Stable Spin Maps, Gromov-Witten Invariants, and Quantum Cohomology

Abstract

We introduce the stack of r-spin maps. These are stable maps into a variety V from n-pointed algebraic curves of genus g, with the additional data of an r-spin structure on the curve. We prove that this stack is a Deligne-Mumford stack, and we define analogs of the Gromov-Witten classes associated to these spaces. We show that these classes yield a cohomological field theory (CohFT) that is the tensor product of the CohFT associated to the usual Gromov-Witten invariants of V and the r-spin CohFT. When r=2, our construction gives the usual Gromov-Witten invariants of V. Restricting to genus zero, we obtain the notion of an r-spin quantum cohomology of V, whose Frobenius structure is isomorphic to the tensor product of the Frobenius manifolds corresponding to the quantum cohomology of V and the r-th Gelfand-Dickey hierarchy (or, equivalently, the A_{r-1} singularity). We also prove a generalization of the descent property which, in particular, explains the appearance of the psi-classes in the definition of gravitational descendants. Finally, we compute the small phase space potential function when r=3 and V=CP^1.