Abstract: In the 90s, Vafa and Witten studied generating series formed from “counts” of solutions to certain gauge theoretic equations on a real 4-manifold. For a complex projective surface, the associated series display remarkable modular properties.
 
A mathematical definition of Vafa and Witten’s solution counts was proposed by Tanaka-Thomas using the language of algebraic geometry. I will recall their definition and explain work in preparation with M. Kool and T. Laarakker in which we express a contribution to the Vafa-Witten invariants in terms of a certain affine quiver variety, the so called "instanton moduli space" of torsion framed sheaves on P^2. In particular, I'll explain how to translate properties of instanton moduli space into formulas for Vafa-Witten invariants predicted by Göttsche, Kool and Laarakker.