On conformal field theory and stochastic Loewner evolution

Abstract We describe Stochastic Loewner Evolution on arbitrary Riemann surfaces with boundary using Conformal Field Theory methods. We propose in particular a CFT construction for a probability measure on (clouded) paths, and check it against known restriction properties. The probability measure can be thought of as a section of the determinant bundle over moduli spaces of Riemann surfaces. Loewner evolutions have a natural description in terms of random walk in the moduli space, and the stochastic diffusion equation translates to the Virasoro action of a certain weight-two operator on a uniformised version of the determinant bundle.

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

[2]  Denis Bernard,et al.  Conformal Transformations and the SLE Partition Function Martingale , 2004 .

[3]  K. Gawȩdzki,et al.  Wess-Zumino-Witten conformal field theory for simply laced groups at level one , 1992 .

[4]  Wendelin Werner,et al.  Conformal Restriction, Highest-Weight Representations and SLE , 2003 .

[5]  M. Bauer,et al.  Conformal Field Theories of Stochastic Loewner Evolutions , 2002, hep-th/0210015.

[6]  S. Shenker,et al.  The Analytic Geometry of Two-Dimensional Conformal Field Theory , 1987 .

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

[8]  K. Ranganathan,et al.  Connections on the state-space over conformal field theories , 1993, hep-th/9304053.

[9]  J. Cardy.  Lectures on Conformal Invariance and Percolation , 2001 .

[10]  Denis Bernard,et al.  SLE martingales and the Virasoro algebra , 2003 .

[11]  J. Cardy.  Boundary conditions, fusion rules and the Verlinde formula , 1989 .

[12]  Shlomo Nir,et al.  NATO ASI Series , 1995 .

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

[14]  G. Segal.  The Definition of Conformal Field Theory , 1988 .

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

[16]  M. Flohr,et al.  Conformal Field Theory , 2006 .

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

[18]  E. Frenkel,et al.  Vertex Algebras and Algebraic Curves , 2000, math/0007054.

[19]  A. Beilinson,et al.  Determinant bundles and Virasoro algebras , 1988 .

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

[21]  Wendelin Werner,et al.  Conformal fields, restriction properties, degenerate representations and SLE , 2002 .

[22]  M. Kontsevich.  Virasoro algebra and Teichmüller spaces , 1987 .

[23]  David Mumford,et al.  Stability of projective varieties , 1977 .

[24]  M. Cornalba,et al.  The Picard groups of the moduli spaces of curves , 1987 .

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

[26]  D. Bernard,et al.  SLEκ growth processes and conformal field theories , 2002 .