Charged soliton of the three-dimensional CS+BI Abelian gauge theory

Abstract

In this paper, we construct a charged soliton with a finite energy and no delta function source in a pure Abelian gauge theory. Specifically, we first consider the three-dimensional Abelian gauge theory, with a Maxwell term and a level N CS term. We find a static solution that carries charge N, angular momentum N2 and whose radius is N independent. However, this solution has a divergent energy. In analogy to the replacement of the four-dimensional Maxwell action with the BI action, which renders the classical energy of a point charge finite, for the three-dimensional theory which includes a CS term such a replacement leads to a finite energy for the solution of above. We refer to this soliton as a CSBIon solution, representing a finite energy version of the fundamental (sourced) charged electron of Maxwell theory in four dimensions. In three dimensions the BI+CS action has a static charged solution with finite energy and no source, hence a soliton solution. The CSBIon, similar to its Maxwellian predecessor, has a charge N, angular momentum proportional to N and an N-independent radius. We also present other nonlinear modifications of Maxwell theory that admit similar solitons. The CSBIon may be relevant in various holographic scenarios. In particular, it may describe a D6-brane wrapping an S4 in a compactified D4-brane background. We believe that the CSBIon may play a role in condensed matter systems in 2+1 dimensions like graphene sheets.