Symmetry protection of critical phases and global anomaly in $1+1$ dimensions

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-$k^{\mathrm{\prime }}$ 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.