Higher-dimensional lifts of Killing–Yano forms with torsion

Abstract

Using a Kaluza-Klein-type lift, it is shown how Killing-Yano forms with torsion can remain symmetries of a higher-dimensional geometry, subject to an algebraic condition between the Kaluza-Klein field strength and the Killing-Yano form. The lift condition's significance is highlighted, and is satisfied by examples of black holes in supergravity.Killing-Yano (KY) p-forms [1] generalize Killing vectors to higher-rank antisymmetric tensors(1.1)The Kerr-Newman black hole spacetime admits a KY 2-form [2,3], which underlies many of the solution's remarkable properties. Higher-dimensional generalizations in Einstein gravity possess additional KY p-forms and retain many of the remarkable properties of the 4dimensional Kerr spacetime; see [4,5] for reviews. Like Killing vectors, KY forms give symmetries. However, they are "hidden symmetries" of phase space, rather than configuration space; see [6] for a review. Some particular consequences of KY forms are: constants of motion for charged particle motion [7]; the existence of an operator commuting with the Dirac operator [8]; and enhanced worldline supersymmetry of a spinning particle [9]. There are generalizations of the Kerr-Newman black hole that are charged, rotating black hole solutions of supergravities. These theories contain the 3-form Kalb-Ramond field strength H of string theory. Some of the known solutions admit generalizations of KY forms in which the connection is modified to include a torsion [10], identified with the 3-form H, for example in [11,12,13,14]. Classifying supersymmetric solutions of supergravity leads to G-structures, on which KY forms with torsion also appear naturally [15,16]. Separately, KY 3-forms arise in 11-dimensional supergravity reduced on S 7 ; for squashed S 7 a KY 3-form constructed from a Killing spinor gives a Ricci-flattening torsion [17], while the 70 KY 3-forms of round S 7 are associated with 35 massless and 35 massive pseudoscalars [18,19]. See [20] for a review of KY forms in supersymmetric theories and G-structures.One reason for studying black holes in supergravity, rather than more general theories involving gravity, is that it seems to be easier to find exact solutions that are charged, rotating black holes. A charged, rotating exact black hole solution of higher-dimensional Einstein-Maxwell theory is not known, whereas examples are known explicitly in supergravity. Although there are solution generating techniques arising from string theory dualities, these do not explain the construction of asymptotically anti-de Sitter (AdS) black holes in gauged supergravity. However, behind all of these known solutions are Killing tensors. In general, these are symmetric Killing-Stäckel tensors rather than antisymmetric KY forms with torsion; see e.g. [21]. Killing tensors may provide guidance for finding new solutions. Whereas a general charged, rotating black hole in 5-dimensional minimal gauged supergravity was originally found using inspired guesswork [22], it can be derived assuming a certain type of Killing tensor [23]. Some ge...

Black holes, hidden symmetries, and complete integrability

“…A subfamily of these metrics provided a new class of Calabi-Yau with torsion metrics (Houri et al 2012), see also Houri et al (2013) for the generalized Sasaki-Einstein metrics, and Hinoue et al (2014) for a generalization of the Wahlquist metric. Further developments on the GKY tensors can be found in Chow (2015Chow ( , 2016. We also refer to the wonderful review on applications of Killing-Yano tensors to string theory by Chervonyi and Lunin (2015).…”

Rational first integrals of geodesic equations and generalised hidden symmetries

“…The authors replaced the Levi-Civita connection in the Killing-Yano equation by connections with totally skew-symmetric torsion ‡. Recently, the generalised (conformal) Killing-Yano tensors have been studied well in the context of string theories [24,25,26,27,28,29]. On the other hand, such a generalisation does not affect the Killing equation.…”