Exact solutions of SL(N,R)-invariant chiral equations one- and two-dimensional subspaces

Abstract

A methodology for integrating the chiral equation (ρg,zg−1),z̄+ (ρg,z̄g−1),z=0 is developed, when g is a matrix of the SL(N,R) group. In this work the ansätze g=g(λ) where λ satisfy the Laplace equation and g=g(λ,τ) are made, where λ and τ are geodesic parameters of an arbitrary Riemannian space. This reduces the chiral equation to an algebraic problem and g can be obtained by integrating a homogeneous linear system of differential equations. As an example of the first ansatz, all the matrices for N=3 and one example for N=8, which corresponds to exact solutions of the d=5 and d=10 Kaluza–Klein theory, respectively are given. For the second ansatz the chiral equations are integrated for the subgroups SL(2,R), SO(2,1R), Sp(2,R), and the Abelian subgroups.