Massless renormalization group flow in SU(N)k perturbed conformal field theory

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.