Soliton equations, vertex operators, and simple singularities

We prove the equivalence of two hierarchies of soliton equations associated to a simply-laced finite Dynkin diagram. The first was defined by Kac and Wakimoto using the principal realization of the basic representations of the corresponding affine Kac-Moody algebra. The second was defined in arXiv:math/0307176 using the Frobenius structure on the local ring of the corresponding simple singularity. We also obtain a deformation of the principal realization of the basic representation over the space of miniversal deformations of the corresponding singularity. As a by-product, we compute the operator product expansions of pairs of vertex operators defined in terms of Picard-Lefschetz periods for more general singularities. Thus, we establish a surprising link between twisted vertex operators and deformation theory of singularities.