Counting sheaves on curves

Abstract: Counting vector bundles on algebraic curves has been a classical problem. These counting invariants were first computed by Witten using physical methods, and his formulae were later proved by Jeffrey and Kirwan. In this talk, I will introduce a new viewpoint towards this problem, using Joyce’s enumerative invariants obtained via his vertex algebra. This also extends the invariants to the case where the rank and degree are not coprime. Computing these invariants using Joyce’s wall-crossing formulae reveals an interesting new structure. Namely, the invariants can be expressed as a divergent infinite sum, which can be assigned a finite value via a regularization process.