CHAPTER 1 Introduction 1.1. Background There are four types of simple non-exceptional Lie Algebra:Am, Bm, Cm, and Dm which Cartan subalgebra are sl(m + 1), so(2m + 1), sp(m), and so(2m) re- spectively. To each of them, a Toda system is associated. In geometry, solutions of Toda system are closely related to holomorphic curves in projective spaces. For example, the Toda system of type Am can be derived from the classical Pl¨ucker formulas, and any holomorphic curve gives rise to a solution u of the Toda system, whose branch points correspond to the singularities of u. Conversely, we could inte- grate the Toda system, and any solution u gives rise to a holomorphic curve in CPn at least locally. See [16] and reference therein. It is very interesting to note that the reverse process holds globally if the domain for the equation is S2 or C. Any solution u of type Am Toda system on S2 or C could produce a global holomorphic curve into CPn. This holds even when the solution u has singularities. We refer the readers to [16] for more precise statements of these results. In physics, the Toda system also plays an important role in non-Abelian gauge field theory. One example is the relativistic Chern-Simons model proposed by Dunne [7–9] in order to explain the physics of high critical temperature supercon- ductivity. See also [13], [14] and [15]. The model is defined in the (2+1) Minkowski space R1,2, the gauge group is a compact Lie group with a semi-simple Lie algebra G. The Chern-Simons Lagrangian density L is defined by: L = −kμνρtr(∂μAνAρ + 2 3 AμAνAρ) tr((Dμφ)†Dμφ) V (φ, φ†) for a Higgs field φ in the adjoint representation of the compact gauge group G, where the associated semi-simple Lie algebra is denoted by G and the G−valued gauge field on 2+1 dimensional Minkowski space R1,2 with metric diag{-1,1,1}. Here k 0 is the Chern-Simons coupling parameter, tr is the trace in the matrix representation of G and V is the potential energy density of the Higgs field V (φ, φ†) given by V (φ, φ†) = 1 4k2 tr(([[φ, φ†],φ] v2φ)†([[φ, φ†],φ] v2φ)), where v 0 is a constant which measures either the scale of the broken symmetry or the subcritical temperature of the system. In general, the Euler-Lagrangian equation corresponding L is very difficult to study. So we restrict to consider solutions to be energy minimizers of the Lagrangian functional, and a self-dual system of first order derivatives could be derived from 1
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