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 ...

We make a detailed investigation of all spaces Q q 1 ···q N n 1 ···n N of the form of U (1) bundles over arbitrary products i CP n i of complex projective spaces, with arbitrary winding numbers q i over each factor in the base. Special cases, including Q 11 11 (sometimes known as T 11 ), Q 111 111 and Q 32 21 , are relevant for compactifications of type IIB and D = 11 supergravity. Remarkable "conspiracies" allow consistent Kaluza-Klein S 5 , S 4 and S 7 sphere reductions of these theories that retain all the Yang-Mills fields of the isometry group in a massless truncation. We prove that such conspiracies do not occur for the reductions on the Q q 1 ···q N n 1 ···n N spaces, and that it is inconsistent to make a massless truncation in which the non-abelian SU (n i + 1) factors in their isometry groups are retained. In the course of proving this we derive many properties of the spaces Q q 1 ···q N n 1 ···n N of more general utility. In particular, we show that they always admit Einstein metrics, and that the spaces where q i = (n i + 1)/ℓ all admit two Killing spinors. We also obtain an iterative construction for real metrics on CP n , and construct the Killing vectors on Q q 1 ···q N n 1 ···n N in terms of scalar eigenfunctions on CP n i .We derive bounds that allow us to prove that certain Killing-vector identities on spheres, necessary for consistent Kaluza-Klein reductions, are never satisfied on Q q 1 ···q N n 1 ···n N .

A recent paper on the large-scale structure of the Universe presented evidence for a rectangular three-dimensional lattice of galaxy superclusters and voids, with lattice spacing ∼ 120 M pc, and called for some "hitherto unknown process" to explain it. Here we report that a rectangular three-dimensional lattice of intersecting domain walls, with arbitrary spacing, emerges naturally as a classical solution of M-theory.