Modular Symmetry and Fractional Charges in N = 2 Supersymmetric Yang-Mills and the Quantum Hall Effect

Abstract

Abstract. The parallel rôles of modular symmetry in N = 2 supersymmetric Yang-Mills and in the quantum Hall effect are reviewed. In supersymmetric Yang-Mills theories modular symmetry emerges as a version of Dirac's electric -magnetic duality. It has significant consequences for the vacuum structure of these theories, leading to a fractal vacuum which has an infinite hierarchy of related phases. In the case of N = 2 supersymmetric Yang-Mills in 3 + 1 dimensions, scaling functions can be defined which are modular forms of a subgroup of the full modular group and which interpolate between vacua. Infra-red fixed points at strong coupling correspond to θ-vacua with θ a rational number that, in the case of pure SUSY Yang-Mills, has odd denominator. There is a mass gap for electrically charged particles which can carry fractional electric charge. A similar structure applies to the 2 + 1 dimensional quantum Hall effect where the hierarchy of Hall plateaux can be understood in terms of an action of the modular group and the stability of Hall plateaux is due to the fact that odd denominator Hall conductivities are attractive infra-red fixed points. There is a mass gap for electrically charged excitations which, in the case of the fractional quantum Hall effect, carry fractional electric charge.

Holography of dyonic dilaton black branes

“…Perhaps suitable modifications of the bulk theory, with an additional potential for the dilaton-axion preserving the SL(2, Z) symmetry and/or with disorder put in, might prove interesting in this respect. These modifications might lead to similarities with systems exhibiting the quantum Hall effect, and the transformation law of the conductivity could then tie in with some of the existing discussion in this subject on RG flows between different fixed points characterised by the various subgroups of SL(2, Z) [32,33,34,35,36]. We briefly explored the addition of a modularinvariant potential in §7, but it seems likely that a deeper investigation of the case with potentials could be fruitful.…”

Study of anisotropic black branes in asymptotically anti-de Sitter

“…At the quantum level, this SL(2, R) symmetry will be enhanced to SL(2, Z). For the connection of SL(2, Z) to Quantum Hall Effects, see [43][44][45][46][47] and review [48].…”

A renormalisation group equation for transport coefficients in (2 + 1)-dimensions derived from the AdS/CMT correspondence

“…This may be expected as there is no fermionic matter in the bulk, but there is no charged matter at all in the bulk -it is interpreted a classical effective theory for the electro-magnetic field after charged matter is integrated out. The quantum Hall effect has been related to an emergent Sl(2, R) symmetry in the infra-red [38][39][40][41] and, while the bulk theory used here does enjoy an electromagnetic duality symmetry, giving rise to S-duality of the…”