DefinitionDefinition: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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Strongly rational VOA convention

Definition

For this pair, a VOA $V$ is strongly rational when all five conditions hold. “Simple” means that $V$ has no nonzero proper subspace stable under every mode $a_{(m)}$ , for all $a \in V$ and $m \in Z$ :

$V$ is simple.
$V$ is of CFT type: $V_{n} = 0$ for $n < 0$ and $V_{0} = C 1$ , with the remaining grading conventions of Conformal vertex algebra of CFT type.
$V$ is self-contragredient: its restricted graded dual $V^{'}$ , with the opposite-field action of Contragredient module of a generalized module, is isomorphic to $V$ as a $V$ -module.
$V$ is rational in the admissible-module convention of Rationality in the admissible-module convention.
$V$ is $C_{2}$ -cofinite in the sense of C2-cofiniteness.

This convention states these five hypotheses separately. It does not include a conclusion about a module category.

Facts & Assumptions

Given: $V$ is a conformal vertex algebra over $C$ .

[F1]

Conformal vertex algebra of CFT type: CFT type includes the nonnegative integral conformal grading, finite-dimensional graded pieces and $V_{0} = C 1$ .

[F2]

Rationality in the admissible-module convention: rationality means complete reducibility of admissible modules.

[F3]

C2-cofiniteness: $C_{2}$ -cofiniteness means $V / C_{2} (V)$ is finite dimensional.

[F4]

Contragredient module of a generalized module: the contragredient construction uses the restricted graded dual and the opposite-field formula; the definition alone does not prove the module identities.

[F5]

The restricted dual with opposite action is a contragredient module proves that for every grading-restricted generalized module over a conformal vertex algebra of CFT type, the opposite-field restricted dual is a grading-restricted generalized module and componentwise evaluation gives a module isomorphism to its double contragredient. Its current proof uses the proved opposite-field Jacobi, derivative and Möbius identities.

Proof

1.1givenF1F2F3F4algebra

The definition is the conjunction of simplicity, CFT type, self-contragredience, admissible rationality and $C_{2}$ -cofiniteness, with the relevant conventions fixed by [F1]–[F4].

2.1givenF1F4F5step 1.1algebra∎

The CFT-type grading in [F1] makes $V$ itself a grading-restricted module: its weights are nonnegative integers and every graded piece is finite dimensional. Therefore [F5] applies to $M = V$ and makes the restricted dual in [F4] an actual $V$ -module. The module isomorphism required in condition 3 is consequently well-defined. Its existence is a separate condition on $V$ ; neither [F5] nor this definition asserts that every such $V$ is self-contragredient or implies any of the other four conditions.

Proof status

This is the selected five-condition convention, matching Huang’s Theorem 1.1. The current contragredient supplier proves the module structure needed to formulate self-contragredience; that earlier missing-identity hold no longer applies. This scoped supplier-use correction does not establish generic HLZ convergence or coherence, or any rigidity, ribbon or modularity conclusion.

Strongly rational VOA convention

Definition

Facts & Assumptions

Proof

Proof status

Depends on

Dependency tree · 2 levels

Sources