Renormalization of modular invariant Coulomb gas and Sine-Gordon theories, and quantum Hall flow diagram

Using the renormalisation group (RG) we study two dimensional electromagnetic coulomb gas and extended Sine-Gordon theories invariant under the modular group SL(2,Z). The flow diagram is established from the scaling equations, and we derive the critical behaviour at the various transition points of the diagram. Following proposal for a SL(2,Z) duality between different quantum Hall fluids, we discuss the analogy between this flow and the global quantum Hall phase diagram.