An−1 Singularities and nKdV Hierarchies

According to a conjecture of E. Witten [18] proved by M. Kontsevich [11], a certain generating function for intersection indices on the Deligne – Mumford moduli spaces of Riemann surfaces coincides with a certain taufunction of the KdV hierarchy. The generating function is naturally generalized under the name the total descendent potential in the theory of Gromov – Witten invariants of symplectic manifolds. The papers [5, 4] contain two equivalent constructions, motivated by some results in Gromov – Witten theory, which associate a total descendent potential to any semisimple Frobenius structure. In this paper, we prove that in the case of K.Saito’s Frobenius structure [14] on the miniversal deformation of the An−1-singularity, the total descendent potential is a tau-function of the nKdV hierarchy. We derive this result from a more general construction for solutions of the nKdV hierarchy from n − 1 solutions of the KdV hierarchy. 1.