Quantum cohomology of symplectic flag manifolds

Abstract

We compute the quantum cohomology of symplectic flag manifolds. The ring structure was known in the special cases of symplectic Grassmannians and complete symplectic flag manifolds. Starting from a GLSM construction, our computation gives the ring structure of quantum cohomology for all symplectic flag manifolds. Symplectic flag manifolds can be described by non-abelian GLSMs with superpotential. Although the ring relations cannot be directly read off from the equations of motion on the Coulomb branch due to the complication introduced by the non-abelian gauge symmetry, it can be shown that they can be extracted from the localization formula in a gauge-invariant form. We show that, when restricted to symplectic Grassmannians and complete symplectic flag manifolds, our general result reduces to the previously established results derived by other means. We also explain why a (0,2) deformation of the GLSM does not give rise to a deformation of the quantum cohomology.

Remarks on 2d Unframed Quiver Gauge Theories

“…GLSMs with Abelian gauge groups have been well studied in relation to toric geometry and mirror symmetry and then extended to non-Abelian gauge groups. Quiver GLSMs have recently been studied from various perspectives [2][3][4][5][6][7][8][9][10][11][12], leading to new mathematical structures and physical insights that would not be attainable by considering only a single gauge node.…”