Duality in strongly interacting systems: N = 2SUSY Yang-Mills and the quantum Hall effect

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 = 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