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We compute the quantum cohomology of 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 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. Our result is general for all symplectic flag manifolds, which reduces to previously established results on symplectic Grassmannians and complete symplectic flag manifolds 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.
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-twi
We discuss the complex geometry of two complex five-dimensional Kahler manifolds which are homogeneous under the exceptional Lie group $G_2$. For one of these manifolds rigidity of the complex structure among all Kahlerian complex structures was prov
We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine FK algeb
The disk partition function of certain 3d N=2 supersymmetric gauge theories computes a quantum K-theoretic ring for Kahler manifolds X. We study the 3d gauge theory/quantum K-theory correspondence for global and local Calabi-Yau manifolds with severa
We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive fo