Localizing non-linear {{mathcal {N}}}=(2,2) sigma model on S^2

Abstract

We present a systematic study of {{mathcal {N}}}=(2,2) supersymmetric non-linear sigma models on S^2 with the target being a Kähler manifold. We discuss their reformulation in terms of cohomological field theory. In the cohomological formulation we use a novel version of 2D self-duality which involves a U(1) action on S^2. In addition to the generic model we discuss the theory with target space equivariance corresponding to a supersymmetric sigma model coupled to a non-dynamical supersymmetric background gauge multiplet. We discuss the localization locus and perform a one-loop calculation around the constant maps. We argue that the theory can be reduced to some exotic model over the moduli space of holomorphic disks.