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

TL;DR

This paper characterizes the most general first-order symmetry operators for the Dirac equation across all dimensions, linking Killing-Yano forms and Schouten-Nijenhuis brackets, and explores their algebraic properties and implications for separability in complex spacetimes.

Contribution

It introduces a comprehensive classification of Dirac symmetry operators using Killing-Yano and conformal Killing-Yano forms, and establishes a new Killing-Yano bracket related to the Schouten-Nijenhuis bracket.

Findings

Derived the general form of symmetry operators for the Dirac equation.

Established the relation between Killing-Yano brackets and Schouten-Nijenhuis brackets.

Identified a complete set of mutually commuting operators for separability in Kerr-NUT-(A)dS spacetimes.

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].