The relativistic non-Abelian Chern-Simons equations

We study the r × r system of nonlinear elliptic equations (formula presented) where λ > O is a constant parameter, K - (K_ab) is the Cartan matrix of a semi-simple Lie algebra, and op is the Dirac measure concentrated at p ∈ R². This system of equations arises in the relativistic non-Abelian Chern-Simons theory and may be viewed as a non-integrable deformation of the integrable Toda system. We establish the existence of a class of solutions known as topological multivortices. The crucial step in our method is the use of the decomposition theorem of Cholesky for positive definite matrices so that a variational principle can be formulated.