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. 131.
ISSN 18150659
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. Infrared fixed points at
strong coupling correspond to θvacua 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 infrared 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 NonPerturbative 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 longterm 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; 
Academic Unit: 
Faculty of Science and 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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