Anomalies and fermion zero modes on strings and domain walls

We show that the mathematical relation between non-abelian anomalies in 2n dimensions, the parity anomaly in 2n+1 dimensions, and the Dirac index density in 2n+2 dimensions can be understood in terms of the physics of fermion zero modes on strings and domain walls. We show that the Dirac equation possesses chiral zero modes in the presence of strings in 2n+2 dimensions (such as occur in axion theories) or domain walls in 2n+1 dimensions. We show that the anomalies due to the chiral zero modes are exactly cancelled by anomalies due to the coupling of axion-like fields to the Dirac index density or by anomalies due to the induced topological mass term.