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Henry Liu: Lectures

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.