SLE for theoretical physicists

Abstract This article provides an introduction to Schramm (stochastic)–Loewner evolution (SLE) and to its connection with conformal field theory, from the point of view of its application to two-dimensional critical behaviour. The emphasis is on the conceptual ideas rather than rigorous proofs.

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

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

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

[4]  Jorgen Rasmussen,et al.  On stochastic evolutions and superconformal field theory , 2005 .

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

[6]  Leo P. Kadanoff,et al.  The Loewner Equation: Maps and Shapes , 2004 .

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

[8]  Cardy Linking numbers for self-avoiding loops and percolation: application to the spin quantum hall transition , 2000, Physical review letters.

[9]  M. B. Hastings,et al.  Laplacian growth as one-dimensional turbulence , 1998 .

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

[11]  M B Hastings Exact multifractal spectra for arbitrary laplacian random walks. , 2002, Physical review letters.

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

[13]  Leo P. Kadanoff,et al.  Exact Solutions for Loewner Evolutions , 2004 .

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

[15]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

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

[17]  Denis Bernard,et al.  Loewner Chains , 2004 .

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

[19]  M. Lapidus,et al.  Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot , 2004 .

[20]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[21]  Wouter Kager,et al.  A Guide to Stochastic Löwner Evolution and Its Applications , 2003 .

[22]  Gregory F. Lawler,et al.  Conformally Invariant Processes in the Plane , 2005 .

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

[24]  Julien Dubedat SLE(κ,ρ) martingales and duality , 2003 .

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

[26]  Jussi Kalkkinen,et al.  On conformal field theory and stochastic Loewner evolution , 2004 .

[27]  Vincent Beffara Hausdorff dimensions for SLE6 , 2004 .

[28]  R. P. Langlands,et al.  On the universality of crossing probabilities in two-dimensional percolation , 1992 .

[29]  Oded Schramm A Percolation Formula , 2001 .

[30]  John Cardy Crossing formulae for critical percolation in an annulus , 2002 .

[31]  Edwin D. Mares,et al.  On S , 1994, Stud Logica.

[32]  Wendelin Werner Girsanov's transformation for SLE(kappa,rho) processes, intersection exponents and hiding exponents , 2003 .

[33]  John Cardy Boundary Conformal Field Theory , 2004 .

[34]  Duplantier Conformally invariant fractals and potential theory , 2000, Physical review letters.

[35]  Karl Löwner Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I , 1923 .

[36]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[37]  J. Cardy,et al.  Calogero–Sutherland model and bulk-boundary correlations in conformal field theory , 2003, hep-th/0310291.

[38]  李幼升,et al.  Ph , 1989 .

[39]  John Cardy Critical percolation in finite geometries , 1992 .

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

[41]  B. M. Fulk MATH , 1992 .

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

[43]  Denis Bernard,et al.  CFTs of SLEs: the radial case , 2004 .

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

[45]  Alexander M. Polyakov,et al.  Infinite conformal symmetry of critical fluctuations in two dimensions , 1984 .

[46]  J. Kondev LIOUVILLE FIELD THEORY OF FLUCTUATING LOOPS , 1997 .

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

[48]  Oded Schramm,et al.  Harmonic explorer and its convergence to SLE4 , 2003 .

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

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

[51]  Bertrand Duplantier Conformal Fractal Geometry and Boundary Quantum Gravity , 2003 .