$A_{n-1}$ singularities and $n$KdV hierarchies

According to a conjecture of E. Witten [21] proved by M. Kontsevich [12], a certain generating function for intersection indices on the Deligne–Mumford moduli spaces of Riemann surfaces coincides with a certain tau-function 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 [6], [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 [17] on the miniversal deformation of the $A_{n-1}$-singularity, the total descendent potential is a tau-function of the $n$KdV hierarchy. We derive this result from a more general construction for solutions of the $n$KdV hierarchy from $n-1$ solutions of the KdV hierarchy.