46 research outputs found
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