Motivated by the physical theory of S-duality, Vafa-Witten gave conjectural expressions for generating series of certain solutions to gauge-theoretic equations on complex surfaces in terms of modular forms. Recently,  Tanaka-Thomas gave an algebro-geometric definition of Vafa-Witten invariants in terms of moduli spaces of stable Higgs pairs on a surface. The invariants are formed from contributions of components; S-duality translates to interesting transformations between these contributions.

One component, the so-called “vertical component,” is a nested Hilbert scheme of curves and points on a surface. To warm up, I’ll give a brief introduction to the Hilbert scheme of points on a surface. I’ll then explain work in preparation with M. Kool and T. Laarakker in which we express the invariants of the vertical component in terms of a quiver variety, the instanton moduli space of torsion-free framed sheaves on CP^2. One consequence is an explicit formula for rank 2 vertical refined Vafa-Witten invariants in terms of Jacobi forms.