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Conformal loop ensembles and the stress-energy tensor. (English) Zbl 1263.81253
Summary: We give a construction of the stress-energy tensor of conformal field theory (CFT) as a local “object” in conformal loop ensembles CLE$$_{\kappa }$$, for all values of $$\kappa$$ in the dilute regime $$8/3 < \kappa \leq 4$$ (corresponding to the central charges $$0 < c \leq 1$$ and including all CFT minimal models). We provide a quick introduction to CLE, a mathematical theory for random loops in simply connected domains with properties of conformal invariance, developed by S. Sheffield and W. Werner [Ann. Math. (2) 176, No. 3, 1827–1917 (2012; Zbl 1271.60090)]. We consider its extension to more general regions of definition and make various hypotheses that are needed for our construction and expected to hold for CLE in the dilute regime. Using this, we identify the stress-energy tensor in the context of CLE. This is done by deriving its associated conformal Ward identities for single insertions in CLE probability functions, along with the appropriate boundary conditions on simply connected domains; its properties under conformal maps, involving the Schwarzian derivative; and its one-point average in terms of the “relative partition function”. Part of the construction is in the same spirit as, but widely generalizes, that found in the context of SLE$$_{8/3}$$ by the author et al. [Commun. Math. Phys. 268, No. 3, 687–716 (2006; Zbl 1121.81108)], which only dealt with the case of zero central charge in simply connected hyperbolic regions. We do not use the explicit construction of the CLE probability measure, but only its defining and expected general properties.

##### MSC:
 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 82B27 Critical phenomena in equilibrium statistical mechanics 81T08 Constructive quantum field theory 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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