Description
I'll present Tanaka-Thomas's algebro-geometric approach to Vafa-Witten invariants of projective surfaces. The invariants are defined by integration over moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components; S-duality implies conjectural symmetries between these contributions.
I'll then explain work in progress with M. Kool and T. Laarakker on the "vertical" or "monopole" component, which can be regarded as a nested Hilbert scheme on a surface. Namely, we apply a recent blow-up identity of Kuhn-Leigh-Tanaka to obtain constraints on Vafa-Witten invariants of the vertical component predicted by Göttsche-Kool-Laarakker. One consequence is a complete formula for refined invariants of this component in rank 2.
