Title: Virtual refinements of the Vafa-Witten formula,
Abstract: We conjecture a formula for the generating function of virtual
$\chi_y$-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with
holomorphic 2-form. Specializing the conjecture to minimal surfaces of general
type and to virtual Euler characteristics, we recover (part of) a formula of C.
Vafa and E. Witten.
These virtual $\chi_y$-genera can be written in terms of descendent Donaldson
invariants. Using T. Mochizuki's formula, the latter can be expressed in terms
of Seiberg-Witten invariants and certain explicit integrals over Hilbert
schemes of points. These integrals are governed by seven universal functions,
which are determined by their values on $\mathbb{P}^2$ and $\mathbb{P}^1 \times
\mathbb{P}^1$. Using localization we calculate these functions up to some
order, which allows us to check our conjecture in many cases.
In an appendix by H. Nakajima and the first named author, the virtual Euler
characteristic specialization of our conjecture is extended to include
$\mu$-classes, thereby interpolating between Vafa-Witten's formula and Witten's
conjecture for Donaldson invariants.