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

Dolan, Brian P. (2007) Modular Symmetry and Fractional Charges in N = 2 Supersymmetric Yang–Mills and the Quantum Hall Effect. Symmetry, Integrability and Geometry : Methods and Applications, 3 (10). pp. 1-31. ISSN 1815-0659

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Abstract

The parallel roles 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.

Item Type: Article
Additional Information: This paper (doi:10.3842/SIGMA.2007.010) is a contribution to the Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 22–24, 2006, Budapest, Hungary). The full collection is available at http://www.emis.de/journals/SIGMA/LOR2006.html. It is a pleasure to thank my long-term collaborator Cliff Burgess for many useful discussions on the quantum Hall effect. This work was partly supported by Enterprise Ireland Basic Research grant SC/2003/415.
Keywords: duality; modular symmetry; supersymmetry; quantum Hall effect;
Subjects: Science & Engineering > Mathematical Physics
Item ID: 2839
Identification Number: 10.3842/SIGMA.2007.010
Depositing User: Dr. Brian Dolan
Date Deposited: 16 Nov 2011 16:29
Journal or Publication Title: Symmetry, Integrability and Geometry : Methods and Applications
Refereed: Yes
URI:

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