Conformal Field Theories of Stochastic Loewner Evolutions
Abstract
Stochastic Loewner evolutions (SLE) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLE evolutions and conformal field theories (CFT) which is based on a group theoretical formulation of SLE processes and on the identification of the proper hull boundary states. This allows us to define an infinite set of SLE 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 SLE evolutions. We point out a relation between SLE processes and two dimensional gravity and conjecture a reconstruction procedure of conformal field theories from SLE data.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2003
 DOI:
 10.1007/s002200030881x
 arXiv:
 arXiv:hepth/0210015
 Bibcode:
 2003CMaPh.239..493B
 Keywords:

 High Energy Physics  Theory;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 Mathematics  Mathematical Physics;
 Mathematics  Probability
 EPrint:
 38 pages, 3 figures, to appear in Commun. Math. Phys