Full universal enveloping vertex algebras from factorisation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917126228410368 |
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| author | Vicedo, Benoit |
| author_facet | Vicedo, Benoit |
| contents | 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 $\mathbb{F}^{\mathfrak a,α}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08412 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Full universal enveloping vertex algebras from factorisation Vicedo, Benoit High Energy Physics - Theory Quantum Algebra 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 $\mathbb{F}^{\mathfrak a,α}$. |
| title | Full universal enveloping vertex algebras from factorisation |
| topic | High Energy Physics - Theory Quantum Algebra |
| url | https://arxiv.org/abs/2501.08412 |