently, the spin representations of the symmetric group). 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. We also give a complete description of fully extended 2d spin TQFTs, following the cobordism hypothesis of Lurie.
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