Cohomology from a point?

0:00Two branches

A three-dimensional gauge theory with enough supersymmetry has two spaces of vacua, the Higgs branch and the Coulomb branch. For a quiver gauge theory, the Higgs branch is a Nakajima quiver variety: a quotient built from linear maps along the arrows of a directed graph, with extra framing spaces attached. The Coulomb branch got a mathematical definition only in 2016, from Braverman, Finkelberg and Nakajima, as a convolution algebra. The simplest example has one vertex, a line, and n framing arrows. Its Higgs branch is the cotangent bundle of projective space of dimension n − 1. Its Coulomb branch is the surface uv = xⁿ: a cone with one singular point.

0:45Hikita’s fat point

Hikita noticed a striking link. The cohomology of that projective space is generated by one class x of degree two, with the single relation xⁿ = 0. Now let a circle act on the Coulomb branch, turning u one way and v the other. The only fixed point is the origin. But as a scheme, setting u and v to zero leaves xⁿ = 0: a fat point whose functions are one, x, and so on up to xⁿ⁻¹. Those are exactly the cohomology classes, degree by degree. Hikita conjectured that this happens for every symplectic dual pair, and proved it for the Hilbert scheme of points in the plane. The set of fixed points sees almost nothing; the nilpotent functions carry the cohomology. The equivariant version turns on masses. The relation becomes uv = ∏(x − mᵢ), and for distinct masses the fat point splits into n separate points. Nakajima proposed that equivariant cohomology should match the functions on this deformed fixed scheme. Later work sharpened it: the match should respect one shared ring of coefficients, Chern classes of tautological bundles on one side, equivariant parameters on the other.

2:00The claim

Earlier proofs covered particular quivers. Krylov and Shlykov did the framed Jordan quiver, and Dumanski and Krylov did finite types A, D and E6, and E7 and E8 with framing restrictions. Result 044 is a 41-page OpenAI preprint, dated September 24. It claims the equivariant cohomological Hikita conjecture for every finite quiver, loops and multiple arrows allowed, any dimension and framing vectors, any commuting flavor torus, and any stability whose semistable points are stable with free gauge action. The isomorphism is of graded algebras over the common coefficient ring, it keeps the nilpotents, and it includes the empty case. It does not claim the quantized version, or other gauge groups.

2:50How the proof goes

The proof passes through a middle ring: functions on the fixed scheme of the full topological torus. It and the Higgs cohomology are both quotients of one coefficient ring, by Weekes’s theorem that minuscule monopoles generate, and by Kirwan surjectivity, proved by McGerty and Nevins. So the task is to show that the two kernels agree. Step one. Every relation among the coefficients is a sum of words in monopoles with total charge zero. Stability gives each step a degree, and the degrees add up to zero, so some step has degree at most zero: put it first. A stable pair on the projective line forces that degree to be at least zero, and degree zero would force charge zero. So the first step is excluded, and the word vanishes in cohomology. The middle ring maps onto cohomology. Step two: size. With an extra loop at every vertex, the loop weights cancel the denominators in the monopole formulas. A filtration by multipartitions bounds the middle ring degree by degree, and the bound equals the Betti numbers from Hausel’s formula. A surjection onto a space of the same size is an isomorphism. A deformation then removes the loops, hyperkähler rotation reaches every regular stability, and a last algebraic step adds every flavor. Step three: the circle. A regular stability can be orthogonal to a dimension vector, so killing nonzero weights might not kill every charge. Adjoin a framing vertex: a surviving monopole would have a surviving complementary monopole, and a Hall algebra theorem of Jindal and Neguț splits both charges into indecomposable pieces of weight zero. Their direct sum is a semistable point with extra symmetry, contradicting free action.

4:38Where it stands

This is a preprint, not yet peer reviewed, with no Lean proof, and it has not been revised. It cites no other paper from the release, but it leans on recent work, including that Hall algebra theorem, a July 2026 preprint. We found no public assessment. If it holds, then for every quiver gauge theory with free stable action, the cohomology ring of the Higgs branch, relations and all, can be read off a fat point on the Coulomb branch, and with masses turned on, the equivariant ring too.