Moduli spaces of higher spin curves and integrable hierarchies. (English) Zbl 1015.14028

Jarvis, Tyler J.; Kimura, Takashi; Vaintrob, Arkady

Compos. Math. 126, No. 2, 157-212 (2001).

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

Keywords:

moduli space of stable \(r\)-spin curves; generalized Witten conjecture; Gelfand-Dickey hierarchies; Frobenius manifold; Gromov-Witten invariants; quantum cohomology