Nonabelian Relativistic Model

Abstract

This Chapter extends the relativistic self-dual Chern-Simons model to include nonabelian local gauge symmetry. This extension introduces many new features related to the interplay of the self-duality with the algebraic structure of the fields [169,170,41,62]. There is once again a Bogomol’nyi lower bound on the energy which is saturated by solutions to a set of first-order self-duality equations. This is true for a general matter coupling, but (as in the corresponding nonrelativistic models) the self-duality equations exhibit additional special properties with adjoint matter coupling, for which one can define matter and gauge fields in the same (arbitrary) representation of the gauge Lie algebra. In this case, the self-dual potential has an intricate vacuum structure, with vacua that are classified by embeddings of SU(2) into the gauge algebra [140,64,65]. In these vacua, there are massive excitations for both gauge and scalar fields, and the masses follow unusual patterns which reflect the self-dual nature of the system and its associated N = 2 supersymmetry. In each vacuum each mass is paired, sometimes as complex scalar fields but in other cases as pairs of mass-degenerate real gauge and real scalar fields. These real fields have masses given by a simple universal mass formula in terms of the exponents of the gauge algebra.