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Phys-83 · physics · mathematics · Physics

Conformal nets and representation equivalence for strongly rational unitary VOAs

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every simple unitary strongly rational complex vertex operator algebra, do polynomial energy bounds and CKLW strong locality hold, yielding an irreducible completely rational conformal net? Require complete unitarity, strong integration of all finite-length unitary grading-restricted modules, and the stated braided unitary tensor equivalence from VOA modules to finite-index net representations.

Short English statement

For every simple unitary strongly rational complex vertex operator algebra, do polynomial energy bounds and CKLW strong locality hold, yielding an irreducible completely rational conformal net? Require complete unitarity, strong integration of all finite-length unitary grading-restricted modules, and the stated braided unitary tensor equivalence from VOA modules to finite-index net representations.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-83-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-83

Simple strongly rational unitary VOAs are claimed to generate completely rational nets, with unitary modules and a braided unitary equivalence to finite-index sectors. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Conformal nets and representation equivalence for strongly rational unitary VOAs

OpenAI's claim

OpenAI presents a proof 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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