Fermion on curved spaces, symmetries, and quantum anomalies

DOI10.3842/SIGMA.2006.083zbMATH Open1188.83046arXivhep-th/0609207OpenAlexW2234126987MaRDI QIDQ2472012

Publication date: 19 February 2008

Published in: SIGMA. Symmetry, Integrability and Geometry: Methods and Applications (Search for Journal in Brave)

Abstract: 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.

Full work available at URL: https://arxiv.org/abs/hep-th/0609207

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zbMATH Keywords

gravitational anomalies Dirac type operators spinning particles axial anomalies

Mathematics Subject Classification ID

Quantum field theory on curved space or space-time backgrounds (81T20) Anomalies in quantum field theory (81T50) Index theory and related fixed-point theorems on manifolds (58J20) Methods of quantum field theory in general relativity and gravitational theory (83C47)