Nonabelian Nonrelativistic Model

Abstract

This Chapter deals with the nonabelian generalization of the abelian nonrelativistic self-dual Chern-Simons models discussed in Chapter 2. This nonabelian generalization is of interest for the understanding of nonabelian fractional statistics, a rich subject with potential applications in various quantum Hall systems [257,31]. As in the abelian models, there is a Bogomol’nyi lower bound for the energy which is saturated by static zero energy configurations which satisfy a set of first order self-duality equations. These self-duality equations may be reduced, by certain algebraic ansatze, to many examples of two dimensional integrable partial differential equations such as the classical Toda or affine Toda equations. Furthermore, with adjoint matter coupling the self-duality equations may be converted into the (Euclidean) two dimensional chiral model equation by a judicious choice of gauge. This fact leads to a complete classification of finite charge solutions in terms of Uhlenbeck’s classification of finite action solutions to the chiral model (or “harmonic map”) equations. This analysis is very similar in both spirit and style to the classification of finite action (“instanton”) solutions to the self-dual Yang-Mills equations (1.3) in four dimensional Euclidean space. In fact, the nonabelian nonrelativistic self-dual Chern-Simons equations may be obtained from the four dimensional self-dual Yang-Mills equations by a dimensional reduction.