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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 review the status of solitons in superstring theory, with a view to understanding the strong coupling regime. These solitonic solutions are non-singular field configurations which solve the empty-space low-energy field equations (generalized, whenever possible, to all orders in α ′ ), carry a non-vanishing topological "magnetic" charge and are stabilized by a topological conservation law. They are compared and contrasted with the elementary solutions which are singular solutions of the field equations with a σ-model source term and carry a non-vanishing Noether "electric" charge. In both cases, the solutions of most interest are those which preserve half the spacetime supersymmetries and saturate a Bogomol'nyi bound. They typically arise as the extreme mass=charge limit of more general two-parameter solutions with event horizons. We also describe the theory dual to the fundamental string for which the roles of elementary and soliton solutions are interchanged. In ten spacetime dimensions, this dual theory is a superfivebrane and this gives rise to a string/fivebrane duality conjecture according to which the fivebrane may be regarded as fundamental in its own right, with the strongly coupled string corresponding to the weakly coupled fivebrane and vice-versa. After compactification to four spacetime dimensions, the fivebrane appears as a magnetic monopole or a dual string according as it wraps around five or four of the compactified dimensions. This gives rise to a four-dimensional string/string duality conjecture which subsumes a Montonen-Olive type duality in that the magnetic monopoles of the fundamental string correspond to the electric winding states of the dual string. This leads to a duality of dualities whereby under string/string duality the strong/weak coupling S-duality trades places with the minimum/maximum length Tduality. Since these magnetic monopoles are extreme black holes, a prediction of S-duality is that the corresponding electric massive states of the fundamental string are also extreme black holes. This is indeed the case.

In 1973 two Salam protégés (Derek Capper and the author) discovered that the conformal invariance under Weyl rescalings of the metric tensor g µν (x) → Ω 2 (x)g µν (x) displayed by classical massless field systems in interaction with gravity no longer survives in the quantum theory. Since then these Weyl anomalies have found a variety of applications in black hole physics, cosmology, string theory and statistical mechanics. We give a nostalgic review.

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Membrane/fivebrane duality in D = 11 implies Type IIA string/Type IIA fivebrane duality in D = 10, which in turn implies Type IIA string/heterotic string duality in D = 6.To test the conjecture, we reproduce the corrections to the 3-form field equations of the D = 10 Type IIA string (a mixture of tree-level and one-loop effects) starting from the Chern-Simons corrections to the 7-form Bianchi identities of the D = 11 fivebrane (a purely tree-level effect). K3 compactification of the latter then yields the familiar gauge and Lorentz Chern-Simons corrections to 3-form Bianchi identities of the heterotic string. We note that the absence of a dilaton in the D = 11 theory allows us to fix both the gravitational constant and the fivebrane tension in terms of the membrane tension. We also comment on an apparent conflict between fundamental and solitonic heterotic strings and on the puzzle of a fivebrane origin of S-duality.

In six spacetime dimensions, the heterotic string is dual to a Type IIA string. On further toroidal compactification to four spacetime dimensions, the heterotic string acquires an SL(2, Z) S strong/weak coupling duality and an SL(2, Z) T × SL(2, Z) U target space duality acting on the dilaton/axion, complex Kahler form and the complex structure fields S, T, U respectively. Strong/weak duality in D = 6 interchanges the roles of S and T in D = 4 yielding a Type IIA string with fields T, S, U. This suggests the existence of a third string (whose six-dimensional interpretation is more obscure) that interchanges the roles of S and U. It corresponds in fact to a Type IIB string with fields U, T, S leading to a fourdimensional string/string/string triality. Since SL(2, Z) S is perturbative for the Type IIB string, this D = 4 triality implies S-duality for the heterotic string and thus fills a gap left by D = 6 duality. For all three strings the total symmetry is SL(2, Z) S × O(6, 22; Z) T U . The O(6, 22; Z) is perturbative for the heterotic string but contains the conjectured nonperturbative SL(2, Z) X , where X is the complex scalar of the D = 10 Type IIB string. Thus four-dimensional triality also provides a (post-compactification) justification for this conjecture. We interpret the N = 4 Bogomol'nyi spectrum from all three points of view. In particular we generalize the Sen-Schwarz formula for short multiplets to include intermediate multiplets also and discuss the corresponding black hole spectrum both for the N = 4 theory and for a truncated S-T -U symmetric N = 2 theory. Just as the first two strings are described by the four-dimensional elementary and dual solitonic solutions, so the third string is described by the stringy cosmic string solution. In three dimensions all three strings are related by O(8, 24; Z) transformations.

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