Conformal loop ensembles and the stress-energy tensor

TL;DR

This paper constructs the stress-energy tensor within conformal loop ensembles (CLE) for all dilute regime values of , generalizing previous work and connecting CLE with conformal field theory (CFT) concepts.

Contribution

It provides a novel construction of the stress-energy tensor in CLE for all dilute regime rom 8/3 to 4or the first time, linking CLE with CFT stress-energy tensor properties.

Findings

Derived conformal Ward identities for CLE

Identified properties of the stress-energy tensor under conformal maps

Expressed the one-point average in terms of the relative partition function

Abstract

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 <= 4 (corresponding to the central charges 0 < c <= 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 Sheffield and Werner (2006). 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, Riva and Cardy (2006), 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.