A Lorentzian construction of timelike Liouville field theory on the cylinder
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917542658834432 |
|---|---|
| author | Chatterjee, Sourav |
| author_facet | Chatterjee, Sourav |
| contents | Timelike Liouville field theory is a candidate model for positive curvature two-dimensional quantum gravity, but a mathematically controlled Lorentzian formulation has remained elusive. In this paper we construct the theory on the cylinder $\mathbb{R}\times \mathbb{S}^1$ in the integer screening sector for a natural algebra of renormalized exponential observables. Starting from a renormalized finite-volume torus regularization, we construct infinite-volume Euclidean correlation functions, prove analytic continuation in the time variables, and identify the resulting Lorentzian boundary values by explicit contour formulas. This yields exact Lorentzian correlators for a natural class of exponential observables. We then prove locality: spacelike separated vertex operators commute in the Lorentzian theory. For smeared observables generated by the integer-charge fields $e^{2nbϕ}$, these Lorentzian expectation values define a vacuum functional on an ordered $*$-algebra and support an AQFT-type quantization without positivity. More precisely, we obtain isotone local algebras, a complete locally convex space $\mathcal H$ with dense algebraic subspace $\mathcal H_0$ carrying a nondegenerate Hermitian form (shown to be indefinite for $b<8^{-1/2}$), a continuous cyclic representation, operator-topologically closed represented local algebras, an action of cylinder translations by continuous linear homeomorphisms, and locality for the represented local net. The construction does not produce a Hilbert space or a Haag-Kastler net of local von Neumann algebras in the usual sense, but it shows that a substantial part of the Euclidean-to-Lorentzian and algebraic reconstruction mechanism survives in this nonpositive setting for timelike Liouville theory on the cylinder. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29203 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Lorentzian construction of timelike Liouville field theory on the cylinder Chatterjee, Sourav Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Probability 81T08, 83C45, 81T40 Timelike Liouville field theory is a candidate model for positive curvature two-dimensional quantum gravity, but a mathematically controlled Lorentzian formulation has remained elusive. In this paper we construct the theory on the cylinder $\mathbb{R}\times \mathbb{S}^1$ in the integer screening sector for a natural algebra of renormalized exponential observables. Starting from a renormalized finite-volume torus regularization, we construct infinite-volume Euclidean correlation functions, prove analytic continuation in the time variables, and identify the resulting Lorentzian boundary values by explicit contour formulas. This yields exact Lorentzian correlators for a natural class of exponential observables. We then prove locality: spacelike separated vertex operators commute in the Lorentzian theory. For smeared observables generated by the integer-charge fields $e^{2nbϕ}$, these Lorentzian expectation values define a vacuum functional on an ordered $*$-algebra and support an AQFT-type quantization without positivity. More precisely, we obtain isotone local algebras, a complete locally convex space $\mathcal H$ with dense algebraic subspace $\mathcal H_0$ carrying a nondegenerate Hermitian form (shown to be indefinite for $b<8^{-1/2}$), a continuous cyclic representation, operator-topologically closed represented local algebras, an action of cylinder translations by continuous linear homeomorphisms, and locality for the represented local net. The construction does not produce a Hilbert space or a Haag-Kastler net of local von Neumann algebras in the usual sense, but it shows that a substantial part of the Euclidean-to-Lorentzian and algebraic reconstruction mechanism survives in this nonpositive setting for timelike Liouville theory on the cylinder. |
| title | A Lorentzian construction of timelike Liouville field theory on the cylinder |
| topic | Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Probability 81T08, 83C45, 81T40 |
| url | https://arxiv.org/abs/2605.29203 |