Supersymmetric Yang-Mills and the Quantum Hall Effect

It is argued that there are strong similarities between the infra-red physics of N=2 supersymmetric Yang-Mills and that of the quantum Hall effect, both systems exhibit a hierarchy of vacua with a subgroup of the modular group mapping between them. The β-function for pure SU(2) N = 2 supersymmetric Yang-Mills in 4-dimensions is re-examined and an earlier suggestion in the literature, that was singular at strong coupling, is modified to a form that is well behaved at both weak and strong coupling and describes the crossover in an analytic fashion. Similarities between the phase diagram and the flow of SUSY Yang-Mills and that of the quantum Hall effect are then described, with the Hall conductivity in the latter playing the role of the θ-parameter in the former. Hall plateaux, with odd denominator filling fractions, are analogous to fixed points at strong coupling in N=2 SUSY Yang-Mills, where the massless degrees of freedom carry an odd monopole charge. PACS Nos. 11.10.Hi, 11.30.Pb, 12.60.Jv, 73.40.Hm