Spin Hurwitz numbers and topological quantum field theory

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-Pandharipande in order to study the moduli of holomorphic differentials on a Riemann surface. They have also been related to Gromov-Witten invariants of of complex 2-folds by work of Lee-Parker and Maulik-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-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-Teleman.

38 pages, substantially rewritten and reorganized following referees' advice. Diagrams added. The key results are unchanged

References in corpus (5)

• The Classification of Two-Dimensional Extended Topological Field Theories
• D-branes and K-theory in 2D topological field theory
• Short-range entanglement and invertible field theories
• Recursion Formulas for Spin Hurwitz Numbers
• An obstruction bundle relating Gromov-Witten invariants of curves and Kahler surfaces

Cited by in corpus (18)

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• Short-range entanglement and invertible field theories
• Cut-and-join structure and integrability for spin Hurwitz numbers
• Around spin Hurwitz numbers
• Tori Detect Invertibility of Topological Field Theories
• Generalized Brézin-Gross-Witten tau-function as a hypergeometric solution of the BKP hierarchy
• Stable pairs with descendents on local surfaces I: the vertical component
• Connection between cut-and-join and Casimir operators
• Topological field theory on r-spin surfaces and the Arf invariant
• Spin Hurwitz theory and Miwa transform for the Schur Q-functions
• Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewanski conjecture
• Reflection Structures and Spin Statistics in Low Dimensions
• Recursion Formulas for Spin Hurwitz Numbers
• Kac-Schwarz Operators of Type $B$, Quantum Spectral Curves, and Spin Hurwitz Numbers
• Symmetric Group Gauge Theories and Simple Gauge/String Dualities
• Connected $(n,m)$-Point Functions of Diagonal $2$-BKP Tau-Functions, and Spin Double Hurwitz Numbers
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
• Symplectic Geometry of Moduli Spaces of Hurwitz Covers