Every conformal net has an associated unitary VOA
André G. Henriques, James E. Tener
TL;DR
This work proves one direction of the conjectured equivalence between unitary VOAs and chiral conformal nets by constructing for every conformal net ${\mathcal A}$ a unitary VOA $V_{\mathcal A}$ whose local observables recover ${\mathcal A}$ via ${\mathcal A}_{V_{\mathcal A}}={\mathcal A}$; representations with discrete $L_0$-spectrum yield unitary $V_{\mathcal A}$-modules. The construction rests on a rich geometric toolkit: worm- and point-insertions, the semigroup of annuli and its central extensions, and holomorphicity properties that encode vertex-algebra axioms in a geometric setting. The paper also establishes that the VOA obtained is AQFT-local and that the inverse construction from a strongly local (AQFT-local) VOA recovers the original conformal net, aligning the two formalisms in a precise, functorial sense. These results lay a rigorous foundation toward the long-sought VOA-net correspondence and provide a concrete pathway to analyze representations of nets via VOA modules. The framework sharpens our understanding of how conformal symmetry and locality translate between algebraic and geometric viewpoints, with potential implications for 2D conformal field theory and related TQFT constructions.
Abstract
Unitary vertex operator algebras (VOAs) and conformal nets are the two most prominent mathematical axiomatizations of two-dimensional unitary chiral conformal field theories. They are conjectured to be equivalent, but a rigorous comparison has proven challenging. We resolve one direction of the conjecture by showing that every conformal net has an associated unitary VOA. We also show that every representation of a conformal net in which the generator of rotation acts with discrete spectrum and finite-dimensional eigenspaces yields a unitary module of the corresponding VOA. A talk describing our results is available at: https://www.youtube.com/watch?v=f_LhNSeiiaE .