  We study the $SU(2)_k$ Wess-Zumino-Novikov-Witten (WZNW) theory perturbed by
the trace of the primary field in the adjoint representation, a theory
governing the low-energy behaviour of a class of strongly correlated electronic
systems. While the model is non-integrable, its dynamics can be investigated
using the numerical technique of the truncated conformal spectrum approach
combined with numerical and analytical renormalization groups (TCSA+RG). The
numerical results so obtained provide support for a semiclassical analysis
valid at $k\gg 1$. Namely, we find that the low energy behavior is sensitive to
the sign of the coupling constant, $\lambda$. Moreover for $\lambda>0$ this
behavior depends on whether $k$ is even or odd. With $k$ even, we find
definitive evidence that the model at low energies is equivalent to the massive
$O(3)$ sigma model. For $k$ odd, the numerical evidence is more equivocal, but
we find indications that the low energy effective theory is critical.
