Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets

Abstract

In this paper we derive the most general first-order symmetry operator commuting with the Dirac operator in all dimensions and signatures. Such an operator splits into Clifford even and Clifford odd parts which are given in terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous forms respectively. We study commutators of these symmetry operators and give necessary and sufficient conditions under which they remain of the first-order. In this specific setting we can introduce a Killing-Yano bracket, a bilinear operation acting on odd Killing-Yano and even closed conformal Killing-Yano forms, and demonstrate that it is closely related to the Schouten-Nijenhuis bracket. An important non-trivial example of vanishing Killing-Yano brackets is given by Dirac symmetry operators generated from the principal conformal Killing-Yano tensor [hep-th/0612029]. We show that among these operators one can find a complete subset of mutually commuting operators. These operators underlie separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all dimensions [arXiv:0711.0078].

37 pages, no figures

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• Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux
• Geometry of Lax pairs: particle motion and Killing-Yano tensors
• Excitations of the Myers-Perry Black Holes
• Gauged twistor spinors and symmetry operators
• Electron in higher-dimensional weakly charged rotating black hole spacetimes
• Covariant Approach of the Dynamics of Particles in External Gauge Fields, Killing Tensors and Quantum Gravitational Anomalies
• Second order symmetry operators for the massive Dirac equation
• First Order Symmetry Operators for the Linearized Field Equation of Metric Perturbations
• Hidden Symmetries of the Dirac equation in curved spacetime
• Separation of variables in the WZW models
• On First Order Symmetry Operators for the Field Equations of Differential Forms
• Algebra structure of conformal Killing-Yano forms in geometries with skew-symmetric torsion
• Higher-dimensional black holes with multiple equal rotations