Classical solutions of the vacuum Maxwell's equations exhibit a SO(2) duality symmetry, which is enhanced to Sl(2,R) when dilaton and axion fields are included. Quantum effects break this symmetry but semi-classically Sl(2,Z) symmetry, or a sub-group thereof, survives in Dirac-Schwinger-Zwanziger quantisation. Even this symmetry is expected to be broken in the full theory of quantum electrodynamics, but a modular sub-group survives as an infinite discrete symmetry of the vacua of N{script} = 2 supersymmetric Yang-Mills theory. An analogous situation occurs in the quantum Hall effect, where different quantum Hall states are related by a modular symmetry which is a sub-group of Sl(2,Z). The similarities between the quantum Hall effect and supersymmetric Yang-Mills are reviewed and a possible link via the gauge/gravity correspondence is described. Scaling exponents in the quantum Hall effect are derived using the gauge-gravity correspondence. Â© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.