W-Constraints for the Total Descendant Potential of a Simple Singularity

TL;DR

This paper demonstrates that the total descendant potential of simple singularities is a highest weight vector for the associated W-algebra, connecting singularity theory with affine Lie algebra representations.

Contribution

It constructs a W-twisted representation of the Heisenberg vertex algebra using period integrals and proves the descendant potential's highest weight property.

Findings

Total descendant potential is a highest weight vector for the W-algebra.

Constructs a global W-twisted representation via period integrals.

Establishes a link between singularity invariants and affine Lie algebra representations.

Abstract

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finite-dimensional Lie algebra, and W its Weyl group. The set of g-invariants in the basic representation of the affine Kac-Moody algebra g^ is known as a W-algebra and is a subalgebra of the Heisenberg vertex algebra F. Using period integrals, we construct an analytic continuation of the twisted representation of F. Our construction yields a global object, which may be called a W-twisted representation of F. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest weight vector for the W-algebra.