Uniqueness Theorem of W -Constraints for Simple Singularities

In a recent paper, Bakalov and Milanov (Compositio. Math. 149: 840–888, 2013) 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 with the root lattice of this singularity and a twisted module of the 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.