Γ(2) modular symmetry, renormalization group flow and the quantum Hall effect

We construct a family of holomorphic beta-functions whose renormalization group (RG) flow preserves the Gamma(2) modular symmetry and reproduces the observed stability of the Hall plateaus. The semicircle law relating the longitudinal and Hall conductivities that has been experimentally observed is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram. The generic scale dependence of the conductivities is found to agree qualitatively with the present experimental data. The existence of a crossing point occurring in the crossover of the permitted transitions is discussed.