Embedding AdS black holes in ten and eleven dimensions

Abstract

We construct the non-linear Kaluza-Klein ansätze describing the embeddings of the U (1) 3 , U (1) 4 and U (1) 2 truncations of D = 5, D = 4 and D = 7 gauged supergravities into the type IIB string and M-theory. These enable one to oxidise any associated lower dimensional solutions to D = 10 or D = 11. In particular, we use these general ansätze to embed the charged AdS 5 , AdS 4 and AdS 7 black hole solutions in ten and eleven dimensions.The charges for the black holes with toroidal horizons may be interpreted as the angular momenta of D3-branes, M2-branes and M5-branes spinning in the transverse dimensions, in their near-horizon decoupling limits. The horizons of the black holes coincide with the worldvolumes of the branes. The Kaluza-Klein ansätze also allow the black holes with spherical or hyperbolic horizons to be reinterpreted in D = 10 or D = 11. IntroductionAnti-de Sitter black hole solutions of gauged extended supergravities [1] are currently attracting a good deal of attention [2,3,4,5,6,7,8,9,10,11,12] due, in large part, to the correspondence between anti-de Sitter space and conformal field theories on its boundary [13,14,15,16]. These gauged extended supergravities can arise as the massless modes of various Kaluza-Klein compactifications of both D = 11 and D = 10 supergravities. The three examples studied in the paper will be gauged D = 4, N = 8 SO(8) supergravity [17, 18] arising from D = 11 supergravity on S 7 [19, 20] whose black hole solutions are discussed in [7]; gauged D = 5, N = 8 SO(6) supergravity [21, 22] arising from Type IIB supergravity on S 5 [23, 24, 25] whose black hole solutions are discussed in [2, 6]; and gauged D = 7, N = 4 SO(5) supergravity [21, 26] arising from D = 11 supergravity on S 4 [27]whose black hole solutions are given in section 4.2 and in [9,28]. 1 In the absence of the black holes, these three AdS compactifications are singled out as arising from the near-horizon geometry of the extremal non-rotating M2, D3 and M5 branes [29,30,31,32]. One of our goals will be to embed these known lower-dimensional black hole solutions into ten or eleven dimensions, thus allowing a higher dimensional interpretation in terms of rotating M2, D3 and M5-branes.Since these gauged supergravity theories may be obtained by consistently truncating the massive modes of the full Kaluza-Klein theories, it follows that all solutions of the lower-dimensional theories will also be solutions of the higher-dimensional ones [33,34]. In principle, therefore, once we know the Kaluza-Klein ansatz for the massless sector, it ought to be straightforward to read off the higher dimensional solutions. It practice, however, this is a formidable task. The correct massless ansatz for the S 7 compactification took many years to finalize [35,36], and is still highly implicit, while for the S 5 and S 4 compactifications, the complete massless ansätze are still unknown. For our present purposes, it suffices to consider truncations of the gauged supergravities to include only gauge fields in the Cartan subalgebras ...

Attractors, black objects and holographic RG flows in 5d maximal gauged supergravities

“…The 2 + corresponds to the N = 2 supersymmetric truncation formulated in [44], whose possible solutions were already explored in detail and are of the same type of the abelian solutions in the 4 + theory. Similarly the abelian solutions in the 2 − truncation correspond to those of the 4 − .…”

At the horizon of a supersymmetricAdS5black hole: Isometries and half-BPS giants

“…We will now lift this to a ten-dimensional solution. The lift formula is [25] (see also [26]) 13) where µ 1 = sin α, µ 2 = cos α sin β, µ 3 = cos α cos β with 0 ≤ α ≤ π/2, 0 ≤ β ≤ π/2, 0 ≤ ξ i ≤ 2π and together they parametrise S 5 . Note that we define the Hodge star of a p-form ω in n-dimensions as…”

Black hole mass and Hamilton-Jacobi counterterms

“…In addition to the metric, (2.7), the four-charge black holes have gauge potentials and scalars given by [23]- [25] …”

“…However, it is consistent to consider a subset of solutions, including the electrically charged black holes of present interest. Because of the slightly unusual nature of the truncated theory, the potential has a more complicated structure [31,25] 20) where X 1 and X 2 are unconstrained fields. In terms of canonically normalized scalars, we may take the representation…”