Conformal loop ensembles and the stress-energy tensor. II. Construction of the stress-energy tensor

Abstract

This is the second part of a work aimed at constructing the stress-energy tensor of conformal field theory (CFT) as a local "object" in conformal loop ensembles (CLE). This work lies in the wider context of re-constructing quantum field theory from mathematically well-defined ensembles of random objects. In the present paper, based on results of the first part, we identify the stress-energy tensor in the dilute regime of CLE. This is done by deriving both its conformal Ward identities for single insertion in CLE probability functions, and its properties under conformal transformations involving the Schwarzian derivative. We also give the one-point function of the stress-energy tensor in terms of a notion of partition function, and we show that this agrees with standard CFT arguments. The construction is in the same spirit as that found in the context of SLE(8/3) by the author, Riva and Cardy (2006), which had to do with the case of zero central charge. The present construction generalises this to all central charges between 0 and 1, including all minimal models. This generalisation is non-trivial: the application of these ideas to the CLE context requires the introduction of a renormalised probability, and the derivation of the transformation properties and of the one-point function do not have counterparts in the SLE context.