### September 12, 2022

TITLE: Multiplicative vertex algebras and wall-crossing in equivariant K-theory

ABSTRACT: K-theory is an interesting multiplicative refinement of

cohomology, and many cohomological objects arising in enumerative

geometry have K-theoretic analogues — modular forms become Jacobi

forms, Yangians become quantum affine algebras, etc. I will explain

how this sort of refinement goes for vertex algebras. As an

application, Joyce’s recent “universal wall-crossing” machine, which

operates by making the homology of certain moduli stacks into vertex

algebras, can be lifted to equivariant K-theory, e.g. thereby proving

the main conjecture on semistable invariants in refined Vafa-Witten

theory. In a different direction, I expect there to be some hidden

multiplicative vertex algebra structure on the aforementioned quantum

affine algebras, which can be viewed as symmetry algebras controlling

various enumerative and physical theories.