Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

Tyler J. Jarvis; Takashi Kimura; Arkady Vaintrob

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Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

Tyler J. Jarvis·Takashi Kimura·Arkady Vaintrob

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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.

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BibTeX RIS

Tyler J. Jarvis, Takashi Kimura, Arkady Vaintrob. 2000-02-09. Moduli Spaces of Higher Spin Curves and Integrable Hierarchies. https://arxiv.org/abs/math/9905034

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