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Math-2411 · mathematics · Algebra

Strong Locality and Reconstruction of Vertex Operator Algebras

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Is every unitary vertex operator algebra strongly local? Does every irreducible conformal net arise from a unitary vertex operator algebra through the conformal-net construction?

Formulation references: [S1], [S2].

Short English statement

Is every unitary vertex operator algebra strongly local? Does every irreducible conformal net arise from a unitary vertex operator algebra through the conformal-net construction?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2411-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2411

The theorem proves strong locality and a conformal-net construction for simple unitary strongly rational VOAs. The problem statement asks all unitary VOAs and also the reverse realization of every irreducible conformal net; neither unrestricted claim follows. This entry records partial support; complete coverage of the question is not established by this catalogue.

Strong Locality and Reconstruction of Vertex Operator Algebras

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Simple strongly rational unitary VOAs are claimed to generate completely rational nets, with unitary modules and a braided unitary equivalence to finite-index sectors.

Mathematical statement of the claimed result

Result 1.
Take a vertex operator algebra $V$ over $\mathbb C$; require simplicity, unitarity and strong rationality. Then the source claims polynomial energy bounds and strong locality of the smeared fields, producing an irreducible conformal net $A_V$. It also asserts complete unitarity of $V$, strong integrability for unitary grading-restricted finite-length modules, and complete rationality of $A_V$. With invariant positive Hermitian forms on simple modules, the representation categories are claimed to obey \[ \operatorname{Rep}(V)\simeq\operatorname{Rep}^{u}(V) \xrightarrow{F_{\mathrm{CWX}}^{V}}\operatorname{Rep}^{f}(A_V), \] a braided unitary tensor equivalence. The net category uses Connes fusion and DHR braiding with matching circle orientation. Strong rationality here includes simplicity, CFT type, self-contragredience, rationality, and $C_2$-cofiniteness; the source does not extend this theorem to every unitary VOA.

Outline of the argument

Energy bounds establish strong locality of the generating fields. Annular charged-field estimates integrate modules, positive fusion forms give unitary tensor structure, and rationality identifies net sectors; the cited extension theorems then relate local net extensions to conformal VOA extensions.

Formal verification

For family 280, the documented entry establishes energy bounds, strong locality, and an irreducible conformal-net construction. Complete rationality, unitary module classification, tensor equivalence, and extension classification are omitted. The listed Comparator files are VertexAlgebraNet.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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