VOA–Conformal Nets Dictionary

Develop a complete correspondence between the vertex operator algebra (VOA) formalism and the conformal nets formalism that provides a full “dictionary” translating objects, constructions, and results between the two frameworks, sufficient to compare and relate phenomena such as the moonshine anomaly across both settings.

Background

The paper discusses the moonshine anomaly and notes that Johnson-Freyd obtains it via conformal nets, whereas Mason’s conjecture is formulated in the setting of vertex operator algebras. Although these two frameworks are believed to be equivalent, the authors emphasize that a comprehensive translation between them is not yet available.

A full VOA–conformal nets dictionary would enable rigorous comparison and identification of results proven in one formalism with those stated or conjectured in the other, clarifying the relationship between constructions central to moonshine and categorical group extensions. This gap hinders direct confirmation that anomalies and related structures coincide across the two approaches.

References

Johnson-Freyd's moonshine anomaly is obtained using conformal nets, while Mason's conjecture lives in the world of vertex operator algebras. The referee pointed out that, while these two formalisms are conjecturally equivalent to each other, the full VOA-conformal nets dictonary has yet to be worked out.

— Looking for a Refined Monster  (2405.16410 - Ganter, 2024) in Introduction, Footnote (following discussion of Johnson-Freyd’s moonshine anomaly and Mason’s conjecture)

The systematic comparison between unitary VOAs and conformal nets was initiated in , and the authors conjectured that there is a bijective correspondence between unitary VOAs and conformal nets. In , we established one direction of this conjectured correspondence, by showing that there is a unitary vertex operator algebra associated to every conformal net.

— A genus zero functorial CFT associated to the vacuum sector of a conformal net  (2608.24437 - Henriques et al., 25 Aug 2026) in Section 1, subsection “Connections to vertex operator algebras”