Full universal enveloping vertex algebras from factorisation

TL;DR

This paper systematically constructs full conformal field theories on arbitrary 2D manifolds using prefactorisation algebras, unifying key examples like Kac-Moody, Virasoro, and beta-gamma systems, and deriving new formal properties.

Contribution

It introduces a coordinate-invariant prefactorisation algebra framework for full vertex algebras, unifying major examples and deriving change of variable formulas and operator formalism.

Findings

Unified construction of full vertex algebras from prefactorisation algebras.

Derived analogue of Huang's change of variable formula.

Constructed Hermitian forms and operator formalism for full vertex algebras.

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 $\Sigma$. 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 $\mathbb{F}^{\mathfrak a,\alpha}$.