SLE martingales and the Virasoro algebra

We present an explicit relation between representations of the Virasoro algebra and polynomial martingales in stochastic Loewner evolutions (SLE). We show that the Virasoro algebra is the spectrum generating algebra of SLE martingales. This is based on a new representation of the Virasoro algebra, inspired by the Borel-Weil construction, acting on functions depending on coordinates parametrizing conformal maps. Fractal critical clusters are the cornerstones of criticality, especially in two dimensions, see eg refs.[1, 2, 3]. Stochastic Loewner evolutions [4, 5, 6] are random processes adapted to a probabilistic description of such fractals. The aim of this Letter is to elaborate on the connection between stochastic Loewner evolutions (SLE) and conformal field theories (CFT) developed in ref.[7]. We shall construct new representations of the Virasoro algebra which allow us to show explicitely that the Virasoro algebra is the generating algebra of (polynomial) martingales for the SLE processes. Physically, martingales are observables conserved in mean. They are essential ingredients for estimating probability of events. Basic definitions of the stochastic Loewner evolutions and of their martingales are recalled in the two first sections. The new representations of Email: [email protected] Member of the CNRS; email: [email protected]