Conformal field theories of stochastic Loewner evolutions

Stochastic Loewner evolutions (SLEkappa) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLEkappa evolutions and conformal field theories (CFT) which is based on a group theoretical formulation of SLEkappa\ processes and on the identification of the proper hull boundary states. This allows us to define an infinite set of SLEkappa zero modes, or martingales, whose existence is a consequence of the existence of a null vector in the appropriate Virasoro modules. This identification leads, for instance, to linear systems for generalized crossing probabilities whose coefficients are multipoint CFT correlation functions. It provides a direct link between conformal correlation functions and probabilities of stopping time events in SLEkappa evolutions. We point out a relation between SLEkappa processes and two dimensional gravity and conjecture a reconstruction procedure of conformal field theories from S L E-kappa data.