Title: The conformal field theory of conformal loop ensembles

Abstract:
It is by now well known that conformal field theory (CFT) is closely linked with Schramm-Loewner evolution. In this talk I will review my work on the more recent subject of linking CFT with conformal loop ensembles (CLE). CLE can be seen as an extension of SLE, a measure description of all random cluster boundaries in the scaling limit (loops in plane domains). The emphasis is on associating local variables with local fields, such that expectations of their products reproduce CFT correlation functions. I will explain how this is done for the stress-energy tensor and many of its descendants, using the Virasoro vertex operator algebra structure of CFT correlation functions on arbitrary domains. I will then explain how this gives rise to null-vector equations in CLE.