C.N. Pope

Embedding AdS black holes in ten and eleven dimensions

Abstract We construct the non-linear Kaluza-Klein ansatze 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 ansatze 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 world-volumes of the branes. The Kaluza-Klein ansatze also allow the black holes with spherical or hyperbolic horizons to be reinterpreted in D = 10 or D = 11.

The general Kerr–de Sitter metrics in all dimensions

We give the general Kerr–de Sitter metric in arbitrary space–time dimension D≥4, with the maximal number [(D−1)/2] of independent rotation parameters. We obtain the metric in Kerr–Schild form, where it is written as the sum of a de Sitter metric plus the square of a null-geodesic vector, and in generalised Boyer–Lindquist coordinates. The Kerr–Schild form is simpler for verifying that the Einstein equations are satisfied, and we have explicitly checked our results for all dimensions D≤11. We discuss the global structure of the metrics, and obtain formulae for the surface gravities and areas of the event horizons. We also obtain the Euclidean-signature solutions, and we construct complete non-singular compact Einstein spaces on associated SD−2 bundles over S2, infinitely many for each odd D≥5.

Dualisation of dualities

We analyse the global (rigid) symmetries that are realised on the bosonic fields of the various supergravity actions obtained from eleven-dimensional supergravity by toroidal compactification followed by the dualisation of some subset of fields. In particular, we show how the global symmetries of the action can be affected by the choice of this subset. This phenomenon occurs even with the global symmetries of the equations of motion. A striking regularity is exhibited by the series of theories obtained respectively without any dualisation, with the dualisation of only the Ramond-Ramond fields of the type IIA theory, with full dualisation to lowest degree forms, and finally for certain inverse dualisations (increasing the degrees of some forms) to give the type IIB series. These theories may be called the GLA, D, E and GLB series, respectively. It turns out that the scalar Lagrangians of the E series are sigma models on the symmetric spaces K(E11−D)⧹E11t-D (where K(G) is the maximal compact subgroup of G) and the other three series lead to models on homogeneous spaces K(G) ⧹G⋉Rs. These can be understood from the E series in terms of the deletion of positive roots associated with the dualised scalars, which implies a group contraction. We also propose a constrained Lagrangian version of the even-dimensional theories exhibiting the full duality symmetry and begin a systematic analysis of abelian subalgebras.

The first law of thermodynamics for Kerr–anti-de Sitter black holes

We obtain expressions for the mass and angular momenta of rotating black holes in anti-de Sitter backgrounds in four, five and higher dimensions. We verify explicitly that our expressions satisfy the first law of thermodynamics, thus allowing an unambiguous identification of the entropy of these black holes with of the area. We find that the associated thermodynamic potential equals the background-subtracted Euclidean action multiplied by the temperature. Our expressions differ from many given in the literature. We find that in more than four dimensions, only our expressions satisfy the first law of thermodynamics. Moreover, in all dimensions we show that our expression for the mass coincides with that given by the conformal conserved charge introduced by Ashtekar, Magnon and Das. We indicate the relevance of these results to the AdS/CFT correspondence.

W∞ and the Racah-Wigner algebra

We examine the structure of a recently constructed W∞ algebra, an extension of the Virasoro algebra that describes an infinite number of fields with all conformal spins 2,3,…, with central terms for all spins. By examining its underlying SL(2, R) structure, we are able to exhibit its relation to the algebras of SL(2, R) tensor operators. Based upon this relationship, we generalise W∞ to a one-parameter family of inequivalent Lie algebras W∞(μ), which for general μ requires the introduction of formally negative spins. Furthermore, we display a realisation of the W∞(μ) commutation relations in terms of an underlying associative product, which we denote with a lone star. This product structure shares many formal features with the Racah-Wigner algebra in angular-momentum theory. We also discuss the relation between W∞ and the symplectic algebra on a cone, which can be viewed as a co-adjoint orbit of SL(2, R).

Hopf Fibration of Eleven-dimensional Supergravity

The authors show that all known solutions of the N=2 non-chiral d=10 supergravity theory can, by a simple procedure, be obtained from the known Freund-Rubin type solutions of D=11 supergravity. This is a consequence of the fact that each solution of the d=10 theory depends crucially on the presence of a topologically non-trivial U(1) field strength Fmn. The mass spectrum of a particular d=10 solution is obtained by truncating the spectrum of the corresponding d=11 solution to the subset of fields neutral under the non-trivial U(1) symmetry. They investigate the compactification on CP3, and show that this yields a theory with N=6 or N=0 supersymmetry according to the orientation, and they relate this to the S7 compactification of d=11 supergravity. Starting from the squashed S7 in d=11 they derive a new solution of the d=10 theory, namely (AdS)4 times CP3 with a squashed non-Einstein metric. The supersymmetry is either N=1 or N=0 depending on the orientation. They conjecture that none of the supergravity theories in 4<d<10 dimensions obtained from d=11 by dimensional reduction admit non-trivial compactifications to four dimensions.

A construction of Killing spinors on Sn

We derive simple general expressions for the explicit Killing spinors on the n-sphere, for arbitrary n. Using these results we also construct the Killing spinors on various AdS×Sphere supergravity backgrounds, including AdS5×S5, AdS4×S7, and AdS7×S4. In addition, we extend previous results to obtain the Killing spinors on the hyperbolic spaces Hn.

Dualisation of dualities II: twisted self-duality of doubled fields and superdualities

We introduce a doubled formalism for the bosonic sector of the maximal supergravities, in which a Hodge dual potential is introduced for each bosonic field (except for the metric). The equations of motion can then be formulated as a twisted self-duality condition on the total field strength G, which takes its values in a Lie superalgebra. This doubling is invariant under dualisations; it allows a unification of the gauge symmetries of all degrees, including the usual U-dualities that have degree zero. These “superdualities” encompass the dualities for all choices of polarisation (i.e. the choices between fields and their duals). All gauge symmetries appear as subgroups of finite-dimensional supergroups, with Grassmann coefficients in the differential algebra of the spacetime manifold.

Kerr-AdS/CFT correspondence in diverse dimensions

It was proposed recently that the near-horizon states of an extremal four-dimensional Kerr black hole could be identified with a certain chiral conformal field theory whose Virasoro algebra arises as an asymptotic symmetry algebra of the near-horizon Kerr geometry. Supportive evidence for the proposed duality came from the equality of the microscopic entropy of the CFT, calculated by means of the Cardy formula, and the Bekenstein-Hawking entropy of the extremal Kerr black hole. In this paper we examine the proposed Kerr/CFT correspondence in a broader context. In particular, we show that the microscopic entropy and the Bekenstein-Hawking entropy agree also for the extremal Kerr-AdS metric in four dimensions, and also for the extremal Kerr-AdS metrics in dimensions 5, 6 and 7. General formulae for all higher dimensions are also presented.

T-duality in the Green–Schwarz formalism, and the massless/massive IIA duality map

We derive a component-field expansion of the Green-Schwarz action for the type IIA string, in an arbitrary background of massless NS-NS and R-R bosonic fields, up to quadratic order in the fermionic coordinates θ. Using this action, we extend the usual derivation of Buscher T-duality rules to include not only NS-NS, but also R-R fields. Our implementation of the T-duality transformation rules makes use of adapted background-field parametrizations, which provide a more geometrically natural and elegant description for the duality maps than the ones previously presented. These T-duality rules allow us to derive the Green-Schwarz action for the type IIB string in an arbitrary background of massless NS-NS and R-R bosonic fields, up to O(θ 2 ). Implemention of another T-duality transformation on this type IIB action then allows us also to derive the Green-Schwarz action for the massive IIA string. By further considering T-duality transformations for backgrounds with the two U(1) isometries of a 2-torus, we give a string-theoretic derivation of the direct T-duality relation between the massless and massive type IIA strings. In addition, we give an explicit construction of the D = 8 SL(3,IR) × SL(2,IR) invariant supergravity with two mass parameters that form a doublet under the SL(2,IR) factor.