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 \(\mathbb {F}^{\mathfrak {a},\alpha }\) .
