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Moduli spaces of higher spin curves and integrable hierarchies. (English) Zbl 1015.14028
Summary: 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 KdV 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 $$A_{r-1}$$. We prove analogs of the puncture, dilation, and topological recursion relations by drawing an analogy with the construction of Gromov-Witten invariants and quantum cohomology.

MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14H10 Families, moduli of curves (algebraic) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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