Symmetry Protection of Critical Phases and a 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 Z_{2} symmetry found by Gepner and Witten. In the presence of both the SU(2) and Z_{2} symmetries, a renormalization-group flow is possible between level-k and level-k^{'} Wess-Zumino-Witten theories only if k≡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.