Harmonic measure of critical curves and conformal field theory

Ilya Gruzberg
University of Chicago

Fractal geometry of critical curves appearing in 2D critical systems is characterized by their harmonic measure. For systems described by conformal field theories (CFTs) with central charge $c \leqslant 1$, scaling exponents of the harmonic measure have been computed by B. Duplantier by relating the problem to boundary 2D gravity. I will present a simple argument connecting the harmonic measure of critical curves to operators obtained by fusion of CFT primary fields, and compute characteristics of the fractal geometry by means of regular methods of CFT. The method is not limited to theories with $c \leqslant 1$.

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