Fermion on Curved Spaces, Symmetries, and Quantum Anomalies

• Main Author: Mihai Visinescu
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2006-11-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: spinning particles; Dirac type operators; gravitational anomalies; axial anomalies
• Online Access: http://www.emis.de/journals/SIGMA/2006/Paper083/

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.