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arXiv · 0909.4032

Soliton equations, vertex operators, and simple singularities

E. Frenkel·A. Givental·T. Milanov
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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.

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E. Frenkel, A. Givental, T. Milanov. 2009-09-22. Soliton equations, vertex operators, and simple singularities. https://arxiv.org/abs/0909.4032

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