Sigma models in which the integer coefficient of the Wess-Zumino term vanis
hes are easy to construct. This is the case if all flavor symmetries are ve
ctorlike. We show that there is a subset of SU(N) x SU(N) vectorlike sigma
models in which the Wess-Zumino term vanishes for reasons of symmetry as we
ll. However, there is no chiral sigma model in which the Wess-Zumino term v
anishes for reasons of symmetry. This can be understood in the sigma model
basis as a consequence of an index theorem for the axialvector coupling mat
rix. We prove this index theorem directly from the SU(N) x SU(N) algebra, (
C) 1998 Elsevier Science B.V. All rights reserved.