  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)XSU(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)XSU(N) algebra.
