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

We suggest the possibility that the two-dimensional $\mathrm{SU}{(2)}_{k}$ Wess-Zumino-Witten (WZW) theory, which has global $\mathrm{SO}(4)$ symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with $\mathrm{SO}(5)$ 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 $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality ${d}_{c}g2$. We suggest that ${d}_{c}l3$ 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 $\mathrm{SU}(2)$ magnets is a ``pseudocritical'' point with approximate $\mathrm{SO}(5)$, controlled by a fixed point slightly outside the physical parameter space.