Using arguments from theoretical physics, Vafa and Witten gave a generating function for the Euler numbers of moduli spaces of rank 2 coherent sheaves on algebraic surfaces.

These moduli spaces are in general very singular, but they carry a perfect obstruction theory (they are virtually smooth). This gives virtual versions of many invariants of smooth projective varieties. Such virtual invariants occur everywhere in modern enumerative geometry, like Gromov-Witten invariants and Donaldson Thomas invariants, when attempting to make sense of the predictions from physics. We conjecture that the Vafa-Witten formula is true for the virtual Euler numbers. We confirm this conjecture in many examples. Then we give refinements of the conjecture.

Our approach is based on Mochizuki's formula which reduces virtual intersection numbers on moduli spaces of sheaves to intersection numbers on Hilbert schemes of points.