The equivariant cohomological Hikita conjecture

The explanation

The complex projective plane has three cells: a point, a line and a plane. Its cohomology, a ring that counts its holes, is spanned by 1, x and x², with x³ = 0. So the ring is ℂ[x]/(x³), of dimension 3. Now take a different space, the set of triples (u, v, w) with uv = w³. Let a circle of symmetries scale u up and v down, leaving w fixed. Divide out every function that changes under this scaling. That sets u and v to zero, and so forces w³ = uv = 0. What remains is ℂ[w]/(w³), again with basis 1, w and w². The two rings match, though one came from holes and the other from functions.

That match is a small case of Hikita's conjecture. It relates the cohomology of one space to the functions on a fixed part of a partner spaceSource 1. The partner is called a Coulomb branch. The fixed part is a scheme, meaning it keeps nilpotent functions like w, whose cube is zero. These nilpotents carry the higher cohomology classes, like x and x²Source 2. Ignoring them would leave just one point, and lose the match.

The spaces come from quivers. A quiver is a directed graph: dots joined by arrows, possibly with loops. Put a vector space at each dot and a linear map along each arrow. The Higgs side is a Nakajima variety, a smooth space of such data up to change of basis. One dot with three framing directions gives the cotangent bundle of the plane, which has the same holes as the plane. The Coulomb side, defined by Braverman, Finkelberg and Nakajima, is built from loop groups. Both sides receive the same ring of coefficients. The question is whether those coefficients obey the same relations on both sidesSource 3.

Hikita proposed this in 2017 and proved it for the Hilbert scheme of points in the planeSource 4. Krylov and Shlykov later proved the Gieseker case, one dot with one loopSource 5. Dumanski and Krylov treated many finite-type quivers. Their result covers types A, D and E₆, with restrictions on framing in types E₇ and E₈Source 6. A general quiver, with loops and many arrows, remained open.

This paper claims the equivariant cohomological Hikita conjecture for every finite quiver, including loops and multiple arrowsSource 7. It needs no finite-type or similar conditionSource 8. It does need a stability choice under which the space is smooth and the symmetries act freely. Equivariant means the statement also tracks an extra torus of symmetries, the flavor torus. The claimed match is an isomorphism of graded algebras. It keeps the nilpotents and even covers the empty case, where both rings are zeroSource 9.

The obstacle is that matching sizes is not enoughSource 10. Both rings may have the same dimension and still have different relations. Counting fixed points, or working after inverting general parameters, misses the nilpotent structureSource 11. The paper shows one ring maps onto the other, using a test on stable pairs. It then matches their sizes, using Hausel's formula for the Betti numbersSource 12. A map onto a ring of equal size has no kernel left. A separate step compares two different fixed schemes on the Coulomb side.

The scope has edges. The paper does not claim the quantized form of the conjecture. It does not extend to arbitrary gauge groups beyond quiversSource 13.

How sure is this? No Lean formalization is listed for this result. The argument rests on many deep inputs from representation theory. No specialist review has been published.

What it doesn't show. This does not prove the quantized Hikita conjecture or cover gauge groups beyond quiver theories.