In Chapter 15 we have seen how scalar fields coupled to gravity arise naturally in KK compactification. In Part III we are going to see that scalar fields are also present, even before compactification, in some higher-dimensional supergravity theories that are the low-energy effective field theories of certain superstring theories. In all these examples the scalar fields couple in a characteristic way to vector (or p-form in higher dimensions) field strengths. In this chapter we are going to study first, in Section 16.1, a simple model that synthesizes the main features of those theories.

The a-model describes a real scalar coupled to gravity and to a vector field strength. The coupling is exponential and depends on a parameter a (hence the name “a-model” that we are giving it here). Since the scalar can be identified in some cases with the string dilaton (or with the KK scalar, which is also called the dilaton sometimes), these models are also generically referred to as dilaton gravity. We will be able to obtain BH-type solutions for general values of a and in any dimension d ≥ 4; however, only a handful of values of a actually occur in the theories of interest, although they occur in many different ways (embeddings [854]).

After studying the main properties of these dilaton BHs, we are going to study in Section 26.1 a more complex (four-dimensional) model that involves several scalar and vector fields. We are going to obtain extreme BH solutions that can be understood as composite BHs. This interpretation will open the door to the construction of four-dimensional extreme BH solutions in string theory as composite objects, the building blocks being p-branes and other extended objects that we will study in Chapter 26.

In Section 16.2 we add to the a-model with a = 1 and d = 4 a second scalar that couples not to the vector field kinetic term F2 but to F ⋆ F and also couples to the dilaton.

Kaluza's [848] and Nordström's [994] original idea/observation that electromagnetism could be seen as part of five-dimensional gravity, combined with Klein's curling up of the fifth dimension in a tiny circle [862], constitutes one of the most fascinating and recurring themes of modern physics. Kaluza–Klein (KK) theories are interesting both in their own right (in spite of their failure to produce realistic four-dimensional theories [1274], at least when the internal space is a manifold) and because of the usefulness of the techniques of dimensional reduction for treating problems in which the dynamics in one or several directions is irrelevant. We saw an example in Chapter 9, when we related four-dimensional instantons to monopoles.

On the other hand, the effective field theories of some superstring theories (which are supergravity theories) can be obtained by dimensional reduction of 11-dimensional supergravity, which is the low-energy effective field theory of (there is no real consensus on this point) M theory or one of its dual versions. In turn, string theory needs to be “compactified” to take a four-dimensional form and, to obtain the four-dimensional low-energy effective actions, one can apply the dimensional-reduction techniques.

Here we want to give a simple overview of the physics of compact dimensions and the techniques used to deal with them (dimensional reduction, etc.) in a non-stringy context. We will deal only with the compactification of pure gravity and vector fields, leaving aside compactification in the presence of more general matter fields (including fermions) until Part III. We will also leave aside many subjects such as spontaneous compactification and the issue of constructing realistic KK theories, which are covered elsewhere [476, 1271]. In addition to establishing the basic results, we want to study classical solutions of the original and dimensionally reduced theories and to see how KK techniques can be used to generate new solutions of both of them.

In the preceding chapter we obtained and studied the Schwarzschild solution of the vacuum Einstein equations and arrived at the BH concept. However, many of the general features of BHs that we discussed, such as the no-hair conjecture, make reference to BHs in the presence of matter fields. In this chapter we are going to initiate the study and construction of BH solutions of the Einstein equations in the presence of matter fields, starting with the simplest ones: massless scalar and vector fields.

The (unsuccessful) search for BH solutions of gravity coupled to a scalar field will allow us to deepen our understanding of the no-hair conjecture.

The (successful) search for BH solutions of gravity coupled to a vector field will allow us to find the simplest BH solution different from the Schwarzschild solution: the Reissner– Nordström (RN) solution. Simple as it is, it has very interesting features, in particular the existence of an extreme limit with a regular horizon and zero Hawking temperature that will be approached with positive specific heat, as in standard thermodynamical systems. Later on we will relate some of these properties to the unbroken supersymmetry of the extreme RN (ERN) solution, which will allow us to reinterpret it as a self-gravitating supersymmetric soliton interpolating between two vacua of the theory.

The ERN BH is the archetype of the more complicated self-gravitating supersymmetric solitons that we are going to encounter later on in the context of superstring low-energy effective actions (actually, one of our goals will be to recover it as a superstring solution), and many of its properties will be shared by them. Furthermore, the four-dimensional Einstein–Maxwell system exhibits electric–magnetic duality in its simplest form. Electric–magnetic duality will play a crucial role in many of the subsequent developments either as a classical solution-generating tool or as a tool that relates the weak- and strong-coupling regimes of QFTs.

Introduction

In the previous chapters we have studied the upper-left- and upper-right-hand boxes of Fig. 20.1 that concern the standard perturbative formulation of string theory and the effective actions of the ten-dimensional string theories (and M theory). We have also learned a bit about the existence of some non-perturbative states in the string spectrum, in particular D-branes and KK and winding modes in compactified theories (the lower-lefthand box of Fig. 20.1). We have studied in the three cases the existence of dualities that related various theories and how these dualities are realized in the worldsheet action (when this is possible, i.e. for T duality) and in the effective actions. We have also mentioned that S dualities and T dualities imply the existence of new solitonic states in the string spectrum.

In this chapter and the next we are going to study systematically the lower-right-hand and central boxes of Fig. 20.1, that is, the solitonic solutions of the string effective field theories and their worldvolume actions. We will study the implications that the various dualities have for them (which are evidently related to the effects of dualities on the effective actions) and for the non-perturbative string spectrum. This chapter will be devoted to a general introduction to extended objects, and in Chapter 25 we will deal specifically with those that occur in string/M theory.

These are subjects with many facets that are related in many ways to each other and to the subjects of the previous chapters. Therefore, it is hopeless to try to give a complete, or even half-complete, account of them in the space that we have at our disposal. Our aim will be to cover the basic material and the essential results and solutions in a unified system of conventions (like the rest of the book), giving pointers to the literature for further developments.

In our study of several solutions in previous chapters we have mentioned that some special properties that arise for special values of the parameters (mass, charge) are related to supersymmetry; more precisely, to the existence of (unbroken) supersymmetry. Those statements were a bit surprising because we were dealing with solutions of purely bosonic theories (Einstein–Maxwell, Kaluza–Klein, …).

The goal of this chapter is to explain the concept and implications of unbroken supersymmetry and how it can be applied in purely bosonic contexts, including pure GR. Supersymmetry will be shown to have a very deep meaning, underlying more familiar symmetries that can be constructed as squares of supersymmetries. At the very least, supersymmetry can be considered as an extremely useful tool that simplifies many calculations and demonstrations of very important results in GR that are related directly or indirectly to the positivity of energy (a manifest property of supersymmetric theories).

As a further reason to devote a full chapter to this topic, unbroken supersymmetry is a crucial ingredient in the stringy calculation of the BH entropy by the counting of microstates. It ensures the stability of the solution and the calculation under classical and quantum perturbations.

To place this subject in a wider context, we will start by giving in Section 17.1 a general definition of residual (unbroken) symmetry and we will relate it to the definition of a vacuum. Vacua are characterized by their symmetries, which determine the conserved charges of point-particles moving in them and, ultimately, the spectra of quantum-field theories (QFTs) defined on them. These definitions will be applied in Section 17.2 to supersymmetry as a particular case. In this section we will have to develop a new tool, the covariant Lie derivative, which will be used to find the unbroken-supersymmetry algebra of any given solution according to Figueroa-O'Farrill's prescription in Ref. [546].

In the previous chapters of this book we have reviewed several methods to construct BH and black-p-brane solutions of supergravity or supergravity-like theories, some of which are related to superstring theory. In Chapter 19 we showed how the results of the general classification of supersymmetric solutions in Chapter 18 can be used to construct systematically all the supersymmetric black-hole solutions (SBHSs) of some general families of four- and five-dimensional supergravity theories. However, SBHSs (and supersymmetric black-p-brane solutions (SBBSs)) occupy a very small region in the space of all BH and black-brane solutions or even in the smaller space of extremal solutions because, as we are going to study in this chapter, generically there are many more extremal non-supersymmetric than supersymmetric solutions in a given theory. On top of this, it is expected that all the extremal solutions arise as limits of some non-extremal (and, therefore, necessarily non-supersymmetric) family of solutions. The techniques we have developed to construct SBHSs and SBBSs within the framework of solutions with unbroken supersymmetry only allow us to explore certain corners of the space of BH and black-brane solutions, the most interesting of which are in the interior.

It is, therefore, highly desirable to find another framework in which non-supersymmetric (extremal and non-extremal) solutions can be studied. In this chapter we are going to review one such framework: the so-called FGK formalism, which was proposed by Ferrara, Gibbons, and Kallosh in Ref. [522] for four-dimensional asymptotically flat, single, static, extremal and non-extremal BHs and has been generalized to BHs in higher dimensions [941] and to black p-branes in arbitrary number of dimensions [56].

The FGK formalism will allow us to derive a number of general results that apply to all the black solutions of a large number of theories (those with actions of the generic form Eq. (2.147) in d = 4 or Eqs. (2.178) and (2.184) in other dimensions). In particular, we will prove the general existence of the attractor mechanism [531, 1157, 526, 527] in the extremal solutions, supersymmetric or not.

Following our general plan, in Chapter 25 we have started to see classical solutions that describe the long-range fields generated by configurations of extended objects in string/M theory. In general, the solutions do not reflect some of the characteristics of the brane configuration which may be understood as “hair”, but in many cases of interest (in general, in the presence of unbroken supersymmetry), given a classical supergravity solution, we can tell which brane configurations give rise to it. This is in itself a very interesting development, but there is more, because, if the brane configurations only involve D-branes, they can be associated with two-dimensional CFTs (string theories) over which we have good control. Furthermore, each of the branes considered here (D- or not D-) has a worldvolume supersymmetric field theory associated with it. All this allows us to relate supergravity configurations to QFTs whose degrees of freedom can be understood as the microscopical degrees of freedom of the quantum (super)gravity theory contained in string/M theory. This is, roughly speaking, the basis of the AdS/CFT correspondence and generalizations [921, 234] and also the basis for the microscopical computations of BH entropies [1160], the subject of this chapter.

In this chapter we are going to present N = 2A/B, d = 10 SUEGRA solutions associated with configurations of extended objects of type-II superstring theories that lead to BH solutions of maximal d = 5, 4 SUEGRAs (N = 4, d = 5 and N = 8, d = 4) (Section 26.2) upon toroidal compactification. The association can be understood as a strong-weak-coupling limit (see Fig. 26.1). We will carefully relate the solutions' integration constants to the physical parameters of the stringy sources and then, using our knowledge of the QFTs associated with those sources in the extreme and supersymmetric cases, we will count the states of these QFTs at each energy level, and the corresponding entropy will be shown to coincide with one quarter of the area of the BH's horizon (Section 26.3).

A minimal-action principle is a basic ingredient of any field theory. With it (with an action) we can systematically find conserved currents and charges, canonically conjugate momenta, and a Hamiltonian (which is necessary for canonical quantization), etc. On the other hand, it is easier to deal with actions than with equations of motion; it is easier to include new fields and couplings in the action respecting certain symmetries than to invent new consistent equations of motion for them and modifications of the equations of motion of the old fields.

In this chapter we are going to study in detail several action principles for GR and for more general theories that we will be concerned with later on. First, we will study the standard second-order Einstein–Hilbert action that we found as the result of imposing self-consistency on the Fierz–Pauli theory coupled to matter. We will derive the Einstein equations from it and we will find the right boundary term that will allow us to impose boundary conditions on the variations of the metric δgμν only, not on its derivatives. We will do the same for theories including a scalar and in a conformal frame that is not Einstein's. In these theories, an extra scalar factor K (which could be e−2φ in the string effective action) multiplies the Ricci scalar and obtaining the gravitational equations becomes more involved.

We are also going to study the behavior of the Einstein–Hilbert action under GCTs; we will obtain the Bianchi (gauge) identity and Noether current associated with them and see how they are modified by the addition of boundary terms to the action.

Then we will study the first-order formalism, in which the metric gμν and the connection Γμνρ are considered as independent variables, and the first-order formalism for the Vielbein eμa and the spin connection ωμab, with and without fermions, which will be seen to induce torsion.

In this chapter we start the study of a number of important classical solutions of GR. There is no doubt that the most important solution is that due to Schwarzschild, which describes the static, spherically symmetric gravitational field in the absence of matter that one finds outside any static, spherically symmetric object (star, planet, etc.). It is this, the simplest non-trivial solution, that leads to the concept of a black hole (BH), which affords a privileged theoretical laboratory for Gedankenexperiment in classical and quantum gravity.

It is, in fact, a firmly established belief in the scientific community that macroscopic BHs (of the size studied by astrophysicists) are the endpoints of the gravitational collapse of stars, which, after a long time, gives rise to Schwarzschild BHs if the stars do not rotate. There should be many macroscopic Schwarzschild BHs in our Universe, since many stars have enough mass to undergo gravitational collapse, and there is evidence of supermassive BHs in the centers of galaxies. It has been suggested that smaller BHs could have been produced in the Big Bang. Here we are going to be interested in BHs of all sizes, independent of their origin (primordial, quantum-mechanical, astrophysical, …).

We begin by deriving the Schwarzschild solution and studying its classical properties in order to find its physical interpretation. The physical interpretation of vacuum solutions of the Einstein equations is a most important and complicated point [232, 233] since the source, located by definition in the region in which the vacuum Einstein equations are not solved, is unknown. In the case of the Schwarzschild solution, we will be led to the new concepts of the event horizon and BHs. Some of the classical properties of BHs can be formulated as laws of thermodynamics, but, classically, the analogy cannot be complete. It is the existence of Hawking radiation, a quantum phenomenon, that makes the analogy complete and allows us to take it seriously, raising at the same time the problem of the statistical interpretation of the BH (Bekenstein–Hawking) entropy and the BH information problem.