Localization of twisted $\mathcal{N}{=}(0,2)$ gauged linear sigma models in two dimensions

TL;DR

This work extends supersymmetric localization to two-dimensional $N=(0,2)$ GLSMs on curved backgrounds by implementing a pseudo-topological $A/2$ twist that preserves a single supercharge on any Riemann surface. The authors derive a generalized Jeffrey-Kirwan-Grothendieck (JKG) residue formula to compute genus-zero correlators on the Coulomb branch, unifying abelian and non-abelian models and yielding quantum sheaf cohomology relations in new settings (e.g., Grassmannians with deformed tangent bundles). They show the correlators are holomorphic in the $E$-type data and largely independent of nonlinear deformations, with $J$-potentials contributing only through $R$-charge constraints; this reproduces known results on the $(2,2)$ locus and extends them to a broad class of non-abelian theories. The paper provides explicit abelian examples (projective spaces, Hirzebruch surfaces, quintic) and non-abelian cases (Grassmannians, complete intersections) that validate the framework and connect to Čech-cohomology approaches, highlighting the practical utility of localization for quantum sheaf cohomology in $(0,2)$ theories.

Abstract

We study two-dimensional $\mathcal{N}{=}(0,2)$ supersymmetric gauged linear sigma models (GLSMs) using supersymmetric localization. We consider $\mathcal{N}{=}(0,2)$ theories with an $R$-symmetry, which can always be defined on curved space by a pseudo-topological twist while preserving one of the two supercharges of flat space. For GLSMs which are deformations of $\mathcal{N}{=}(2,2)$ GLSMs and retain a Coulomb branch, we consider the $A/2$-twist and compute the genus-zero correlation functions of certain pseudo-chiral operators, which generalize the simplest twisted chiral ring operators away from the $\mathcal{N}{=}(2,2)$ locus. These correlation functions can be written in terms of a certain residue operation on the Coulomb branch, generalizing the Jeffrey-Kirwan residue prescription relevant for the $\mathcal{N}{=}(2,2)$ locus. For abelian GLSMs, we reproduce existing results with new formulas that render the quantum sheaf cohomology relations and other properties manifest. For non-abelian GLSMs, our methods lead to new results. As an example, we briefly discuss the quantum sheaf cohomology of the Grassmannian manifold.