We present an approach that gives rigorous construction of a class of crossing invariant functions in c = 1 CFTs from the weakly invariant distributions on the moduli SLspace A404(s,C) of SL(2, C) flat connections on the sphere with four punctures. By using this approach we show how to obtain correlation functions in the Ashkin-Teller and the RunkelWatts theory. Among the possible crossing-invariant theories, we obtain also the analytic Liouville theory, whose consistence was assumed only on the basis of numerical tests.
