The equivariant cohomological Hikita conjecture for quiver gauge theories
Hikita conjectured that for a symplectic dual pair the cohomology ring of the symplectic resolution on one side is the coordinate ring of the scheme-theoretic fixed locus of a torus on the other, proving it for Hilbert schemes of points and some type A examples. For a quiver gauge theory the Higgs side is a Nakajima quiver variety and the dual is the Braverman-Finkelberg-Nakajima Coulomb branch . The equivariant form (Nakajima, recorded by Kamnitzer-Tingley-Webster-Weekes-Yacobi, formulated by Dumanski-Krylov) asks for compatibility with the tautological coefficient maps from . Is as graded algebras over the common coefficient ring, for every quiver?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Geometric representation theory; symplectic duality; Coulomb branches
- Posed by
- Tatsuyuki Hikita (arXiv:1501.02430, IMRN 2017); equivariant form in Kamnitzer-Tingley-Webster-Weekes-Yacobi (2019), Conj. 8.9-8.10; form proved: Dumanski-Krylov (2025), Conj. 2.1
- Year posed
- 2015
- Years open
- 11y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims the equivariant cohomological Hikita conjecture for all quiver gauge theories with regular stability: the flavor-equivariant cohomology of the Nakajima variety is canonically the coordinate ring of the scheme-theoretic stability-cocharacter fixed locus of the flavor-deformed Coulomb branch, including the empty case. It does NOT prove the quantized Hikita statement, nor extend to arbitrary reductive gauge groups, where Hikita-type identifications can fail, nor treat non-regular stability.
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. Single manuscript; no Lean formalization.
Verification
No independent mathematician has checked this yet. Theorem 1.1 was read against Dumanski-Krylov Conjecture 2.1, Equation (2.3), which the paper says it proves: for any finite quiver (loops and multiple arrows allowed), dimension and framing vectors, commuting flavor torus and regular stability (semistable equals stable, free action), both coefficient maps are surjective with equal kernels, giving the graded isomorphism with nilpotents retained. The regularity hypothesis is the paper's own scope limit. Inputs include McGerty-Nevins Kirwan surjectivity and Weekes's generation theorem. No Lean formalization.