Abstract: I will explain how a recent “universal wall-crossing”
framework of Joyce works in equivariant K-theory, which I view as a
multiplicative refinement of equivariant cohomology. Enumerative
invariants, possibly of strictly semistable objects living on the walls, are controlled by a certain (multiplicative version of) vertex lgebra structure on the K-homology groups of the ambient stack. In very special settings like refined Vafa-Witten theory, one can obtain
some explicit formulas. For moduli stacks of quiver representations, this geometric vertex algebra should be dual in some sense to the quantum loop algebras that act on the K-theory of stable loci.