Relations in the tautological ring derived from combinatorial classes and hyperelliptic fatgraphs

Abstract

AbstractA closed formula is obtained for the integral$\int_{\Hgbs^1}\kappa_{1}\psi^{2g-2}$of tautological classes over the locus of hyperelliptic Weier points in the moduli space of curves. As a corollary, a relation between Hodge integrals is obtained.The calculation utilizes the homeomorphism between the moduli space of curves$\M_{g,1}$and the combinatorial moduli space$\Mc_{g,1}$, a PL-orbifold whose cells are enumerated by fatgraphs. This cell decomposition can be used to naturally construct combinatorial PL-cycles$W_a\subset\Mc_{g,1}$whose homology classes are essentially the Poin duals of the Mumford–Morita–Miller classes κa. In this paper we construct another PL-cycle$\Hgc \subset \Mc_{g,1}$representing the locus of hyperelliptic Weier points and explicitly describe the chain level intersection of this cycle withW1. Using this description of$\Hgc\cap W_1$, the duality between Witten cyclesWaand the κaclasses, and the Kontsevich--Penner method of integration, scheme of integrating ε classes, the integral$\int_{\Hgbs^1}\kappa_{1}\psi^{2g-2}$is reduced to a weighted sum over graphs and is evaluated by the enumeration of trees.