Q-balls of quasi-particles in a (2, 0)-theory model of the fractional quantum Hall effect

Abstract

A toy model of the fractional quantum Hall effect appears as part of the low-energy description of the Coulomb branch of the A 1 (2, 0)-theory formulated on $$ \left({S}^1\times {\mathrm{\mathbb{R}}}^2\right)/{\mathrm{\mathbb{Z}}}_k $$ , where the generator of $$ {\mathrm{\mathbb{Z}}}_k $$ acts as a combination of translation on S 1 and rotation by 2π/k on $$ {\mathrm{\mathbb{R}}}^2 $$ . At low energy the configuration is described in terms of a 4+1D Super-Yang-Mills theory on a cone $$ \left({\mathrm{\mathbb{R}}}^2/{\mathrm{\mathbb{Z}}}_k\right) $$ with additional 2+1D degrees of freedom at the tip of the cone that include fractionally charged particles. These fractionally charged “quasi-particles” are BPS strings of the (2, 0)-theory wrapped on short cycles. We analyze the large k limit, where a smooth cigar-geometry provides an alternative description. In this framework a W-boson can be modeled as a bound state of k quasi-particles. The W-boson becomes a Q-ball, and it can be described as a soliton solution of Bogomolnyi monopole equations on a certain auxiliary curved space. We show that axisymmetric solutions of these equations correspond to singular maps from AdS 3 to AdS 2, and we present some numerical results and an asymptotic expansion.