A TQFT of Intersection Numbers on Moduli Spaces of Admissible Covers

We construct a two-level weighted TQFT whose structure coefficents are equivariant intersection numbers on moduli spaces of admissible covers. Such a structure is parallel (and strictly related) to the local GromovWitten theory of curves in [BP04]. We compute explicitly the theory using techniques of localization on moduli spaces of admissible covers of a parametrized P. The Frobenius Algebras we obtain are one parameter deformations of the class algebra of the symmetric group Sd. In certain special cases we are able to produce explicit closed formulas for such deformations in terms of the representation theory of Sd. Introduction This paper studies a large class of (equivariant) intersection numbers on moduli spaces of admissible covers. For a smooth algebraic curve X , ramified covers of a given degree of X by smooth curves of a given genus are parametrized by moduli spaces called Hurwitz schemes. A smooth compactification of a Hurwitz scheme can be obtained by allowing suitable degenerations, called admissible covers. Moduli spaces of admissible covers were introduced originally by Harris and Mumford in [HM82]. Intersection theory on these spaces was for a long time extremely hard and mysterious, mostly because they are in general not normal, even if the normalization is always smooth. Only recently in [ACV01], Abramovich, Corti and Vistoli exhibit this normalization as the stack of balanced stable maps of degree 0 from twisted curves to the classifying stack BSd. This way they attain both the smoothness of the stack and a nice modulitheoretic interpretation of it. We abuse notation and refer to the AbramovichCorti-Vistoli (ACV) spaces as admissible covers. At about the same time, Ionel developed a parallel theory in the symplectic category ([Ion02]) and used push-pull techniques on admissible covers to produce new relations in the tautological ring of Mg,n (see also [Ion05]). In [GV03b], admissible cover loci within the boundary of moduli spaces of stable maps play a key role in establishing the result that the degree 3g−3 part