2006/10/19 by Bene, Alex James
#57M99 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.math/0610603
A closed formula is obtained for the integral ∫_\mathcalHg1κ1ψ2g-2 of tautological classes over the locus of hyperelliptic Weierstraß 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 Mg,1 and the combinatorial moduli space Mcombg,1, a PL-orbifold whose cells are enumerated by fatgraphs. This cell decomposition can be used to naturally construct combinatorial PL-cycles Wa\subsetMcombg,1 whose homology classes are essentially the Poincaré duals of the Mumford-Morita-Miller classes κa. In this paper we construct another PL-cycle Hcombg ⊂ Mcombg,1 representing the locus of hyperelliptic Weierstraß points and explicitly describe the chain level intersection of this cycle with W1. Using this description of Hcombg∩ W1, the duality between Witten cycles Wa and the κa classes, and Kontsevich's scheme of integrating ψ classes, the integral ∫_\mathcalHg1κ1ψ2g-2 is reduced to a weighted sum over graphs and is evaluated by the enumeration of trees.