Abstract

I will review some exact results in 2 dimensional gauge theories with (2,2) supersymmetry, namely sphere partition functions and elliptic genera, obtained via supersymmetric localization. The partition function of these theories on a Torus computes the elliptic genus; for theories that flow in the IR to non-linear sigma models (NLSM) on a non-compact Kahler target with an asymptotically linear dilaton profile the elliptic genera are shown to be “Mock modular”. The sphere partition function of these gauge theories is another fascinating object. For theories that flow to NLSM with Calabi-Yau targets the sphere partition function computes the Kahler potential for the Kahler or Complex structure moduli of the target. This can be used for instance to compute some Topological invariants of Calabi-Yau manifolds known as the Gromov-Witten invariants without using Mirror symmetry.