Chiral Anomalies in Field Theories

The role of the contribution from the fermion mass term in the axial vector Ward identity in generating the U(1) axial anomaly, both local and global, is elucidated. Gauge invariance requires the fermion to decouple from the gauge field if it is very heavy. This identifies the Adler-Bell-Jackiw (ABJ) anomaly with the asymptotic limit of the sign reversed mass term. In an instanton background, the chiral limit (m = 0) of the mass term does not vanish but consists of contributions from fermion zero modes. Space time integral of these zero mode contributions exactly cancels, thanks to the Atiyah-Singer index theorem, the integral of the ABJ anomaly and suggests that the Jacobian for global U(1) chiral transformation is trivial even in an instanton background. This can be realised in the representation of the fermion partition function in a Weyl basis. The resolution of the strong CP problem is thus achieved in an axionless physical world. In chiral gauge theories the fermion partition function admits of a gauge invariant representation but only at the cost of locality. Implementation of fermion averaging of the gauge current with the invariant partition function yields the current whose covariant derivative is the covariant anomaly. With the covariant current as input one can derive an integrable current whose covariant derivative is the minimal consistent anomaly obeying the Wess-Zumino consistency condition. The distinction between the two currents disappears if either the covariant or the consistent anomaly vanishes. This is realised only if the fermion belongs to an anomaly-free representation of the gauge group. Invited article for a book to be published by the Indian National Science Academy on the occasion of the International Mathematical Year 2000. 1