Nuclear Physics B250 (1985) 427-436 © North-Holland Publishing Company ANOMALIES AND FERMION ZERO MODES ON STRINGS AND DOMAIN WALLS C.G. CALLAN, Jr. l and J. A. HARVEY 2 Jadwin Hall, Princeton University, Princeton, New Jersey 08544, USA Received 10 September 1984 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. Anomalies arise when the quantum-mechanical vacuum functional of a field theory fails to have all the symmetries of the classical lagrangian from which it is derived. Ifthe symmetry in question is a gauge symmetry associated with a dynamical gauge field, the anomaly is intolerable and must be eliminated. The usual method is direct cancellation by the addition of extra fermion species with appropriate quantum numbers. In this paper, we will show that in certain contexts involving defects (i.e. strings and domain walls), anomalies arise which may be compensated by quantum number flows in the higher-dimensional space in which the defect is embedded, rather than cancelled. This phenomenon gives a particularly simple physical interpretation of the recently discovered mathematical connection between chiral anomalies in 2n +2 dimensions, parity anomalies in 2n + l dimensions and gravitational and non-abelian anomalies in 2n dimensions [1-5]. It also suggests some interesting phenomenological possibilities for the axion strings which may have played a role in the development of density perturbations in the early universe [6]. Our first step will be to show that the Dirac equation for a fermion coupled to a scalar field with either a string or a domain wall structure has chiral zero modes*. We first consider a Dirac fermion in 2n +2 dimensions in the field of an axion string. Fermion zero modes in the presence of gauge strings have been analyzed previously in refs. [7-10]. Their existence is guaranteed by an index theorem analyzed Supported in part by DOE grant no. DE-AC02-76ERR03072 2 Supported in part by NSF grant no. PHY80-19754 " This was noted independently by R. Rhom for the case of axion strings. 427