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

TL;DR

This paper demonstrates that a modified Dirac equation becomes separable in a five-dimensional rotating, charged black hole background when an additional counterterm is included, revealing hidden symmetries related to Killing-Yano tensors.

Contribution

It introduces a modified Dirac operator that allows variable separation in a complex five-dimensional black hole spacetime, extending previous results to charged, rotating solutions in string theory.

Findings

Modified Dirac equation is separable with an added counterterm.

Existence of a symmetry operator related to a generalized Killing-Yano tensor.

Identification of a second-order Stackel-Killing tensor in the spacetime.

Abstract

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.