Soliton equations, vertex operators, and simple singularities

TL;DR

This paper establishes a deep connection between soliton equations, vertex operators, and singularity theory, proving the equivalence of two hierarchies and linking deformation theory with algebraic structures.

Contribution

It proves the equivalence of two soliton hierarchies related to Dynkin diagrams and introduces a deformation of the basic representation over singularity deformations.

Findings

Proved the equivalence of two soliton hierarchies.

Computed operator product expansions for vertex operators.

Linked twisted vertex operators with singularity deformation theory.

Abstract

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.