Equivariant Gromov-Witten theory of one dimensional stacks

Abstract

Gromov-Witten theory constructs moduli spaces of maps from curves to a target space and gives a virtual count of such maps satisfying given conditions by intersecting cycles on these moduli spaces. A primary interest of Gromov-Witten theory is the recursive structure of these moduli spaces and virtual counts. A simple instance of this is Kontsevich's recursive formula for the number of degree $d$ rational curves through $3d-1$ points in the plane. Okounkov and Pandharipande have investigated more complicated recursive structures in the case when the target space is a curve; their first result is that the generating function for the equivariant Gromov-Witten theory of the sphere satisfies a set of differential equations known as the 2-Toda hierarchy.

We extend the above result of Okounkov and Pandharipande to one dimensional toric stacks. An operator formalism for computing the equivariant Gromov-Witten invariants of a one dimensional toric stack is developed. If the stack is effective (that is, if the generic isotropy group is trivial), these operators act on the infinite wedge. If the generic point of the stack has isotropy group $K$, then these operators act on a Fock space associated with the representation theory of wreath products of $K$, which is essential a tensor product of copies of the infinite wedge.

Two applications of this operator formalism are presented. First, we show that the equivariant Gromov-Witten theory of these stacks satisfy the 2-Toda hierarchy. Second, we prove they satisfy the decomposition conjecture, namely, that the Gromov-Witten theories of ineffective orbifolds decompose into copies of the underlying effective orbifold. The main tools used are virtual localization and an orbifold version of the ELSV formula, relating Hurwitz-Hodge integrals to double Hurwitz numbers.