$\mathcal {W}$-constraints for the total descendant potential of a simple singularity. (May 2013) - Bakalov, Bojko
Abstract Simple, or Kleinian, singularities are classified by Dynkin diagrams of type $ADE$ . Let $\mathfrak {g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set of $\mathfrak {g}$ -invariants in the basic representation of the affine Kac–Moody algebra $\hat {\mathfrak {g}}$ is known as a $\mathcal {W}$ -algebra and is a subalgebra of the Heisenberg vertex algebra $\mathcal {F}$ . Using period integrals, we construct an analytic continuation of the twisted representation of $\mathcal {F}$ . Our construction yields a global object, which may be called a $W$ -twisted representation of $\mathcal {F}$ . Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the $\mathcal {W}$ -algebra.