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The one-dimensional vertex operator algebra is rational assuming Choice

Example

Let $V = C 1$ with $T = 0$ , $Y (1, z) = {id}_{V}$ , and conformal vector $ω = 0$ (central charge $0$ ). Assuming the Axiom of Choice (The Axiom of Choice), under the admissible-module rationality convention of Rationality in the admissible-module convention, $V$ is rational.

Facts & Assumptions

Given: The Axiom of Choice and the one-dimensional complex vector space $V = C 1$ , $T = 0$ , $Y (1, z) = {id}_{V}$ , $ω = 0$ , and an arbitrary admissible $V$ -module $M$ .

[F1]

Vertex algebra in the state–field convention: for every $a, v \in V$ , the state field is lower truncated on $v$ .

[F2]

Vertex algebra in the state–field convention: $Y (1, z) = {id}_{V}$ , $T 1 = 0$ , and $Y (a, z) 1 = e^{z T} a$ .

[F3]

Vertex algebra in the state–field convention: $[T, Y (a, z)] = \partial_{z} Y (a, z)$ for every $a \in V$ .

[F4]

Vertex algebra in the state–field convention: for every $a, b \in V$ , some $N \geq 0$ satisfies $(z - w)^{N} [Y (a, z), Y (b, w)] = 0$ .

[F5]

Conformal vertex algebra of CFT type: the conformal modes satisfy the Virasoro relations, $L (- 1) = T$ , and $V_{0} = C 1$ in the CFT-type grading.

[F6]

Module for an ordinary vertex algebra: a module has vacuum action $Y_{M} (1, z) = {id}_{M}$ and satisfies the module Jacobi identity.

[F7]

Admissible module for a vertex operator algebra: an admissible module has $M = ⨁_{n \geq 0} M (n)$ and $a_{(k)}^{M} M (n) \subseteq M (n + r - k - 1)$ for homogeneous $a \in V_{r}$ .

[F8]

Rationality in the admissible-module convention: rationality means every admissible module is a direct sum of irreducible admissible modules.

[F9]

Every vector space has a basis: assuming AC, every vector space over a field has a basis.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function.

Proof

1.1givenF1F2F3F4algebra

Every state is a scalar multiple of $1$ . The identity field is lower truncated by [F1], and [F2] gives the vacuum and creation identities. Translation covariance holds because both sides are zero by [F3]; locality holds because the identity commutes with itself by [F4]. Thus $V$ satisfies the vertex-algebra axioms.

1.2F6F7givenalgebra

Every state is $a = λ 1$ . Linearity and the vacuum property in [F6] give $Y_{M} (a, z) = λ {id}_{M}$ . Hence every graded subspace of $M$ is a $V$ -submodule, and each $M (n)$ in [F7] is a submodule.

2.1givenF5step 1.1algebra

Since $Y (ω, z) = 0$ , all $L (n) = 0$ and the Virasoro relations hold with central charge $0$ ; also $L (- 1) = 0 = T$ . The grading is $V_{0} = V = C 1$ and $V_{n} = 0$ for $n > 0$ , so $V$ is conformal of CFT type by [F5].

3.1A1F7F8F9step 1.2step 2.1∎

If $M = 0$ , it is the empty direct sum of simple modules. Otherwise, for each $n$ with $M (n) \neq 0$ , [F9] gives a nonempty set of bases of $M (n)$ ; [A1] chooses one basis in every such degree. Each basis line is a one-dimensional admissible submodule concentrated in degree $n$ , hence is simple, and the direct sum of all these lines is $M$ . Thus every admissible module is completely reducible, which is rationality by [F8].

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Sources

Christophe Nozaradan, Introduction to Vertex Algebras, §4.2 (standard reference, not scraped)
Tomoyuki Arakawa and Anne Moreau, Arc Spaces and Vertex Algebras, §§2.5–2.7 (standard reference, not scraped)