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

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

2012-03-15

arxiv.org/abs/1203.3414v1

Mathematics

Quantum Algebra

61 pages, 2 figures

Scientific paper

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.

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