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

We study two-dimensional N = (2, 2) supersymmetric gauged linear sigma models (GLSM) on the Omega-deformed sphere, S-Omega(2), which is a one-parameter deformation of the A-twisted sphere. We provide an exact formula for the S-Omega(2) 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 Omega-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 epsilon(Omega)-deformation of the quantum cohomology relations. Finally, we present a complementary Higgs branch localization scheme in the special case of abelian gauge groups.