  We investigate the infrared properties of SU(N)$_k$ conformal field theory
perturbed by its adjoint primary field in 1+1 dimensions. The latter field
theory is shown to govern the low-energy properties of various SU(N) spin chain
problems. In particular, using a mapping onto k-leg SU(N) spin ladder, a
massless renormalization group flow to SU(N)$_1$ criticality is predicted when
N and k have no common divisor. The latter result extends the well-known
massless flow between SU(2)$_k$ and SU(2)$_1$ Wess-Zumino-Novikov-Witten
theories when k is odd in connection to the Haldane's conjecture on SU(2)
Heisenberg spin chains. A direct approach is presented in the simplest N=3 and
k=2 case to investigate the existence of this massless flow.
