Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

Details Moduli Spaces of Higher Spin Curves and Integrable Hierarchies Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

1999-05-05

arxiv.org/abs/math/9905034v4

Compositio Mathematica 126:157-212 (2001)

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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