Fermion on Curved Spaces, Symmetries, and Quantum Anomalies

We review the geodesic motion of pseudo-classical spinning particles in curved spaces. Investigating the generalized Killing equations for spinning spaces, we express the constants of motion in terms of Killing–Yano tensors. Passing from the spinning spaces to the Dirac equation in curved backgrounds we point out the role of the Killing–Yano tensors in the construction of the Dirac-type operators. The general results are applied to the case of the four-dimensional Euclidean Taub–Newman–Unti–Tamburino space. The gravitational and axial anomalies are studied for generalized Euclidean Taub-NUT metrics which admit hidden symmetries analogous to the Runge–Lenz vector of the Kepler-type problem. Using the Atiyah–Patodi–Singer index theorem for manifolds with boundaries, it is shown that the these metrics make no contribution to the axial anomaly.