Note on Wess-Zumino-Witten models and quasiuniversality in 2+1 dimensions

We suggest the possibility that the two-dimensional SU(2)$_k$ Wess-Zumino-Witten (WZW) theory, which has global SO(4) symmetry, can be continued to $2+\epsilon$ dimensions by enlarging the symmetry to SO$(4+\epsilon)$. This is motivated by the three-dimensional sigma model with SO(5) symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\epsilon$ may be fixed by assuming analyticity in $\epsilon$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality $d_c>2$. We suggest that $d_c < 3$ for all $k$, and we compute $d_c$ in the limit of large $k$. The flows support the conjecture that the deconfined phase transition in SU(2) magnets is a ``pseudocritical'' point with approximate SO(5), controlled by a fixed point slightly outside the physical parameter space.