A note on Wess-Zumino terms and discrete symmetries

Sigma models in which the integer coefficient of the Wess-Zumino term vanishes are easy to construct. This is the case if all flavor symmetries are vectorlike. We show that there is a subset of SU(N)×SU(N) vectorlike sigma models in which the Wess-Zumino term vanishes for reasons of symmetry as well. However, there is no chiral sigma model in which the Wess-Zumino term vanishes for reasons of symmetry. This can be understood in the sigma model basis as a consequence of an index theorem for the axialvector coupling matrix. We prove this index theorem directly from the SU(N)×SU(N) algebra.