Crossing Probabilities and Modular Forms

TL;DR

This paper explores the modular properties of crossing probabilities in 2D conformally invariant systems, revealing new modular forms and characterizing known functions like Cardy's percolation probability through modular transformations.

Contribution

It introduces a new class of higher-order modular forms and demonstrates how crossing probabilities can be understood via modular symmetry in conformal field theory.

Findings

Crossing probabilities exhibit modular behavior.

Cardy's crossing probability follows from modular symmetry.

A new type of higher-order modular form is identified.

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 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.