Simple singularities and integrable hierarchies

Abstract

The paper math.AG/0108100 gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In math.AG/0209205, it is proved that the total descendent potential corresponding to K. Saito's Frobenius structure on the parameter space of the miniversal deformation of the A_{n-1}-singularity satisfies the modulo-n reduction of the KP-hierarchy. In this paper, we identify the hierarchy satisfied by the total descendent potential of a simple singularity of the A,D,E-type. Our description of the hierarchy is parallel to the vertex operator construction of Kac -- Wakimoto except that we give both some general integral formulas and explicit numerical values for certain coefficients which in the Kac -- Wakimoto theory are studied on a case-by-case basis and remain, generally speaking, unknown.