Duality and the modular group in the quantum Hall effect

We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalization group flow, we derive many properties of both the integer and fractional quantum Hall effects including: universality; the selection rule | p 1 q 2 - p 2 q 1 | = 1 for transitions between quantum Hall states characterized by filling factors 1 = p 1 / q 1 and 2 = p 2 / q 2 ; critical values of the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalization group flow lead to the semicircle rule for transitions between Hall plateaux.