Quantum cohomology from mixed Higgs-Coulomb branches

TL;DR

This paper extends Coulomb-branch-based methods for computing quantum cohomology rings to cases with mixed Higgs-Coulomb phases, including Landau-Ginzburg orbifolds, and verifies results against known mathematical data.

Contribution

It introduces a systematic approach to quantum cohomology computations in GLSMs with mixed Coulomb and Higgs branches, expanding beyond previous Coulomb-only analyses.

Findings

Successfully describes state spaces and operator products in mixed phases.

Provides detailed comparison between geometric and IR descriptions of quantum cohomology.

Validates methods through numerous examples matching mathematical results.

Abstract

We generalize Coulomb-branch-based gauged linear sigma model (GLSM)-based computations of quantum cohomology rings of Fano spaces. Typically such computations have focused on GLSMs without superpotential, for which the IR phase of the GLSM is a pure Coulomb branch, and quantum cohomology is determined by the critical locus of a twisted one-loop effective superpotential. Here, we systematically extend to cases for which the IR phase is a mixture of Coulomb and Higgs branches, where the latter is a Landau-Ginzburg orbifold. We describe the state spaces and products of corresponding operators in detail, comparing a geometric phase description, where the OPE ring is quantum cohomology, to the IR description in terms of Coulomb and Higgs branch states. As a concrete test of our methods, we compare to existing mathematics results for quantum cohomology rings of hypersurfaces in projective spaces in numerous examples.