Tautological relations via 𝑟-spin structures

Abstract

Relations among tautological classes on M g,n are obtained via the study of Witten's r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl 2 (C) is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten's r-spin class is obtained (along with tautological relations in higher degrees). As an application, the r = 4 relations are used to bound the Betti numbers of R * (M g ). At the second semisimple point, the form of the R-matrix implies a polynomiality property in r of Witten's r-spin class.In the Appendix (with F. Janda), a conjecture relating the r = 0 limit of Witten's r-spin class to the class of the moduli space of holomorphic differentials is presented. 0 Introduction 0.1 Overview Let M g,n be the moduli space of stable genus g curves with n markings. Letbe the subring of tautological classes in cohomology 1 . The subringsare defined together as the smallest system of Q-subalgebras closed under push-forward via all boundary and forgetful maps, see [6,7,12]. There has been substantial progress in the understanding of RH * (M g,n ) since the study began in the 1980s [21]. The subject took a new turn in 2012 with the family of relations conjectured in [25]. We refer the reader to [23] for a survey of recent developments.Witten's r-spin class defines a Cohomological Field Theory (CohFT) for each integer r ≥ 2. Witten's 2-spin theory concerns only the fundamental classes of the moduli spaces of curves (and leads to no new geometry). In our previous paper [24], we used Witten's 3-spin theory to construct a family of relations among tautological classes of M g,n equivalent (in cohomology) to the relations proposed in [25]. Our goal here is to extend our study of tautological relations to Witten's r-spin theory for all r ≥ 3.Taking [24] as a starting point, Janda has completed a formal study of tautological relations obtained from CohFTs. Two results of Janda are directly relevant here:(i) The relations for r = 3 are valid in Chow [14,16].(ii) The relations for r ≥ 4 are implied by the relations for r = 3 [14,15]. By (i) and (ii) together, all of the r-spin relations that we find will be valid in Chow. However, since our methods here are cohomological, we will use the language of cohomology throughout the paper.

Closed extended r-spin theory and the Gelfand–Dickey wave function

“…It is interesting to compare this result with the explicit formula for genus-zero primary closed r-spin intersection numbers [30]. One can immediately see that the structure of the primary open intersection numbers is much simpler than the structure of the primary closed intersection numbers.…”

Cohomological field theories with non-tautological classes

“…There are several constructions of CohFTs with unit -Gromov-Witten theory, Witten's r-spin class, and the Chern characters of the Verlinde bundle all define CohFTs with unit, see [14,16,17,18]. Moreover, once a CohFT is found, others can be constructed via the action of the Givental group, see [17,23,25].…”

Cohomological Field Theory Calculations

“…Much of what I know about the Givental-Teleman classification was learned through writing [22] • Sections 1-2 are based on the papers [32,33] and the Appendix of [33],…”