Duality in supersymmetric Yang Mills and the quantum Hall effect

The evidence for the parallel roles played by the modular group in N = 2 supersymmetric Yang-Mills in (3 + 1) dimensions and the quantum Hall effect in (2 + 1) dimensions is reviewed. In both cases a subgroup of the full modular group acts as a map between different low energy phases of the theory, parametrised by a complex parameter in the upper-half-complex plane whose real part is a topological parameter and whose imaginary part is the coupling associated the kinetic term of the effective U(1) gauge theory. In the case of the quantum Hall effect experimental evidence in favour of the modular action is also reviewed.