Full universal enveloping vertex algebras from factorisation
Benoit Vicedo
TL;DR
This work develops a coordinate-free, geometric framework for full (non-chiral) two-dimensional conformal field theories via prefactorisation algebras. Central to the construction is the unital local Lie algebra L^Σ_α on a real conformal surface and its twisted prefactorisation envelope, unifying the Kac–Moody, Virasoro, and βγ systems into a single universal enveloping vertex algebra F^{𝔞,α}. The authors establish a robust operator formalism, including state-field correspondences, Borcherds-type identities, conformal/anti-conformal states, a Huang-type change of variable formula, and an invariant Hermitian structure on the full vertex algebra, with explicit treatments on S^2 and the orientation double Σ^. They further derive Fourier-mode and operator-analytic realizations, yielding a full VOA-like axiomatization for KM and Virasoro sectors while addressing βγ via a nuanced treatment of global observables. The framework provides a path to analytic Langlands-type perspectives and a coordinate-invariant foundation for full CFTs on arbitrary 2D conformal manifolds, with potential extensions to boundary phenomena and unitarity structures.
Abstract
We give a systematic construction of the symmetries, or observables in the vacuum sector, of a full conformal field theory on an arbitrary real two-dimensional conformal manifold $Σ$. Specifically, we construct a prefactorisation algebra on $Σ$ which locally encodes the full (non-chiral) version $\mathbb{F}^{\mathfrak{a},α} = \mathbb{V}^{\mathfrak{a},α} \otimes \bar{\mathbb{V}}^{\mathfrak{a},α}$ of a universal enveloping vertex algebra $\mathbb{V}^{\mathfrak a,α}$, where $\mathfrak a$ is a finite-dimensional vector space labelling the set of fields and $α$ is a $2$-cocycle controlling the central extension of their Lie brackets. Our construction provides a unified treatment of the three canonical examples of (full) universal enveloping vertex algebras - Kac-Moody, Virasoro and $βγ$ system - using the notion of unital local Lie algebra. By using the coordinate-invariant nature of prefactorisation algebras we derive an analogue of Huang's change of variable formula for full vertex algebras. We give a careful treatment of (both Euclidean and Lorentzian) reality conditions in this formalism which allows us, in the Kac-Moody and Virasoro cases, to construct a Hermitian sesquilinear form on these full vertex algebras by using the factorisation product to the global observables on $S^2$. We also give an explicit derivation of Borcherds type identities and a construction of the operator formalism for $\mathbb{F}^{\mathfrak a,α}$.