Quantum vertex algebras and cohomological Hall algebras

There is an extremely rich history of interaction between string theory and the mathematics of moduli spaces, for instance cohomological Hall algebras/algebras of BPS states, or vertex/chiral algebras.

In this talk, I will explain a link between two of these: Joyce’s vertex algebras attached to the moduli stack of objects in an abelian category, and one dimensional CoHAs. This is based on my recent paper 2110.14356, whose main result says that the cohomologies of such stacks are “quantum vertex algebras”: the factorisation/vertex analogues of quasitriangular bialgebras. The main technical tool is a “bivariant” Euler class which makes torus localisation work in this context. I will discuss applications of these techniques to CoHAs of coherent sheaves on a curve and future directions.