9.2 Black Holes Once Again

gauged-supergravity in D = 4 from the bulk point of view and we have instead D2-branes and Ramond strings from the boundary point of view. Recent work on the AdS/CFT duality with N = 6 conformal field theories in three dimensions was indeed centered on this solution of type IIA supergravity.

9.2 Black Holes Once Again

As announced in the introduction the first type of supergravity solutions we consider are the spherical symmetric black holes in D = 4. The motivations and perspective of this choice were explained above. The technique to obtain such solutions consists in the mapping of the supergravity field equations into those of a σ -model on an appropriate target manifold. This technique allowed to establish a complete integration algorithm that provides all solutions and their full-fledged classification [15–18]. We will not dwell on such integration algorithm and confine ourselves to present the oxidation rules from the σ -model to the actual supergravity configurations. We will also present, without derivation, some examples of exact solutions, our goal being the illustration of the attraction mechanism and the emergence of the quartic invariant as codifier of the black hole entropy.

9.2.1 The σ -Model Approach to Spherical Black Holes

A very powerful token in deriving solutions of supergravity that depend only on one effective parameter, like spherical symmetric black-holes depending only on a radial coordinate r or cosmological configurations depending only on time t , is provided by the reduction of the four-dimensional field equations to those of an effective onedimensional σ -model. In this section we shortly review such a procedure for the spherical black hole case.

Let us consider the generic form of the bosonic Lagrangian of an ungauged D = 4 supergravity as given in (8.4.1). Besides the metric field gμν (x), the theory contains ns scalar fields and nv vector fields. The geometric data specifying the Lagrangian and hence all interactions are the metric hrs (φ) of the ns -dimensional scalar manifold Mscalar which, for N > 2 is necessarily a symmetric coset manifold, while for N = 2 is any special Kähler manifold S K n, and the kinetic nv × nv matrix NΛΣ (φ) which, for all coset manifold cases is given by the Gaillard-Zumino master formula (8.3.67), while for the generic special Kähler case admits the definition given in (8.5.30). In all N = 2 cases the number of vector fields in the theory is nv = n + 1 where n is the complex dimension of the scalar manifold (ns = 2n), while in the case of other theories the relation between nv and ns is different. Notwithstanding this difference, we can always introduce a 2nv × 2nv field dependent matrix M4 defined as follows:

and we can introduce the following set of 2 + ns + 2nv fields depending on a parameter τ which later will be interpreted as the inverse of the radial coordinate τ 1/r:

the fields {U, a, φ, Z} are interpreted as the coordinates of a new (2 + ns + 2nv)- dimensional manifold Q, whose metric we declare to be the following:

ds2 = 1 dU 2 + hrs dφr dφs + e−2U da + ZT C dZ 2 + 2e−U dZT M4 dZ

Q 4

(9.2.3)

having denoted by C the constant symplectic invariant metric in 2nv dimensions that underlies the construction of the matrix NΛΣ .

Solutions of the one-dimensional σ -model are just geodesics of the above metric

since the matrix M4 is negative definite. Hence the geodesics can be time-like, nulllike or space-like depending on the three possible cases:

where the dot denotes derivative with respect to the affine parameter τ . Every solution of the Euler Lagrangian equations:

Φ(τ ) ≡ (U, a, φr , ZM )

defines a geodesic and provides a solution of the original supergravity field equations according to an oxidation rule that we will specify few lines below. Spacelike geodesics correspond to unphysical supergravity solutions with naked singularities and are excluded. Time-like geodesics correspond to non-extremal black-holes while null-like geodesics yield extremal black-holes.