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

Abstract

We study two-dimensional $\mathcal{N}=(2,2)$ 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.

101 pages plus appendices; v2: minor changes, typos corrected, references added; v3: typos corrected

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