Result 044, Algebraic and complex geometry

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.

Proof

The bigger picture

Why it matters

Two geometric spaces built from the same directed graph can encode information in very different ways. The manuscript claims an exact bridge between symmetry-sensitive topology on one side and algebraic functions on the other.

What changes?

For any finite quiver (a directed graph, allowing loops and multiple arrows), any dimension and framing vectors specifying sizes and auxiliary data, and any commuting flavor torus, the unreviewed manuscript reports a canonical correspondence. It assumes a stability character for which every semistable point is stable and the gauge action is free. Under these conditions, the Nakajima variety's equivariant cohomology, a symmetry-sensitive algebra of topological information, is canonically isomorphic to the coordinate ring of the stability-cocharacter-fixed scheme in the flavor-deformed Coulomb branch.

What does that help mathematicians do?

This identifies graded algebras over their shared coefficient ring, not merely dimensions or points. The coordinate ring records algebraic functions on the fixed scheme, the locus unchanged by the specified one-parameter symmetry. Retaining nilpotents, nonzero elements whose powers vanish, preserves structure invisible in a point-by-point description. Researchers can therefore translate questions about cohomology classes and their products into questions about that fixed scheme. The claimed correspondence also includes the empty case.

Are there practical applications?

The immediate value is foundational: the result would provide a common algebraic description across all finite quivers satisfying the stated stability and freeness conditions. Allowing loops, repeated arrows and arbitrary dimension and framing data makes this a uniform framework for comparing these geometric constructions, rather than a correspondence restricted to particular graph shapes or sizes.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The equivariant cohomological Hikita conjecture for arbitrary quivers

September 24, 2026 41 pages

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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Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.