Spin Hurwitz numbers and topological quantum field theory. (English) Zbl 1347.81070

Gunningham, Sam

Geom. Topol. 20, No. 4, 1859-1907 (2016).

Summary: 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.

MSC:

81T45	Topological field theories in quantum mechanics
53D30	Symplectic structures of moduli spaces
53D37	Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14H10	Families, moduli of curves (algebraic)
14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)

Keywords:

spin Hurwitz numbers; topological quantum field theory