Tautological relations via r-spin structures

Relations among tautological classes on the moduli space of stable curves 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 sl2 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 the tautological ring of the moduli of nonsingular curves. 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.Comment: Corrected powers of phi in the analysis of the second shift. Appendix on the moduli of holomorphic differentials by F. Janda, R. Pandharipande, A. Pixton, and D.Zvonkine. Final versio

On double Hurwitz numbers with completed cycles

In this paper, we collect a number of facts about double Hurwitz numbers, where the simple branch points are replaced by their more general analogues --- completed (r+1)-cycles. In particular, we give a geometric interpretation of these generalised Hurwitz numbers and derive a cut-and-join operator for completed (r+1)-cycles. We also prove a strong piecewise polynomiality property in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural ELSV/GJV-type formula, that is, an expression in terms of some intrinsic combinatorial constants that might be related to the intersection theory of some analogues of the moduli space of curves. The structure of these conjectural "intersection numbers" is discussed in detail.Comment: 31 page

Top tautological group of Mg,n

We describe the structure of the top tautological group in the cohomology of the moduli space of smooth genus g curves with n marked points