Result 280, Mathematical physics

Unitary vertex operator algebras and conformal nets

Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are unitarizable, and its representation category agrees with the net’s finite-index sectors as a braided unitary tensor category.

Lean formalization Proof

The bigger picture

Why it matters

Conformal field theory has algebraic and operator-based descriptions that can be difficult to reconcile. The manuscript reports that, for a specified class, these descriptions produce matching theories and matching rules for combining their representations.

What changes?

For every simple unitary strongly rational complex vertex operator algebra, the manuscript reports a completely rational conformal net. A vertex operator algebra encodes fields and their algebraic relations; a conformal net assigns operator algebras to regions, making locality explicit. Strong rationality is a restrictive finiteness and regularity assumption, so this does not settle strong locality for all unitary vertex operator algebras. All simple grading-restricted modules, representations with controlled grading, are also claimed to admit compatible positive inner products.

What does that help mathematicians do?

The reported braided unitary tensor equivalence identifies the algebra's grading-restricted finite-length module category with the net's finite-index sectors. This correspondence preserves how representations combine, how they exchange order, and their unitary structure, not just a list of labels. Together with the claimed positivity of canonical fusion forms, it lets researchers translate representation-theoretic questions between the algebraic and operator-based settings without losing these structures.

Are there practical applications?

Its immediate value is foundational: it connects two rigorous frameworks for conformal field theory. Using Gui's extension theorems, the manuscript also identifies normalized irreducible finite-index local extensions of these nets with simple CFT-type conformal extensions of the vertex operator algebras. This translates specified ways of enlarging a theory between the frameworks, rather than demonstrating a new physical application.

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

Strongly rational unitary vertex operator algebras and conformal nets

September 25, 2026 47 pages

Every simple unitary strongly rational vertex operator algebra generates a completely rational conformal net. All its simple grading-restricted modules are unitarizable, its canonical fusion forms are positive, and the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence from its grading-restricted finite-length module category onto the finite-index sectors of the net. Gui's extension theorems then identify normalized irreducible finite-index local extensions of these nets with simple CFT-type conformal extensions of the vertex operator algebras.

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Cite (BibTeX) 
@misc{OAI:Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026,
  author = {{OpenAI}},
  title = {{Strongly rational unitary vertex operator algebras and conformal nets}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026/paper.pdf}{OAI:Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/280.md.

Unitary vertex operator algebras and conformal nets

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The linked formalization proves the first construction step relating strongly rational unitary vertex operator algebras to conformal nets. For a simple unitary strongly rational vertex operator algebra, it establishes polynomial energy bounds and strong locality and constructs an irreducible conformal net with the stated covariance, vacuum, and positive-energy properties.

The selected statement does not include complete rationality of the net, unitarizability of all simple modules, the braided tensor equivalence, or the classification of local extensions described in the paper.

Comparator links

Result Comparator statement
Energy bounds, strong locality, and an irreducible conformal net VertexAlgebraNet.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 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.