Abstract
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 Kupiainen et al. (Commun Math Phys 371:1005–1069, 2019) 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 Oikarinen (Ann Henri Poincaré 20(7):2377–2406, 2019) allows us to control an arbitrary number of stress-energy tensor insertions needed for representation theory.
Highlights
Introduction and Main Result1.1 Local Conformal SymmetryTwo-dimensional conformal field theory (CFT) is characterized by local conformal symmetry, an infinite dimensional symmetry that strongly constrains the theory
The local conformal symmetry arises from the transformation properties of the correlation functions under the action of the groups of smooth diffeomorphisms and local Weyl transformations of the metric
It is necessary to have (1.8) for arbitrary n in order to construct the representation of the Virasoro algebra for Liouville conformal field theory (LCFT)
Summary
Two-dimensional conformal field theory (CFT) is characterized by local conformal symmetry, an infinite dimensional symmetry that strongly constrains the theory. The local conformal symmetry arises from the transformation properties of the correlation functions under the action of the groups of smooth diffeomorphisms and local Weyl transformations of the metric. The former acts by pullback on the metric g → ψ∗g and the latter acts by a local scale transformation g → eφ g with φ ∈ C∞( , R). Every smooth metric g can be obtained from a given one gby the action of diffeomorphisms and Weyl transformations: This fact together with the symmetries (1.2) and (1.3) yields the tools for defining and computing the functional derivatives on the right-hand side of (1.5). This term originates from the fact that compact Riemann surfaces with positive genus have non-trivial moduli spaces, so variation of the metric can vary the conformal class of the surface
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