Tensor products of Frobenius manifolds and moduli spaces of higher spin curves

Abstract

We review progress on the generalized Witten conjecture and some of its major ingredients. This conjecture states that certain intersection numbers on the moduli space of higher spin curves assemble into the logarithm of the tau function of a semiclassical limit of the r-th Gelfand-Dickey (or KdV_r) hierarchy. Additionally, we prove that tensor products of the Frobenius manifolds associated to such hierarchies admit a geometric interpretation in terms of moduli spaces of higher spin structures. We also elaborate upon the analogy to Gromov-Witten invariants of a smooth, projective variety.