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

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