Hodge-type integrals on moduli spaces of admissible covers

Hodge integrals are a class of intersection numbers on moduli spaces of curves involving the tautological classes λi, which are the Chern classes of the Hodge bundle E. In recent years Hodge integrals have shown a great amount of interconnections with Gromov-Witten theory and enumerative geometry. The classical Hurwitz numbers, counting the numbers of ramified Covers of a curve with an assigned set of ramification data, can be computed via Hodge integrals. Simple Hurwitz numbers have been discussed in [ELSV99], [ELSV01] and [GV03]; progress towards double Hurwitz numbers has been made in [GJV03]. Various spectacular computations of Hodge integrals were carried out in the late nineties by Faber and Pandharipande ( [FP00]). Their results have been used to determine the multiple cover contributions in the GW invariants of P, thus extending the well-known Aspinwall-Morrison formula in GromovWitten Theory. Hodge integrals are also at the heart of the theory developed in [BP04], studying the local Gromov-Witten theory of curves. It is this last theory that brought our attention to a similar type of integrals. We study moduli spaces of Admissible Covers, a natural compactification of the Hurwitz scheme. It has been shown in [ACV01] that these spaces are smooth Deligne Mumford stacks. A class of natural intersection numbers on these spaces, parallel (and we believe related) to the structure coefficients of the Topological Quantum Field Theory in [BP04], are obtained in the