Full universal enveloping vertex algebras from factorisation

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 $\Sigma$. Specifically, we construct a prefactorisation algebra on $\Sigma$ which locally encodes the full (non-chiral) version $\mathbb{F}^{\mathfrak{a},\alpha} = \mathbb{V}^{\mathfrak{a},\alpha} \otimes \bar{\mathbb{V}}^{\mathfrak{a},\alpha}$ of a universal enveloping vertex algebra $\mathbb{V}^{\mathfrak a,\alpha}$, where $\mathfrak a$ is a finite-dimensional vector space labelling the set of fields and $\alpha$ 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 $\beta\gamma$ 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 $\mathbb{F}^{\mathfrak a,\alpha}$.