CLASSICAL SOLUTIONS IN (2+1)-DIMENSIONAL TOPOLOGICALLY MASSIVE YANG-MILLS THEORIES

Abstract

The aim of our investigation is to find exact analytical solutions for the SU(2) Yang-Mills (YM) equations with the Chern-Simons (CS) term in (2+1) dimensions. Beginning with a simple static and axially symmetric ansatz, modified Bessel function solutions are obtained. This axially symmetric ansatz essentially linearizes the YM equations with the CS term, and the solution is actually abelian. Generalization of the ansatz can be done, giving modified Bessel function solution again. The solutions have singularity at the origin and vanish to zero like a vortex solution at large distance. We can regard them as the Dirac type vortex since Higgs field is absent. For this class of solutions, time dependence can be introduced by assuming functional dependence in time in the ansatz. The resulting solution exhibits exponential time decay. Complex time dependent solution can also be obtained by defining a suitable complex null vector. More general static and complex solution can be obtained from the real static axially symmetric ansatz. The resulting reduced equations permit different types of solutions, each defined by a parameter. All the axially symmetric ansatze are also valid in Euclidean space. This class of solutions can be regarded as real static vortex-likc solutions for the noncompact gauge group SL(2,C) and may be relevant in (2+1)-dimensional gravity theory. External source densities can be introduced into the equations of motion. Two different types of source densities are discussed. Ansatz expressing the gauge field in terms of the Jacobi elliptic functions is constructed in Euclidean space. Special solutions of the Jacobi elliptic function are displayed. Finally wave-like solutions are exhibited in Minkowski space.