  We derive a selection rule among the $(1+1)$-dimensional SU(2)
Wess-Zumino-Witten theories, based on the global anomaly of the discrete
$\mathbb{Z}_2$ symmetry found by Gepner and Witten. In the presence of both the
SU(2) and $\mathbb{Z}_2$ symmetries, a renormalization-group flow is possible
between level-$k$ and level-$k'$ Wess-Zumino-Witten theories only if $k\equiv
k' \mod{2}$. This classifies the Lorentz-invariant, SU(2)-symmetric critical
behavior into two "symmetry-protected" categories corresponding to even and odd
levels,restricting possible gapless critical behavior of translation-invariant
quantum spin chains.
