# Is there any qualitative difference between the WZW $SO(2)_1$ and the WZW $SU(2)_1$ CFT?

Consider the anisotropic spin-$$\frac{1}{2}$$ Heisenberg chain $$H = \sum_{n=1}^N S^x_n S^x_{n+1}+S^y_n S^y_{n+1} + \Delta S^z_n S^z_{n+1}$$ which for $$\Delta = 0$$ realizes the Wess-Zumino-Witten (WZW) $$SO(2)_1$$ conformal field theory (CFT), whereas at the isotropic point $$\Delta = 1$$ we have the WZW $$SU(2)_1$$ CFT. Moreover, the line $$\Delta \in [0,1]$$ is said to be a fixed point line connecting the two critical models.

So on the level of the central charge and scaling dimensions, both models continuously connect to one another, and hence on that level there is no qualitative difference. However, I was just wondering: is there any qualitative difference between them? For example, even though the scaling dimensions are not qualitatively different, I could imagine that (for example) one model has log-like contributions whereas the other doesn't. After all: even though the above model has a line of fixed points connecting both extremes, $$\Delta=1$$ is the end point of such a line of fixed points, and hence one might expect something funny to happen there.

EDIT: rereading this question I posed two years ago, I regret its poor formulation. The question attempts to ask whether the $$\Delta = 1$$ point has any singular behavior on the compact boson line, e.g., are there loglike corrections at this point which are not present for $$|\Delta|<1$$ etc?

They are very different. First of all the free fermion is fermionic. Then $SU(2)_1$ has non-chiral fields coming from the left and right primary fields with $h=1/4$ (spin 1/2), namely the Hilbert space decomposes as $(\mathcal{H}_0\otimes\bar{\mathcal{H}}_0)\oplus (\mathcal{H}_{1/4}\otimes\bar{\mathcal{H}}_{1/4})$.
Well, the names say it all... the symmetries are different. At the $$SU(2)$$ point, with the additional symmetry (for the Heisenberg chain it is an explicit symmetry, in the bosonized theory it is emergent) there are additional current operators. It is at a self-dual point of the T-duality.