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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 ,
Takashi Kimura  and
Arkady Vaintrob
Tyler J. Jarvis
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A. E-mail: Jarvis@math.byu.edu
Takashi Kimura
Affiliation:
Department of Mathematics, 111 Cummington Street, Boston University, Boston, MA 02215, U.S.A. E-mail: kimura@math.bu.edu
Arkady Vaintrob
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A. E-mail: vaintrob@math.uoregon.edu
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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.

Keywords

higher spin curvesintegrable hierarchiesFrobenius manifoldscohomological field theoriesgeneralized Witten conjecture
Type
Research Article
Information
Compositio Mathematica , Volume 126 , Issue 2 , April 2001 , pp. 157 - 212
Copyright
© 2001 Kluwer Academic Publishers