Quantum Sheaf Cohomology and Duality of Flag Manifolds

TL;DR

This paper investigates the quantum sheaf cohomology of flag manifolds with tangent bundle deformations, revealing how dualities transform these structures and establishing dualities between different supersymmetric gauge theories.

Contribution

It introduces a method to compute quantum sheaf cohomology for nonabelian GLSMs and derives explicit dualities for flag manifolds and Grassmannians.

Findings

Derived quantum sheaf cohomology for products of Grassmannians and flag manifolds.

Proposed a new computational method for (0,2) GLSMs with (2,2) locus.

Established explicit IR dualities between dual deformations of gauge theories.

Abstract

We study the quantum sheaf cohomology of flag manifolds with deformations of the tangent bundle and use the ring structure to derive how the deformation transforms under the biholomorphic duality of flag manifolds. Realized as the OPE ring of A/2-twisted two-dimensional theories with (0,2) supersymmetry, quantum sheaf cohomology generalizes the notion of quantum cohomology. Complete descriptions of quantum sheaf cohomology have been obtained for abelian gauged linear sigma models (GLSMs) and for nonabelian GLSMs describing Grassmannians. In this paper we continue to explore the quantum sheaf cohomology of nonabelian theories. We first propose a method to compute the generating relations for (0,2) GLSMs with (2,2) locus. We apply this method to derive the quantum sheaf cohomology of products of Grassmannians and flag manifolds. The dual deformation associated with the biholomorphic duality gives rise to an explicit IR duality of two A/2-twisted (0,2) gauge theories.