Uniqueness Theorem of W-Constraints for Simple Singularities

TL;DR

This paper proves the uniqueness (up to a constant factor) of solutions to W-constraints associated with simple singularities, confirming a conjecture and deepening understanding of the algebraic structures involved.

Contribution

It establishes the uniqueness of solutions to W-constraints for simple singularities, confirming a conjecture by Bakalov and Milanov.

Findings

Solution to W-constraints is unique up to a constant factor.

Confirms the conjecture by Bakalov and Milanov.

Enhances understanding of W-algebra structures in singularity theory.

Abstract

In a recent paper [3], Bakalov and Milanov proved that the total descendant potential of a simple singularity satisfies the W-constraints, which come from the W-algebra of the lattice vertex algebra associated to the root lattice of this singularity and a twisted module of this vertex algebra. In the present paper, we prove that the solution of these W-constraints is unique up to a constant factor, as conjectured by Bakalov and Milanov in their paper.