Tautological relations via ����-spin structures

Relations among tautological classes on
M ¯ g , n \overline {\mathcal {M}}_{g,n} are obtained via the study of Witten’s r r -spin theory for higher r r . In order to calculate the quantum product, a new formula relating the r r -spin correlators in genus 0 to the representation theory of s l 2 ( C ) {\mathsf {sl}}_2(\mathbb {C}) is proven. The Givental-Teleman classification of CohFT (cohomological field theory) is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R R -matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten’s r r -spin class is obtained (along with tautological relations in higher degrees). As an application, the r = 4 r=4 relations are used to bound the Betti numbers of R ∗ ( M g ) R^*(\mathcal {M}_g) . At the second semisimple point, the form of the R R -matrix implies a polynomiality property in r r of Witten’s r r -spin class.

In Appendix A (with F. Janda), a conjecture relating the r = 0 r=0 limit of Witten’s r r -spin class to the class of the moduli space of holomorphic differentials is presented.