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 őeld theory on an arbitrary real two-dimensional conformal manifold Σ. Speciőcally, we construct a prefactorisation algebra on Σ which locally encodes the full (non-chiral) version F a,α = V a,α ⊗ Va,α of a universal enveloping vertex algebra V a,α , where a is a őnite-dimensional vector space labelling the set of őelds and α is a 2-cocycle controlling the central extension of their Lie brackets.Our construction provides a uniőed treatment of the three canonical examples of (full) universal enveloping vertex algebras ś Kac-Moody, Virasoro and βγ system ś 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 F a,α .Contents 3.5 Invariant bilinear form . . . . . .