This chapter is devoted to the topological and cohomological properties of abelian and non-abelian chiral anomalies in Yang-Mills theories.

First, the Gribov ambiguity and the appearance of anomalies are related to the non-trivial topology of the configuration or Yang-Mills orbit space. This is followed by the explicit path integral calculation of the abelian chiral anomaly in D = 2p dimensions and the non-abelian gauge anomalies (for D = 2) by using Fujikawa's method. It is seen how these results may be interpreted in terms of suitable index theorems on spaces of adequate dimensions (D = 2p and (D + 2) respectively). The consistency conditions for the anomalies and the Schwinger terms are interpreted in terms of a cohomology in which the cocycles are valued in ℱ[A], the space of functionals of the gauge fields. Then it is shown how the cohomological descent procedure starting from the Chern character forms provides a method for obtaining non-trivial candidates for both the non-abelian anomalies and the Schwinger terms.

The question of the ambiguity of the cohomological descent procedure, which gives rise to different (but cohomologous) expressions for the Schwinger terms, the BRST formulation of the gauge cohomology and the Wess-Zumino-Witten terms are also discussed. At the end, some comments on the possible consistency of anomalous gauge theories are made.

The group of gauge transformations and the orbit space of Yang-Mills potentials

The requirement that field theories invariant under rigid transformations (gi ≠ gi(x)) of a group G should also be invariant under local (gi = gi(x)) transformations constitutes the gauge invariance principle.