Vertex Operator Algebras with Haagerup Symmetry

Vertex Operator Algebras with Haagerup Symmetry

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

A vertex operator algebra packages chiral fields and their operator products. Its representation category records modules and tensor or fusion structure where defined. The Haagerup fusion category has the exceptional fusion data of the corresponding subfactor construction. Containing this symmetry data must be specified as a suitable categorical realization, rather than only a numerical coincidence of dimensions.

Catalogue wording retained for traceability.

Construct a vertex operator algebra arising in conformal field theory whose representation category contains the Haagerup fusion category or its symmetry data, or prove that no such vertex operator algebra exists.

Scope and source qualification.

The report leaves the precise realization framework part of the problem. A no-go result in one restricted VOA class would not settle all permitted realizations.

Short English statement

Construct a conformal vertex algebra realizing Haagerup symmetry data, or establish an obstruction to such a realization.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Reconstructed the noncommutative fusion-ring obstruction to full monoidal embedding of the six-object Haagerup category into a braided category; no broad VOA no-go.

Sources and literature check.

Read the supplied definition, cached source Haagerup discussion, and Hong–Rowell–Wang, On exotic modular tensor categories, p. 1053, through the primary author-hosted PDF on 2026-10-11. Only its explicitly displayed six-object fusion rules are used as external input. No comprehensive novelty or current-status search.

A restricted categorical obstruction.

The six-simple-object Haagerup category $\mathcal H$ in [E1, p. 1053] has distinct simples $1,\alpha,\alpha^2,\rho,\alpha\rho,\alpha^2\rho$, with $\alpha^3=1$ and $\alpha\rho=\rho\alpha^2$. Multiplying the latter relation on the left by $\alpha^2$ and on the right by $\alpha$ gives $\rho\alpha=\alpha^2\rho$. Therefore \[ \alpha\otimes\rho\not\cong\rho\otimes\alpha. \] There cannot be a fully faithful strong monoidal functor from this $\mathcal H$ into a braided category: the target braiding would give an isomorphism between the two image products, and full faithfulness would pull it back to the forbidden isomorphism. Indeed fullness lifts both it and its inverse, and faithfulness proves their composites are identities.

Scope of the obstruction.

Under a VOA framework whose relevant module category is braided, literal full tensor-subcategory containment of this six-object category is consequently impossible. This is an elementary reconstruction of a known nonbraided fusion-ring obstruction, not a new nonexistence theorem for Haagerup symmetry. It uses the specific six-object category, not an unspecified Morita-relative realization.

Why the source remains unresolved.

The recorded question explicitly permits symmetry data and leaves the realization framework to be fixed. Defect categories or a construction involving a center need not embed as that full tensor subcategory. The first gap is constructing an allowed broader categorical realization and verifying VOA locality and operator products, or finding an obstruction that genuinely covers all those allowed frameworks. The restricted obstruction does neither.

Verification and adversarial audit.

Checked the fusion-rule consequence, distinct simple objects, and lifting of target braiding by full faithfulness; restricted the conclusion to the specified containment meaning. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked the fusion-rule consequence, distinct simple objects, and lifting of target braiding by full faithfulness; restricted the conclusion to the specified containment meaning. First remaining gap: Construct a source-allowed realization through defects or other symmetry data and verify VOA axioms, or obstruct all such frameworks. The inherited source status is preserved; original-target completion remains zero.

Coordinator checked the noncommuting Haagerup fusion products and the full-faithfulness obstruction to a strong tensor embedding into a braided category; broader defect realizations are retained. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.

Primary-source web search and PDF inspection; Python 3/local file tools for serialization; direct algebraic reasoning, no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.

Final status and first remaining gap.

UNRESOLVED. Construct a source-allowed realization through defects or other symmetry data and verify VOA axioms, or obstruct all such frameworks.

Completion Estimate

COMPLETION: 0% This is an elementary auxiliary calculation or an explicit test of a restricted strategy. No new portion of the full source target is established.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 30006358 (OWR-14299511-007). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Subfactors and Applications $2025$, Oberwolfach Reports, DOI 10.4171/owr/2025/34, pp. 1823--1886. Original report and mathematical context. Incorporated report material remains under the source CC BY-SA 4.0 terms unless otherwise noted. https://doi.org/10.4171/owr/2025/34.

• E1 S.-M. Hong, E. Rowell and Z. Wang, On exotic modular tensor categories, Communications in Contemporary Mathematics 10, Suppl. 1 $2008$, 1049–1074; p. 1053 fusion rules. https://web.math.ucsb.edu/~zhenghwa/data/research/pub/exotic-08.pdf.