Self-Dual Chern-Simons Theories

TL;DR

This paper reviews classical self-dual Chern-Simons theories describing charged scalar fields in 2+1 dimensions, highlighting their solutions, symmetries, and integrability in both nonrelativistic and relativistic cases.

Contribution

It provides a comprehensive analysis of self-dual Chern-Simons models, including their solutions, symmetries, and integrability properties, with new insights into their potential structures and supersymmetric embeddings.

Findings

Nonrelativistic self-dual equations are integrable and all finite charge solutions are obtainable.

Relativistic models have a sixth-order potential with rich degenerate minima.

Self-dual solutions correspond to energy-minimizing configurations satisfying first-order equations.

Abstract

In these lectures I review classical aspects of the self-dual Chern-Simons systems which describe charged scalar fields in $2+1$ dimensions coupled to a gauge field whose dynamics is provided by a pure Chern-Simons Lagrangian. These self-dual models have one realization with nonrelativistic dynamics for the scalar fields, and another with relativistic dynamics for the scalars. In each model, the energy density may be minimized by a Bogomol'nyi bound which is saturated by solutions to a set of first-order self-duality equations. In the nonrelativistic case the self-dual potential is quartic, the system possesses a dynamical conformal symmetry, and the self-dual solutions are equivalent to the static zero energy solutions of the equations of motion. The nonrelativistic self-duality equations are integrable and all finite charge solutions may be found. In the relativistic case the self-dual potential is sixth order and the self-dual Lagrangian may be embedded in a model with an extended supersymmetry. The self-dual potential has a rich structure of degenerate classical minima, and the vacuum masses generated by the Chern-Simons Higgs mechanism reflect the self-dual nature of the potential.