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

The parallel rôles of modular symmetry in $\mathcal N=2$ 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 $\mathcal N=2$ 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 $\theta$-vacua with $\theta$ 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.