Andrei Okounkov

Abstract:   This is based on joint work with Nikita Nekrasov. The partition function of a supersymmetric gauge theory on a manifold of the form $X\times S^1$ is computed by the index of Dirac operator acting on the moduli space of instantons on $X$. After recalling why this is the case, I will explain how to define and compute this index in the case when $X$ is a Calabi-Yau manifold of complex dimension 3. This gives new theoretical and computational tools for the ongoing investigation of the link between gauge theories in 2 and 3 complex dimensions. As a special case of our construction, we deduce the refined vertex formula of Iqbal, Kozcaz, and Vafa.