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 2-spin Witten class.

Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewanski conjecture

“…We address the reader to [EOP08, Gun16, Lee18, Lee20, MMN20, GKL21] for the basic definitions and properties. Different authors use different conventions, our notations are consistent with that of [GKL21].…”

“…Let us remark that while with the motivation coming from [GKL21] we focus on this particular family of spin Hurwitz numbers, we expect that our modification of the methods of [BDBKS20a,BDBKS20b] should immediately work for other families of the generating functions of the spin Hurwitz numbers, analogous to the families investigated in [BDBKS20b]. We also expect that the integrable approach to the topological recursion in the BKP case should be as universal as for the KP case.…”

“…Giacchetto-Kramer-Lewański conjecture and its generalization. Additional input and motivation to study the correlation functions of the corresponding hypergeometric 2-BKP tau-functions comes from a recent work of Giaccheto, Kramer, and Lewański [GKL21]. They study in detail the theory of so-called spin Hurwitz numbers with completed cycles, both single and double, whose elements occur naturally in a number of other works in relation to computation of the volumes of strata in the moduli spaces of holomorphic differentials [EOP08] and Gromov-Witten theory of Kähler surfaces [Lee20,Introduction].…”

Integrals of $ψ$-classes on twisted double ramification cycles and spaces of differentials

“…In this section, we give an explicit proposal for the cycle DR spin g from Assumption 1.3 as well as a conjecture generalizing Conjecture A of [FP18] and [Sch18] to the spin setting. The proposal for DR spin g is inspired by recent work [GKL21] on spin Hurwitz numbers. Its formula consists in a small modification of the original formulas for the double ramification cycle presented above.…”

“…As mentioned before, the formula above is inspired by the paper [GKL21]. There, the authors introduce a spin Chiodo class C ϑ (r, k; a), where the modified vector a is defined by the convention 2 a i + 1 = a i .…”