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The equivariant cohomological Hikita conjecture for arbitrary quivers
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Abstract
We prove the equivariant cohomological Hikita conjecture for arbitrary finite quivers, including loops and multiple arrows. For any dimension and framing vectors, any commuting flavor torus, and a stability character whose semistable locus is stable and has free gauge action, the equivariant cohomology of the Nakajima variety is canonically isomorphic to the coordinate ring of the scheme-theoretic fixed locus of the stability cocharacter on the flavor-deformed Coulomb branch. This is an isomorphism of graded algebras over the common coefficient ring, retaining nilpotents and including the empty case.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Ben Webster (repository owner), with substantial assistance from Claude, Hikita's conjecture for quiver gauge theories: after the slop (LaTeX draft in github.com/bwebste/Hikita-AI, October 2026). A rewrite that credits the theorem and overall strategy to this preprint, takes the comparison map from Botta–Tamagni, and proves the dimension count for quivers with a loop at every vertex by restriction to Levi subgroups instead of Hausel's formula, using an announced comparison theorem without proof; an inline comment says "At the moment I don't want to say that humans have verified the full argument, since that isn't true". [AI-generated per source]
- bwebste/Hikita-AI, NOTES.md (working notes, 6–14 October 2026). Describes a draft
bps-proof.tex, a second proof of the dimension bound via BPS cohomology that avoids added loops, cut ideals and Hausel's formula and is conditional on three inputs from Jindal–Neguț. [AI-generated per source]
Comments and discussion
- Hiraku Nakajima, pull request #2 "Adding HN's texts" to bwebste/Hikita-AI (GitHub, 13 October 2026). Nakajima contributed a rewrite of the geometric part of Section 6 of the preprint; a later commit by him adds citations to Konno's papers on (equivariant) cohomology rings of toric hyperKähler manifolds.
Related prior literature mentioned by others
- T. M. Botta, S. Tamagni, Quasimap critical cohomology, Coulomb branches, and quantum groups (arXiv:2608.29400, August 2026). Cited in the preprint; the Webster rewrite states that the surjection onto equivariant cohomology follows from this work, and its introduction says the preprint "effectively rewrote the argument of Botta and Tamagni's paper" without adequate credit.
- Ilya Dumanski and Vasily Krylov, K-theoretic Hikita conjecture for quiver gauge theories (arXiv:2509.06226, 2025; revised 2026-01-05). Identified in the AI Proved What? exposition as a finite-type predecessor; the manuscript states that its main theorem proves this work's Conjecture 2.1, Equation (2.3).
- Vasily Krylov and Pavel Shlykov, Hikita-Nakajima conjecture for the Gieseker variety (arXiv:2202.09934, 2022; revised 2023-07-01). Identified in the AI Proved What? exposition as the framed Jordan-quiver predecessor; it proves the equivariant correspondence for the Gieseker variety with an explicit isomorphism on generators.
- Tatsuyuki Hikita, An algebro-geometric realization of the cohomology ring of Hilbert scheme of points in the affine plane (arXiv:1501.02430, 2015-01-11). Cited in the AI Proved What? exposition as the Hilbert-scheme precedent for realizing cohomology as functions on a scheme-theoretic fixed locus.
Possibly relevant literature (found by the agent, not checked)
- J. Kamnitzer, M. McBreen, N. Proudfoot, The quantum Hikita conjecture (arXiv:1807.09858, 2018/2020). Formulates the quantum Hikita conjecture; relevant to the preprint's statement that it does not assert the quantized version, and not in its bibliography.
Digests and expositions
- Ben Webster (requester), Hikita-AI: Reader's guide to the OpenAI equivariant Hikita preprint (GitHub repository, from 7 October 2026). A 7-page reader's guide drafted by Claude at Webster's request for participants of an SLMath program, reorganizing the argument without verifying the proofs. [AI-generated per source]
- bwebste/Hikita-AI, VERIFICATION.md (GitHub, 7 October 2026). Section-by-section adversarial reading of §§2–8 by separate Claude instances, which reports no mathematical error or load-bearing gap and lists 16 presentational fixes (e.g. the Thom–Sebastiani step in §4, a Brion citation in §7, an example in §8). [AI-generated per source]
- Hiraku Nakajima (requester), Multipartition spaces, Kirwan's equivariant perfection, and Hausel's formula (note in github.com/bwebste/Hikita-AI, October 2026). Self-contained account of the geometric part of Section 6, proving Proposition 6.2 via Kirwan's equivariant perfection and crediting Konno's earlier presentation of the toric hyperKähler cohomology. [AI-generated per source]
- AI Proved What?, Cohomology from a point? The equivariant cohomological Hikita conjecture (episode 044, October 2026). An animated exposition with transcript explaining the fixed-scheme example, the regularity hypothesis, and the manuscript's proof strategy; the site's production disclosure describes AI-assisted scripts and animations and AI-generated narration. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 044
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026,
author = {{OpenAI}},
title = {{The equivariant cohomological Hikita conjecture for arbitrary quivers}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026/main.pdf}{OAI:The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026}},
year = {2026}
}Family 044: The equivariant cohomological Hikita conjecture
Proves the equivariant cohomological Hikita correspondence for every finite quiver, including loops and multiple arrows, with arbitrary dimension and framing vectors and commuting flavor torus. When every semistable point is stable and the gauge action is free, the equivariant cohomology of the Nakajima variety is canonically the coordinate ring of the scheme-theoretic cocharacter-fixed locus of its flavor-deformed Coulomb branch.
This is the only paper in this family.