Mathematics

Using arguments from theoretical physics, Vafa and Witten gave agenerating function for the Euler numbers ofmoduli spaces of rank 2 coherent sheaves on algebraic surfaces.These moduli spaces are in general very singular, but they carry aperfect obstruction theory (they are virtually smooth).This gives virtual versions of many invariants of smooth projectivevarieties. Such virtual invariants occur everywhere in modernenumerative geometry, like Gromov-Witten invariants and Donaldson Thomasinvariants, when attempting to make sense of the predictions from physics.We conjecture that the Vafa-Witten formula is true for the virtual Eulernumbers. We confirm this conjecture in many examples.Then we give refinements of the conjecture.Our approach is based on Mochizuki's formula which reduces virtualintersection numbers on moduli spaces of sheaves to intersection numberson Hilbert schemes of points.