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

Simple 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. (Joint work with T. Milanov.)