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

TL;DR

This paper applies supersymmetric localization to two-dimensional (0,2) GLSMs, computing genus-zero correlation functions and revealing new insights into quantum sheaf cohomology, especially for non-abelian models like Grassmannians.

Contribution

It introduces new residue formulas for correlation functions in (0,2) GLSMs, extending known results from (2,2) theories to non-abelian cases and generalizing quantum sheaf cohomology computations.

Findings

Derived residue formulas for correlation functions

Extended quantum sheaf cohomology to non-abelian GLSMs

Reproduced and generalized existing abelian GLSM results

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.