  We consider random conformally invariant paths in the complex plane (SLEs).
Using the Coulomb gas method in conformal field theory, we rederive the mixed
multifractal exponents associated with both the harmonic measure and winding
(rotation or monodromy) near such critical curves, previously obtained by
quantum gravity methods. The results also extend to the general cases of
harmonic measure moments and winding of multiple paths in a star configuration.
