16.5. Liouville theory and minimal models

a. In the quantization scheme of the Sine–Gordon model in terms of the Liouville theory, determine the quantized values of the coupling constant g that reproduce the central charges of the minimal models. Prove that the conformal dimension of the vertex operator that perturbes the Liouville action is equal to 1,3.

b. Repeat the same exercise for the two Liouville theories, with complex exponentials, associated to the Bullogh–Dodd model. Show that the perturbations correspond to the operators 1,2 and 2,1 of the minimal models respectively.