Stress-energy in Liouville conformal field theory. (English) Zbl 1446.81036
Summary: We construct the stress-energy tensor correlation functions in probabilistic Liouville conformal field theory (LCFT) on the two-dimensional sphere \({\mathbb{S}}^2\) by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric on \({\mathbb{S}}^2\). In particular we derive conformal Ward identities for these correlation functions. This forms the basis for the construction of a representation of the Virasoro algebra on the canonical Hilbert space of the LCFT. In A. Kupiainen et al. [Commun. Math. Phys. 371, No. 3, 1005-1069 (2019; Zbl 1480.83053)] the conformal Ward identities were derived for one and two stress-energy tensor insertions using a different definition of the stress-energy tensor and Gaussian integration by parts. By defining the stress-energy correlation functions as functional derivatives of the LCFT correlation functions and using the smoothness of the LCFT correlation functions proven in J. Oikarinen [Ann. Henri Poincaré 20, No. 7, 2377-2406 (2019; Zbl 1459.81100)] allows us to control an arbitrary number of stress-energy tensor insertions needed for representation theory.
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 17B68 | Virasoro and related algebras |
| 70S05 | Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems |
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