Crossing Probabilities and Modular Forms

Details Crossing Probabilities and Modular Forms Crossing Probabilities and Modular Forms

2002-09-12

arxiv.org/abs/math-ph/0209023v2

J. Stat. Phys. (2003) 113, pp. 431-454

Physics

Mathematical Physics

16 pages, AMSTeX. Minor corrections, references added

Scientific paper

10.1023/A:1026012600583

We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic L\"owner 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.

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