Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

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.

65 pages, postscript figures, AMS-LaTeX, uses Paul Taylor's diagrams.tex. Exposition improved. Many minor corrections made

References in corpus (3)

• Virasoro constraints and the Chern classes of the Hodge bundle
• Topological recursion relations in genus 2
• A descendent relation in genus 2

Cited by in corpus (7)

• Tautological relations and the r-spin Witten conjecture
• Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations
• Pointed Admissible G-Covers and G-equivariant Cohomological Field Theories
• Geometry of meromorphic functions and intersections on moduli spaces of curves
• Crossed simplicial groups and structured surfaces
• Punctures and p-spin curves from matrix models
• Intersections in genus 3 and the Boussinesq hierarchy