Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

TL;DR

This paper proves the genus zero part of the generalized Witten conjecture, linking moduli spaces of higher spin curves to integrable hierarchies, and constructs a cohomological field theory with applications to Frobenius manifolds.

Contribution

It establishes the genus zero correspondence between intersection numbers on moduli spaces of r-spin curves and solutions to Gelfand-Dickey hierarchies, extending the Witten conjecture.

Findings

Intersection numbers form a generating function solving the semiclassical KdV_r equations.

Constructs a cohomological field theory of rank r-1 in all genera.

Shows the Frobenius manifold structure matches the deformation of A_{r-1} singularity.

Abstract

We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to Gelfand-Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdV_r equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank $r-1$ in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the A_{r-1} singularity. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of Gromov-Witten invariants and quantum cohomology.