Crossing Probabilities and Modular Forms

Abstract

We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic Löwner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In particular, Cardy's function for the percolation crossing probability (including the conformal dimension 1/3), follows from a simple modular argument. A new type of "higher-order modular form" arises and its properties are discussed briefly.

16 pages, AMSTeX. Minor corrections, references added

Cited by in corpus (16)

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• Conformal boundary state for the rectangular geometry
• Anchored Critical Percolation Clusters and 2-D Electrostatics
• Rectangular amplitudes, conformal blocks, and applications to loop models
• A formula for crossing probabilities of critical systems inside polygons
• Proposal for a CFT interpretation of Watts' differential equation for percolation
• Connection probabilities for conformal loop ensembles
• Finite-size corrections for universal boundary entropy in bond percolation
• The density of critical percolation clusters touching the boundaries of strips and squares
• Cluster densities at 2-D critical points in rectangular geometries
• Density of critical clusters in strips of strongly disordered systems
• The computation of generalized percolation critical polynomials by the deletion-contraction algorithm
• A generalization of Cardy's and Schramm's formulae