A topological quantum field theory of intersection numbers on moduli spaces of admissible covers

Abstract

We construct a two-level weighted topological quantum field theory whose structure coefficients are equivariant intersection numbers on moduli spaces of admissible covers. Such a structure is parallel (and strictly related) to the local Gromov–Witten theory of curves of Bryan and Pandharipande. We compute explicitly the theory using techniques of localization on moduli spaces of admissible covers of a parametrized ${\mathbb{P}}^1$. The Frobenius algebras we obtain are one-parameter deformations of the class algebra of the symmetric group $S_d$. In certain special cases we are able to produce explicit closed formulas for such deformations in terms of the representation theory of $S_d$.