Finiteness of Automorphism Groups of Rational Vertex Operator Algebras

Definitions and mathematical statement

The source proposes a conjecture about vertex operator algebras (VOAs). It does not explicitly specify the ground field or the convention for rationality. The following formulation uses the usual complex-VOA and admissible-module conventions; these conventions require confirmation against the source.

Let $V=(V,Y,\mathbf{1},\omega)$ be a vertex operator algebra over ${\mathbb{C}}$, with vacuum $\mathbf{1}$, conformal vector $\omega$, and mode notation \[ Y(a,z)=\sum_{m\in{\mathbb{Z}}}a_m z^{-m-1},\qquad L(0)=\omega_1. \] Being of CFT type means that \[ V=\bigoplus_{n\geq 0}V_n,\qquad V_n=\{v\in V:L(0)v=nv\},\qquad \dim_{{\mathbb{C}}}V_n<\infty,\qquad V_0={\mathbb{C}}\mathbf{1}. \] Define \[ C_2(V)=\operatorname{span}_{{\mathbb{C}}}\{a_{-2}b:a,b\in V\}. \] The algebra is $C_2$-cofinite if $\dim_{{\mathbb{C}}}(V/C_2(V))<\infty$. Under the convention used here, $V$ is rational if every admissible $V$-module is a direct sum of irreducible admissible modules. Here an admissible module is a weak $V$-module with a grading $M=\bigoplus_{r\geq0}M(r)$ satisfying \[ a_m M(r)\subseteq M(r+n-m-1) \quad(a\in V_n,\ m\in{\mathbb{Z}}), \] where $M(s)=0$ for $s<0$.

The VOA automorphism group is \[ \operatorname{Aut}(V)=\left\{g\in\operatorname{GL}_{{\mathbb{C}}}(V): \begin{array}{l} g\mathbf{1}=\mathbf{1},\quad g\omega=\omega,\\ gY(a,z)b=Y(ga,z)gb\quad\text{for all }a,b\in V \end{array}\right\}. \] Conjecture, under the conventions above. For every rational, $C_2$-cofinite vertex operator algebra $V$ of CFT type, \[ V_1=\{0\}\quad\Longrightarrow\quad |\operatorname{Aut}(V)|<\infty. \] The supplied statement imposes no separate simplicity or self-contragredience hypothesis; none is added here. Whether the original source implicitly requires either condition remains unresolved. The conjectural designation and the supplied open-status assessment are source labels, not independently verified conclusions.

Short English statement

The supplied conjecture says that every rational, $C_2$-cofinite VOA of CFT type whose weight-one subspace is zero has only finitely many automorphisms preserving its vacuum, conformal vector, and vertex operations. The complex ground field and admissible-module meaning of rationality are explicit interpretive conventions requiring source confirmation.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30002842$, source problem code catalogue entry, title Finiteness of Automorphism Groups of Rational Vertex Operator Algebras, in the Oberwolfach Reports Open Problems collection. No proposer is identified; the supplied proposed year is 2015.
• Supplied report citation: Subfactors and Conformal Field Theory (2016), Oberwolfach Reports, pp. 849--926, DOI 10.4171/owr/2015/16. https://doi.org/10.4171/owr/2015/16. The record describes these reports as documenting workshops and their open-problem discussions.
• Supplied literature reference: Dong--Griess, "automorphism groups." https://arxiv.org/abs/math/0106051.
• Supplied literature reference: Lam, "VOA automorphism conjecture." https://doi.org/10.4171/owr/2015/16. This label does not by itself establish authorship of the conjecture.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports finiteness for important subclasses such as framed VOAs, and says that current surveys still record the general conjecture as open and that no general proof was found. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying verification note reports manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. This is supplied editorial provenance, not independent source verification or mathematical review of this draft.