Duality in strongly interacting systems: 𝒩 = 2 SUSY 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 = 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.