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Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

Published online by Cambridge University Press:  04 December 2007

Tyler J. Jarvis
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A. E-mail: [email protected]
Takashi Kimura
Affiliation:
Department of Mathematics, 111 Cummington Street, Boston University, Boston, MA 02215, U.S.A. E-mail: [email protected]
Arkady Vaintrob
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A. E-mail: [email protected]
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Abstract

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We prove the genus zero part of the generalized Witten conjecture, relating moduli spaces of higher 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 KdVr 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 singularity Ar−1. 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.

Type
Research Article
Copyright
© 2001 Kluwer Academic Publishers