Hikita Conjecture

The cohomological Hikita conjecture compares the cohomology algebra of a symplectic resolution with the coordinate algebra of a scheme-theoretic fixed locus of its symplectic dual. For quiver data, the equivariant version uses a smooth Nakajima quiver variety and the corresponding parameter-deformed Coulomb branch .

Let be a torus acting linearly on the quiver representation space and commuting with . Choose a regular stability character , so semistability equals stability and the action on the stable zero moment-map locus is free. With the diagonal torus, Weyl group , and , the common coefficient algebra is

(1)

The Kirwan map to and the Coulomb coefficient map to the fixed-scheme coordinate algebra are required to be surjective with the same kernel. They would induce a canonical graded -algebra isomorphism

(2)

If is the weight- piece for the cocharacter defined by , this fixed algebra is

(3)

Nilpotents in this quotient are retained, and linear parameters have degree two.