Classification of Non-abelian Chern-simons Vortices

The two-dimensional self-dual Chern-Simons equations are equivalent to the conditions for static, zero-energy vortex-like solutions of the (2+1) dimensional gauged nonlinear Schrödinger equation with Chern-Simons matter-gauge coupling. The finite charge vacuum states in the Chern-Simons theory are shown to be gauge equivalent to the finite action solutions to the two-dimensional chiral model (or harmonic map) equations. The Uhlenbeck-Wood classification of such harmonic maps into the unitary groups thereby leads to a complete classification of the vacuum states of the ChernSimons model. This construction also leads to an interesting new relationship between SU(N) Toda theories and the SU(N) chiral model. The study of the nonlinear Schrödinger equation in 2+1-dimensional space-time is partly motivated by the well-known success of the 1+1-dimensional nonlinear Schrödinger equation. Here we consider a gauged nonlinear Schrödinger equation in which we have not only the nonlinear potential term for the matter fields, but also we have a coupling of the matter fields to the gauge fields. Furthermore, this matter-gauge dynamics is chosen to be of the Chern-Simons form rather than the conventional Yang-Mills form. With this choice, the ∗Talk presented at the XXII International Conference on Differential Geometric Methods in Theoretical Physics, Ixtapa (Mexico), September 1993.