W-constraints for the total descendant potential of a simple singularity

Abstract

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type [formula omitted] be the corresponding finite-dimensional Lie algebra, and [formula omitted] its Weyl group. The set of [formula omitted]-invariants in the basic representation of the affine Kac–Moody algebra [formula omitted] is known as a [formula omitted]-algebra and is a subalgebra of the Heisenberg vertex algebra [formula omitted]. Using period integrals, we construct an analytic continuation of the twisted representation of[formula omitted]. Our construction yields a global object, which may be called a [formula omitted]-twisted representation of [formula omitted]. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the [formula omitted]-algebra. © 2013, London Mathematical Society. All rights reserved.