SLEκ Growth Processes and Conformal Field Theories

SLEκ stochastic processes describe growth of random curves which, in some cases, may be identified with boundaries of two dimensional critical percolating clusters. By generalizing SLEκ growths to formal Markov processes on the central extension of the 2d conformal group, we establish a connection between conformal field theories with central charges cκ = 1 2(3κ−8)( 6 κ−1) and zero modes – observables which are conserved in mean – of the SLEκ stochastic processes. Critical phenomena are characterized by large scale fluctuations. Conformal field theories [1] are powerful tools for analyzing their multifractal properties, leading for instance to exact results concerning scaling behavior of geometrical models, see eg. refs.[5]. An alternative probabilistic approach has recently been introduced [2]. It consists in formulating conformally covariant processes, so-called SLEκ processes, based on Loewner’s equation. Among the many conformal field theory results, including crossing percolation probabilities [6] or multifractal distributions of electrostatic potential near conformally invariant fractal boundaries [7], some have been rederived in the SLEκ framework, see eg refs.[3, 4] for a review. However, the precise connection between SLEκ models and conformal field theories in the sense Email: [email protected] Member of the CNRS; email: [email protected]