Conformal fields, restriction properties, degenerate representations and SLE

Abstract We relate the Schramm–Loewner Evolution processes (SLE) to highest-weight representations of the Virasoro Algebra. The restriction properties of SLE that have been recently derived in [19] play a crucial role. In this setup, various considerations from conformal field theory can be interpreted and reformulated via SLE. This enables one to make a concrete link between the two-dimensional discrete critical systems from statistical physics and conformal field theory. To cite this article: R. Friedrich, W. Werner, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 947–952.

[1]  V. Kac,et al.  Bombay Lectures On Highest Weight Representations Of Infinite Dimensional Lie Algebras , 1983 .

[2]  Oded Schramm,et al.  Basic properties of SLE , 2001 .

[3]  Wendelin Werner,et al.  One-Arm Exponent for Critical 2D Percolation , 2001 .

[4]  Oded Schramm,et al.  Scaling limits of loop-erased random walks and uniform spanning trees , 1999, math/9904022.

[5]  Saleur,et al.  Exact surface and wedge exponents for polymers in two dimensions. , 1986, Physical review letters.

[6]  O. Schramm,et al.  On the scaling limit of planar self-avoiding walk , 2002, math/0204277.

[7]  Wendelin Werner,et al.  Conformal invariance of planar loop-erased random walks and uniform spanning trees , 2001 .

[8]  P. Goddard,et al.  Kac-Moody and Virasoro algebras : a reprint volume for physicists , 1988 .

[9]  Wendelin Werner,et al.  Values of Brownian intersection exponents, I: Half-plane exponents , 1999 .

[10]  A. Polyakov,et al.  Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory - Nucl. Phys. B241, 333 (1984) , 1984 .

[11]  C. Itzykson,et al.  Conformal invariance and applications to statistical mechanics , 1998 .

[12]  T. Kennedy.  Monte Carlo tests of stochastic Loewner evolution predictions for the 2D self-avoiding walk. , 2001, Physical review letters.

[13]  Yvan Saint-Aubin,et al.  Conformal invariance in two-dimensional percolation , 1994 .

[14]  Wendelin Werner.  Random planar curves and Schramm-Loewner evolutions , 2003 .

[15]  I. Stewart,et al.  Infinite-dimensional Lie algebras , 1974 .

[16]  O. Schramm,et al.  Conformal restriction: The chordal case , 2002, math/0209343.

[17]  S. Smirnov.  Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits , 2001 .

[18]  J. Cardy,et al.  Conformal Invariance and Surface Critical Behavior , 1984 .

[19]  A. Polyakov.  Nonhamiltonian approach to conformal quantum field theory , 1974 .

[20]  V. Kaimanovich.  An introduction to the Stochastic Loewner Evolution , 2004 .

[21]  Wendelin Werner,et al.  CRITICAL EXPONENTS FOR TWO-DIMENSIONAL PERCOLATION , 2001 .

[22]  Wendelin Werner,et al.  Values of Brownian intersection exponents III: Two-sided exponents , 2002 .

[23]  Wendelin Werner,et al.  Values of Brownian intersection exponents, II: Plane exponents , 2000, math/0003156.