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

Henry Liu

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.


Simons- Special Holonomy in Geometry, Analysis, and Physics: Progress and Open Problems