Chiral anomaly and the Wess-Zumino condition

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Abstract
The connection between the Wess-Zumino condition and the chiral anomaly is discussed in the path-integral framework. When the non-Abelian anomaly exists, the Dirac operator detDLSL can be specified in various ways. The different specifications of detDLSL in the path integral lead to different forms of Ward-Takahashi identities and to different definitions of composite current operators involved. The Wess-Zumino condition imposes the generalized Bose symmetry among the gauge vertices, and thus it introduces a particular definition of current operators. For example, the axial-vector hypercharge current of an SU(3) theory, which is constructed to be consistent with the Wess-Zumino condition and global chiral symmetry, may be reduced to a U(1) current of the SU(2) subtheory; this particular construction of the U(1) current, however, does not satisfy the Atiyah-Singer index theorem expected for the SU(2) theory. We also discuss a semiperturbative regularization which automatically gives the non-Abelian anomaly for chiral fermions in the presence of electroweak fields.