On harmonic measure of critical curves

TL;DR

This paper introduces a conformal field theory approach to analyze the harmonic measure of critical curves in 2D systems, extending previous methods to theories with central charge greater than 1.

Contribution

It provides a new, straightforward method to compute fractal geometric properties of critical curves using fusion of primary fields in conformal field theory, beyond the c ≤ 1 case.

Findings

Connects harmonic measure to primary field fusion operators

Enables computation of fractal geometry characteristics with standard CFT methods

Extends analysis to theories with central charge greater than 1

Abstract

Fractal geometry of critical curves appearing in 2D critical systems is characterized by their harmonic measure. For systems described by conformal field theories with central charge $c\leqslant 1$, scaling exponents of harmonic measure have been computed by B. Duplantier [Phys. Rev. Lett. {\bf 84}, 1363 (2000)] by relating the problem to boundary two-dimensional gravity. We present a simple argument that allows us to connect harmonic measure of critical curves to operators obtained by fusion of primary fields, and compute characteristics of fractal geometry by means of regular methods of conformal field theory. The method is not limited to theories with $c\leqslant 1$.