The equivariant cohomological Hikita conjecture
From OpenAI Aftermath
The equivariant cohomological Hikita conjecture belongs to symplectic duality. It predicts that the cohomology of a conical symplectic resolution is isomorphic to the ring of functions on the scheme-theoretic fixed locus of a torus acting on the dual Coulomb branch; in the equivariant form, the isomorphism is to be compatible with the natural coefficient maps on both sides. A manuscript dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims this statement for the Nakajima quiver varieties of every finite quiver, loops and multiple arrows allowed. The collection has no Lean formalization of this result. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- A classical model is the description of the cohomology of type-A Springer fibers through scheme-theoretic intersections, by de Concini and Procesi and by Tanisaki. In the setting of symplectic duality, Hikita formulated the fixed-scheme realization of cohomology and proved it for the Hilbert scheme of points in the plane and for related type-A and hypertoric examples.
- Braverman, Finkelberg and Nakajima constructed Coulomb branches of three-dimensional gauge theories; for quiver gauge theories the corresponding Higgs side is a Nakajima variety. Formulations compatible with the coefficient maps were given by Kamnitzer, Tingley, Webster, Weekes and Yacobi, by Hilburn, Kamnitzer and Weekes (quantized form) and by Krylov and Shlykov, who proved the Gieseker case. The form used is Conjecture 2.1 of Dumanski and Krylov, whose finite-type result covers types A, D and E6, and E7 and E8 under framing restrictions.
- Further results include Setiabrata's surjectivity result for the base-affine example and the refined quantum Hikita theorem of Dinkins, Karpov and Krylov for ADE quivers with minuscule-supported framing and for the Jordan quiver. Outside this setting, proposed Hikita-type identifications can fail (Hoang, Krylov and Matvieievskyi).
The original result
- Claim. Theorem 1.1 of The equivariant cohomological Hikita conjecture for arbitrary quivers states that for every finite quiver, every dimension and framing vector, every flavor torus commuting with the gauge group and every regular stability character (the semistable locus is stable and the gauge action on it is free), both coefficient maps are surjective and have the same kernel. They therefore give a canonical isomorphism of graded algebras between the equivariant cohomology of the Nakajima variety and the coordinate ring of the scheme-theoretic fixed locus of the stability cocharacter on the flavor-deformed Coulomb branch, nilpotents included; the empty case is included. No finite-type, Dynkin or conical hypothesis is imposed. The manuscript does not claim the quantized statement or arbitrary reductive gauge groups.
- Method, as described: a stable-pair test proves an inclusion between the kernels of the two coefficient maps for the fixed scheme of the full topological torus, and a dimension count, using Hausel's formula for the Betti numbers, turns it into equality. A separate argument then identifies the full-torus fixed scheme with the fixed scheme of the stability cocharacter.
- Sources. The manuscript The equivariant cohomological Hikita conjecture for arbitrary quivers at commit
1581d4c. - Formalization. The collection has no Lean formalization or scope note for this family.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 21 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | B. Webster, reader's guide and verification notes in bwebste/Hikita-AI (GitHub; commit a997223) |
Review (AI-assisted reading); exposition | A seven-page reader's guide for participants of the SLMath program on quantum field theory, drafted by Claude at Webster's request. The verification notes report seven independent adversarial readings of Sections 2 to 8 by Claude instances, two of which checked the cited external results against their sources, and find no mathematical error and no load-bearing gap, only presentational issues. | The notes state that this is a machine reading and not a human referee report, and that the guide does not verify the proofs. |
| 2026-10-12 to 2026-10-14 | B. Webster, with texts by H. Nakajima, draft Hikita's conjecture for quiver gauge theories: after the slop (source) in the same repository | Rewrite (draft); critique | A rewrite aimed at a version of the argument that is readable and checkable by humans, written with assistance from Claude under the author's guidance. It includes Nakajima's rewrite of the geometric part of Section 6, on multipartition spaces, Kirwan surjectivity and Hausel's formula, in a note written by Claude Opus 5.5 at his request, and a second proof of the dimension bound through BPS cohomology that is conditional on results of Jindal and Neguț whose proofs are only sketched. Its introduction states that the rewrite is not an endorsement of the release and criticizes the release on research-ethics grounds. | Draft; a note in the source states that humans have not yet verified the full argument. |
| 2026-10-16 | J. Jardine, Lattice24, referee review in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap and one minor note: the step identifying the stability-cocharacter fixed quotient with the full-torus quotient depends on two external inputs that the reviewer could not check, polynomial BPS generation of the loop-nilpotent shuffle algebra (Jindal and Neguț) and a normalized leading limit of Tsymbaliuk's difference-operator maps, whose order table is asserted rather than derived. The reviewer reports that the logic built on these inputs checks out and that every section was read. | Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
