The equivariant A-twist and gauged linear sigma models on the two-sphere

Abstract: We study two-dimensional N=2,2$$ \mathcal{N}=\left(2,\;2\right) $$ supersymmetric gauged linear sigma models (GLSM) on the Ω-deformed sphere, S<inf>Ω</inf> <sup>2</sup>, which is a one-parameter deformation of the A-twisted sphere. We provide an exact formula for the S<inf>Ω</inf> <sup>2</sup> supersymmetric correlation functions using supersymmetric localization. The contribution of each instanton sector is given in terms of a Jeffrey-Kirwan residue on the Coulomb branch. In the limit of vanishing Ω-deformation, the localization formula greatly simplifies the computation of A-twisted correlation functions, and leads to new results for non-abelian theories. We discuss a number of examples and comment on the ϵ<inf>Ω</inf>-deformation of the quantum cohomology relations. Finally, we present a complementary Higgs branch localization scheme in the special case of abelian gauge groups.