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 renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects including: universality; the selection rule |p1q2 − p2q1| = 1 for transitions between quantum Hall states characterised by filling factors ν1 = p1/q1 and ν2 = p2/q2; critical values of the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaux.