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A new spin on Hurwitz theory and ELSV via theta characteristics

Alessandro Giacchetto, Reinier Kramer, Danilo Lewański

Abstract

We study spin Hurwitz numbers, which count ramified covers of the Riemann sphere with a sign coming from a theta characteristic. These numbers are known to be related to Gromov-Witten theory of Kähler surfaces and to representation theory of the Sergeev group, and are generated by BKP tau-functions. We use the latter interpretation to give polynomiality properties of these numbers and we derive a spectral curve which we conjecture computes spin Hurwitz numbers via a new type of topological recursion. We prove that this conjectural topological recursion is equivalent to an ELSV-type formula, expressing spin Hurwitz numbers in terms of the Chiodo class twisted by the $2$-spin Witten class.

A new spin on Hurwitz theory and ELSV via theta characteristics

Abstract

We study spin Hurwitz numbers, which count ramified covers of the Riemann sphere with a sign coming from a theta characteristic. These numbers are known to be related to Gromov-Witten theory of Kähler surfaces and to representation theory of the Sergeev group, and are generated by BKP tau-functions. We use the latter interpretation to give polynomiality properties of these numbers and we derive a spectral curve which we conjecture computes spin Hurwitz numbers via a new type of topological recursion. We prove that this conjectural topological recursion is equivalent to an ELSV-type formula, expressing spin Hurwitz numbers in terms of the Chiodo class twisted by the -spin Witten class.

Paper Structure

This paper contains 37 sections, 71 theorems, 277 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Related developments since prepublication
  3. Open questions
  4. Topological recursion and cohomological field theories
  5. Topological recursion
  6. Cohomological field theories and Givental action
  7. Identification of the two theories
  8. Preliminaries on spin algebra
  9. Spin representations
  10. Supersymmetric functions
  11. Operator formalism
  12. Introduction to spin Hurwitz numbers
  13. Cut-and-join operators and Okounkov--Pandharipande operators
  14. Cut-and-join operators
  15. B-Okounkov--Pandharipande operators
  16. ...and 22 more sections

Key Result

Theorem \oldthetheorem

Single spin Hurwitz numbers are quasi-po-ly-no-mi-al; spin double Hurwitz numbers are piecewise polynomials with explicit wall-crossing formulae.

Theorems & Definitions (170)

  • Theorem \oldthetheorem: \ref{['cor:quasi-poly', 'thm:piecewise', 'thm:wall:crossing']}
  • Theorem \oldthetheorem: SSZ15DKPS23
  • Conjecture \oldthetheorem
  • Theorem \oldthetheorem: AS23
  • Theorem \oldthetheorem: SSZ15DKPS23
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem: EO07
  • Remark \oldthetheorem
  • Definition \oldthetheorem: KM94
  • ...and 160 more