Separability of a modified Dirac equation in a five-dimensional rotating, charged black hole in string theory

The aim of this paper is to investigate the separability of a spin-1/2 spinor field in a five-dimensional rotating, charged black hole constructed by Cvetic and Youm in string theory, in the case when three U(1) charges are set equal. This black hole solution represents a natural generalization of the famous four-dimensional Kerr-Newman solution to five dimensions with the inclusion of a Chern-Simons term to the Maxwell equation. It is shown that the usual Dirac equation can not be separated by variables in this general spacetime with two independent angular momenta. However if one supplements an additional counterterm into the usual Dirac operator, then the modified Dirac equation for the spin-1/2 spinor particles is separable in this rotating, charged Einstein-Maxwell-Chern-Simons black hole background geometry. A first-order symmetry operator that commutes with the modified Dirac operator has exactly the same form as that previously found in the uncharged Myers-Perry black hole case. It is expressed in terms of a rank-three totally antisymmetric tensor and its covariant derivative. This tensor obeys a generalized Killing-Yano equation and its square is a second-order symmetric Stackel-Killing tensor admitted by the five-dimensional rotating, charged black hole spacetime.

💡 Research Summary

The paper investigates whether the Dirac equation for a spin‑½ field can be separated into ordinary differential equations in the background of a five‑dimensional rotating, charged black hole discovered by Cvetič and Youm. This solution generalizes the four‑dimensional Kerr‑Newman black hole to five dimensions, featuring two independent rotation parameters and three equal U(1) charges, and it solves the Einstein‑Maxwell equations supplemented by a Chern‑Simons term.

Initially the authors show that the standard Dirac operator (D=\gamma^{\mu}\nabla_{\mu}) fails to separate in this geometry. The failure is traced to the coupling between the spin connection and the electromagnetic field that is modified by the Chern‑Simons contribution; the resulting Dirac equation contains terms that mix the radial and angular coordinates in an inseparable way.

To overcome this obstacle they introduce a counter‑term proportional to the totally antisymmetric three‑form built from the modified field strength. The modified operator is
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