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

Abstract

Abstract We present a systematic study of $${{\mathcal {N}}}=(2,2)$$ N = ( 2 , 2 ) supersymmetric non-linear sigma models on $$S^2$$ 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 $$S^2$$ 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.