Fermion on Curved Spaces, Symmetries, and Quantum Anomalies

TL;DR

This paper explores the role of Killing-Yano tensors in the motion of spinning particles and Dirac operators in curved spaces, analyzing symmetries and anomalies, especially in Taub-NUT spaces, with implications for quantum anomalies and index theorems.

Contribution

It connects Killing-Yano tensors to constants of motion and Dirac operators in curved backgrounds, applying these concepts to Taub-NUT spaces and analyzing anomalies using index theory.

Findings

Killing-Yano tensors generate conserved quantities for spinning particles.

Taub-NUT metrics with hidden symmetries do not contribute to axial anomalies.

The Atiyah-Patodi-Singer index theorem confirms anomaly cancellation in these spaces.

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.