Generalized Killing–Yano equations inD=5gauged supergravity

Abstract

We propose a generalization of the (conformal) Killing-Yano equations relevant to D = 5 minimal gauged supergravity. The generalization stems from the fact that the dual of the Maxwell flux, the 3-form * F , couples naturally to particles in the background as a 'torsion'. Killing-Yano tensors in the presence of torsion preserve most of the properties of the standard Killing-Yano tensors-exploited recently for the higher-dimensional rotating black holes of vacuum gravity with cosmological constant. In particular, the generalized closed conformal Killing-Yano 2-form gives rise to the tower of generalized closed conformal Killing-Yano tensors of increasing rank which in turn generate the tower of Killing tensors. An example of a generalized Killing-Yano tensor is found for the Chong-Cvetić-Lü-Pope black hole spacetime [hepth/0506029]. Such a tensor stands behind the separability of the Hamilton-Jacobi, Klein-Gordon, and Dirac equations in this background.

Maxwell’s equations in the Myers-Perry geometry

“…For the Schwarzschild-Tangherlini geometry [35], separation of the Klein-Gordon and Maxwell equations is rather straightforward, 2 but the study of gravitational waves in such backgrounds is an active area of research [36][37][38][39][40][41]. In contrast, the vast majority of efforts in studying rotating geometries has been dedicated to scalar and spinor fields [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] with a few notable exceptions [57][58][59][60]. Unfortunately, in this case, the full description of the Maxwell's equations, let alone gravitational waves, has been missing, and the goal of the present article is to close this gap in the literature on electromagnetic waves.…”

“…Such symmetries encoded in the Killing-(Yano) tensors have been explored in the past, both in higher-dimensional general relativity [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] and in string theory [61][62][63][64][65][66][67][68][69][70][71]. This paper will connect the properties of such tensors, in particular, their eigenvectors, to separation of the Maxwell's equations in an arbitrary number of dimensions.…”

“…While most of this article focuses on electromagnetic waves in the background of the Myers-Perry black hole [72], the extensions of our results to the GLPP geometry [73], which generalizes MP solution to non-zero value of the cosmological constant, is straightforward, and such extensions are presented in the appropriate sections. The scalar and spinor excitations of the GLPP black holes have been subjects of intensive studies [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56], 3 and the results are summarized in a very nice recent review [78]. Under very mild assumptions, the GLPP solution with added NUT charge is the most general geometry admitting separation of variables in the wave and Dirac equations [79][80][81][82], and the same uniqueness property is expected to hold for the Maxwell's equations discussed in this article.…”

“…While it is possible to just guess the appropriate vielbein, a more constructive approach is based on characterizing the frames (2.6) by their algebraic properties and finding the generalization by solving appropriate equations. Such approach to special vielbeins was developed in [71] based on earlier work [42][43][44][46][47][48][49][50][51][52][53][54][55][56], and in this section we will review the appropriate construction. Specifically, we introduce the geometry of the Myers-Perry black hole [72], discuss it symmetries encoded in Killing-Yano tensors, and demonstrate that the higher dimensional counterparts of the vierbein (2.6) are uniquely determined by solving equations for such tensors.…”

“…As we will see, this crucial fact is responsible for separation of variables in Klein-Gordon and Maxwell equations. 17 An alternative approach, applicable only to neutral black holes, was introduced earlier in [42][43][44][46][47][48][49][50][51][52][53][54][55][56]. This work is summarized in a very nice recent review [78].…”