Black Holes in Supergravity

A brief review is given of the use of duality symmetries to form orbits of supergravity black-hole solutions and their relation to extremal (i.e. BPS) solutions at the limits of such orbits. An important technique in this analysis uses a timelike dimensional reduction and exchanges the stationary black-hole problem for a nonlinear sigma-model problem. Families of BPS solutions are characterized by nilpotent orbits under the duality symmetries, based upon a tri-graded or pentagraded decomposition of the corresponding duality group algebra. Aside from the general mathematical interest in classifying black hole solutions of any kind, the study of families of such solutions is also of current interest because it touches other important issues in theoretical physics. For example, the classification of BPS and non-BPS black holes forms part of a more general study of branes in supergravity and superstring theory. Branes and their intersections, as well as their worldvolume modes and attached string modes, are key elements in phenomenological approaches to the marriage of string theory with particle physics phenomenology. The related study of nonsingular and horizon-free BPS gravitational solitons is also central to the “fuzzball” proposal of BPS solutions as candidate black-hole quantum microstates. The search for supergravity solutions with assumed Killing symmetries can be recast as a Kaluza-Klein problem [1, 2, 3]. To see this, consider a 4D theory with a nonlinear bosonic symmetry G4 (e.g. the “duality” symmetry E7 for maximal N = 8 supergravity). Scalar fields take their values in a target space Φ4 = G4/H4, where H4 is the corresponding linearly realized subgroup, generally the maximal compact subgroup of G4 (e.g. SU(8) ⊂ E7 for N = 8 SG). The search will be constrained by the following considerations: • We assume that a solution spacetime is asymptotically flat or asymptotically Taub-NUT and that there is a ‘radial’ function r which is divergent in the asymptotic region, g∂μr∂νr ∼ 1 +O(r−1). • Searching for stationary solutions amounts to assuming that a solution possesses a timelike Killing vector field κμ(x). Lie derivatives with respect to κμ are assumed to vanish on all fields. The Killing vector κμ will be assumed to have email: [email protected] W := −gμνκκ ∼ 1 +O(r−1). • We also assume asymptotic hypersurface orthogonality, i.e.κ(∂μκν − ∂νκμ) ∼ O(r−2). In any vielbein frame, the curvature will then fall off as Rabcd ∼ O(r−3). The 3D theory obtained after dimensional reduction with respect to a timelike Killing vector κμ will have an Abelian principal bundle structure, with a metric ds = −W (dt+Bidx) +Wγijdxdx (1) where t is a coordinate adapted to the timelike Killing vector κμ and γij is the metric on the 3-dimensional hypersurface M3 at constant t. If the 4D theory also has Abelian vector fields Aμ, they similarly reduce to 3D as 4 √ 4πGAμdx = U(dt+Bidx) +Aidx (2) The timelike reduced 3D theory will have a G/H∗ coset space structure similar to the G/H coset space structure of a 3D theory reduced with a spacelike Killing vector. Thus, for the spacelike reduction of maximal supergravity down to 3D, one obtains an E8/SO(16) theory from the sequence of dimensional reductions descending fromD = 11 [4]. The resulting 3D theory has this exceptional symmetry because 3D Abelian vector fields can be dualized to scalars; this also happens for the analogous theory subjected to a timelike reduction to 3D. The resulting 3D theory contains 3D gravity coupled to a G/H∗ nonlinear sigma model. Although the numerator group G for a timelike reduction is the same as that obtained in a spacelike reduction, the divisor group H∗ for a timelike reduction is a noncompact form of the spacelike divisor group H [2]. A consequence of this H → H∗ change and the dualization of vectors is the appearance of negative-sign kinetic terms for some 3D scalars. Consequently, maximal supergravity, after a timelike reduction to 3D and the subsequent dualization of 29 vectors to scalars, has a bosonic sector containing 3D gravity coupled to a E8/SO ∗(16) nonlinear sigma model with 128 scalar fields. As a consequence of the timelike dimensional reduction and vector dualizations, however, the scalars do not all have the same signs for their “kinetic” terms: • There are 72 positive-sign scalars: 70 descending directly from the 4D theory, one emerging from the 4D metric and one more coming from the D = 4→ D = 3 Kaluza-Klein vector, subsequently dualized to a scalar. • There are 56 negative-sign scalars: 28 descending directly from the time components of the 28 4D vectors, and another 28 emerging from the 3D vectors obtained from spatial components of the 28 4D vectors, becoming then negative-sign scalars after dualization. The sigma-model structure of this timelike reduced maximal theory is E8/SO ∗(16). The SO∗(16) divisor group is not an SO(p, q) group defined via preservation of an indefinite metric. Instead it is constructed starting from the SO(16) Clifford algebra {ΓI ,ΓJ} = 2δIJ and then by forming the complex U(8)-covariant oscillators ai := 1 2(Γ2i−1 + iΓ2i) and a i ≡ (ai) = 1 2(Γ2i−1 − iΓ2i). These satisfy the standard fermi oscillator annihilation/creation anticommutation relations {ai, aj} = {a, a} = 0 , {ai, a} = δi j (3)