Large N field theories, string theory and gravity

Abstract

We review the holographic correspondence between field theories and string/M theory, focusing on the relation between compactifications of string/M theory on Anti-de Sitter spaces and conformal field theories. We review the background for this correspondence and discuss its motivations and the evidence for its correctness. We describe the main results that have been derived from the correspondence in the regime that the field theory is approximated by classical or semiclassical gravity. We focus on the case of the $\text{N}\text{=4}$ supersymmetric gauge theory in four dimensions, but we discuss also field theories in other dimensions, conformal and non-conformal, with or without supersymmetry, and in particular the relation to QCD. We also discuss some implications for black hole physics.

Introduction

The microscopic description of nature as presently understood and verified by experiment involves quantum field theories. All particles are excitations of some field. These particles are point like and they interact locally with other particles. Even though quantum field theories describe nature at the distance scales we observe, there are strong indications that new elements will be involved at very short distances (or very high energies), distances of the order of the Planck scale. The reason is that at those distances (or energies) quantum gravity effects become important. It has not been possible to quantize gravity following the usual perturbative methods. Nevertheless, one can incorporate quantum gravity in a consistent quantum theory by giving up the notion that particles are point like and assuming that the fundamental objects in the theory are strings, namely one-dimensional extended objects [1], [2]. These strings can oscillate, and there is a spectrum of energies, or masses, for these oscillating strings. The oscillating strings look like localized, particle-like excitations to a low-energy observer. So, a single oscillating string can effectively give rise to many types of particles, depending on its state of oscillation. All string theories include a particle with zero mass and spin two. Strings can interact by splitting and joining interactions. The only consistent interaction for massless spin two particles is that of gravity. Therefore, any string theory will contain gravity. The structure of string theory is highly constrained. String theories do not make sense in an arbitrary number of dimensions or on any arbitrary geometry. Flat space string theory exists (at least in perturbation theory) only in ten dimensions. Actually, 10-dimensional string theory is described by a string which also has fermionic excitations and gives rise to a supersymmetric theory.¹ String theory is then a candidate for a quantum theory of gravity. One can get down to four dimensions by considering string theory on $\text{R}{}^4\text{×M}{}_6$ where M₆ is some six-dimensional compact manifold. Then, low-energy interactions are determined by the geometry of M₆.

Even though this is the motivation usually given for string theory nowadays, it is not how string theory was originally discovered. String theory was discovered in an attempt to describe the large number of mesons and hadrons that were experimentally discovered in the 1960s. The idea was to view all these particles as different oscillation modes of a string. The string idea described well some features of the hadron spectrum. For example, the mass of the lightest hadron with a given spin obeys a relation like m²∼TJ²+const. This is explained simply by assuming that the mass and angular momentum come from a rotating, relativistic string of tension T. It was later discovered that hadrons and mesons are actually made of quarks and that they are described by QCD.

QCD is a gauge theory based on the group SU(3). This is sometimes stated by saying that quarks have three colors. QCD is asymptotically free, meaning that the effective coupling constant decreases as the energy increases. At low energies QCD becomes strongly coupled and it is not easy to perform calculations. One possible approach is to use numerical simulations on the lattice. This is at present the best available tool to do calculations in QCD at low energies. It was suggested by 't Hooft that the theory might simplify when the number of colors N is large [3]. The hope was that one could solve exactly the theory with N=∞, and then one could do an expansion in 1/N=1/3. Furthermore, as explained in the next section, the diagrammatic expansion of the field theory suggests that the large N theory is a free string theory and that the string coupling constant is 1/N. If the case with N=3 is similar to the case with N=∞ then this explains why the string model gave the correct relation between the mass and the angular momentum. In this way the large N limit connects gauge theories with string theories. The 't Hooft argument, reviewed below, is very general, so it suggests that different kinds of gauge theories will correspond to different string theories. In this review we will study this correspondence between string theories and the large N limit of field theories. We will see that the strings arising in the large N limit of field theories are the same as the strings describing quantum gravity. Namely, string theory in some backgrounds, including quantum gravity, is equivalent (dual) to a field theory.

We said above that strings are not consistent in four flat dimensions. Indeed, if one wants to quantize a four-dimensional string theory an anomaly appears that forces the introduction of an extra field, sometimes called the “Liouville” field [4]. This field on the string worldsheet may be interpreted as an extra dimension, so that the strings effectively move in five dimensions. One might qualitatively think of this new field as the “thickness” of the string. If this is the case, why do we say that the string moves in five dimensions? The reason is that like any string theory, this theory will contain gravity, and the gravitational theory will live in as many dimensions as the number of fields we have on the string. It is crucial then that the five-dimensional geometry is curved, so that it can correspond to a four-dimensional field theory, as described in detail below.

The argument that gauge theories are related to string theories in the large N limit is very general and is valid for basically any gauge theory. In particular we could consider a gauge theory where the coupling does not run (as a function of the energy scale). Then, the theory is conformally invariant. It is quite hard to find quantum field theories that are conformally invariant. In supersymmetric theories it is sometimes possible to prove exact conformal invariance. A simple example, which will be the main example in this review, is the supersymmetric SU(N) (or U(N)) gauge theory in four dimensions with four spinor supercharges $\text{(}\text{N}\text{=4)}$. Four is the maximal possible number of supercharges for a field theory in four dimensions. Besides the gauge fields (gluons) this theory contains also four fermions and six scalar fields in the adjoint representation of the gauge group. The Lagrangian of such theories is completely determined by supersymmetry. There is a global $\text{SU(4)}\text{R}$-symmetry that rotates the six scalar fields and the four fermions. The conformal group in four dimensions is $\text{SO(4,}\text{2)}$, including the usual Poincaré transformations as well as scale transformations and special conformal transformations (which include the inversion symmetry x^μ→x^μ/x²). These symmetries of the field theory should be reflected in the dual string theory. The simplest way for this to happen is if the five-dimensional geometry has these symmetries. Locally there is only one space with $\text{SO(4,}\text{2)}$ isometries: five-dimensional anti-de-Sitter space, or AdS₅. Anti-de Sitter space is the maximally symmetric solution of Einstein's equations with a negative cosmological constant. In this supersymmetric case we expect the strings to also be supersymmetric. We said that superstrings move in ten dimensions. Now that we have added one more dimension it is not surprising any more to add five more to get to a ten dimensional space. Since the gauge theory has an SU(4)≃SO(6) global symmetry it is rather natural that the extra five-dimensional space should be a five sphere, S⁵. So, we conclude that $\text{N}\text{=4}\text{U(N)}$ Yang–Mills theory could be the same as ten-dimensional superstring theory on AdS₅×S⁵ [5]. Here we have presented a very heuristic argument for this equivalence; later we will be more precise and give more evidence for this correspondence.

The relationship we described between gauge theories and string theory on Anti-de-Sitter spaces was motivated by studies of D-branes and black holes in strings theory. D-branes are solitons in string theory [6]. They come in various dimensionalities. If they have zero spatial dimensions they are like ordinary localized, particle-type soliton solutions, analogous to the 't Hooft–Polyakov [7], [8] monopole in gauge theories. These are called D-zero-branes. If they have one extended dimension they are called D-one-branes or D-strings. They are much heavier than ordinary fundamental strings when the string coupling is small. In fact, the tension of all D-branes is proportional to 1/g_s, where g_s is the string coupling constant. D-branes are defined in string perturbation theory in a very simple way: they are surfaces where open strings can end. These open strings have some massless modes, which describe the oscillations of the branes, a gauge field living on the brane, and their fermionic partners. If we have N coincident branes the open strings can start and end on different branes, so they carry two indices that run from one to N. This in turn implies that the low-energy dynamics is described by a U(N) gauge theory. D-p-branes are charged under (p+1)-form gauge potentials, in the same way that a 0-brane (particle) can be charged under a one-form gauge potential (as in electromagnetism). These (p+1)-form gauge potentials have (p+2)-form field strengths, and they are part of the massless closed string modes, which belong to the supergravity (SUGRA) multiplet containing the massless fields in flat space string theory (before we put in any D-branes). If we now add D-branes they generate a flux of the corresponding field strength, and this flux in turn contributes to the stress energy tensor so the geometry becomes curved. Indeed it is possible to find solutions of the supergravity equations carrying these fluxes. Supergravity is the low-energy limit of string theory, and it is believed that these solutions may be extended to solutions of the full string theory. These solutions are very similar to extremal charged black hole solutions in general relativity, except that in this case they are black branes with p extended spatial dimensions. Like black holes they contain event horizons.

If we consider a set of N coincident D-3-branes the near horizon geometry turns out to be AdS₅×S⁵. On the other hand, the low-energy dynamics on their worldvolume is governed by a U(N) gauge theory with $\text{N}\text{=4}$ supersymmetry [9]. These two pictures of D-branes are perturbatively valid for different regimes in the space of possible coupling constants. Perturbative field theory is valid when g_sN is small, while the low-energy gravitational description is perturbatively valid when the radius of curvature is much larger than the string scale, which turns out to imply that g_sN should be very large. As an object is brought closer and closer to the black brane horizon its energy measured by an outside observer is redshifted, due to the large gravitational potential, and the energy seems to be very small. On the other hand low-energy excitations on the branes are governed by the Yang–Mills theory. So, it becomes natural to conjecture that Yang–Mills theory at strong coupling is describing the near horizon region of the black brane, whose geometry is AdS₅×S⁵. The first indications that this is the case came from calculations of low-energy graviton absorption cross sections [10], [11], [12]. It was noticed there that the calculation done using gravity and the calculation done using super Yang–Mills theory agreed. These calculations, in turn, were inspired by similar calculations for coincident D1–D5 branes. In this case the near horizon geometry involves AdS₃×S³ and the low-energy field theory living on the D-branes is a (1+1)-dimensional conformal field theory. In this D1–D5 case there were numerous calculations that agreed between the field theory and gravity. First black hole entropy for extremal black holes was calculated in terms of the field theory in [13], and then agreement was shown for near extremal black holes [14], [15] and for absorption cross sections [16], [17], [18]. More generally, we will see that correlation functions in the gauge theory can be calculated using the string theory (or gravity for large g_sN) description, by considering the propagation of particles between different points in the boundary of AdS, the points where operators are inserted [19], [20].

Supergravities on AdS spaces were studied very extensively; see [21], [22] for reviews.

One of the main points of this review will be that the strings coming from gauge theories are very much like the ordinary superstrings that have been studied during the last 20 years. The only particular feature is that they are moving on a curved geometry (anti-de Sitter space) which has a boundary at spatial infinity. The boundary is at an infinite spatial distance, but a light ray can go to the boundary and come back in finite time. Massive particles can never get to the boundary. The radius of curvature of anti-de Sitter space depends on N so that large N corresponds to a large radius of curvature. Thus, by taking N to be large we can make the curvature as small as we want. The theory in AdS includes gravity, since any string theory includes gravity. So in the end we claim that there is an equivalence between a gravitational theory and a field theory. However, the mapping between the gravitational and field theory degrees of freedom is quite non-trivial since the field theory lives in a lower dimension. In some sense the field theory (or at least the set of local observables in the field theory) lives on the boundary of spacetime. One could argue that in general any quantum gravity theory in AdS defines a conformal field theory (CFT) “on the boundary”. In some sense the situation is similar to the correspondence between three dimensional Chern–Simons theory and a WZW model on the boundary [23]. This is a topological theory in three dimensions that induces a normal (non-topological) field theory on the boundary. A theory which includes gravity is in some sense topological since one is integrating over all metrics and therefore the theory does not depend on the metric. Similarly, in a quantum gravity theory we do not have any local observables. Notice that when we say that the theory includes “gravity on AdS” we are considering any finite energy excitation, even black holes in AdS. So this is really a sum over all spacetimes that are asymptotic to AdS at the boundary. This is analogous to the usual flat space discussion of quantum gravity, where asymptotic flatness is required, but the spacetime could have any topology as long as it is asymptotically flat. The asymptotically AdS case as well as the asymptotically flat cases are special in the sense that one can choose a natural time and an associated Hamiltonian to define the quantum theory. Since black holes might be present this time coordinate is not necessarily globally well-defined, but it is certainly well-defined at infinity. If we assume that the conjecture we made above is valid, then the U(N) Yang–Mills theory gives a non-perturbative definition of string theory on AdS. And, by taking the limit N→∞, we can extract the (ten-dimensional string theory) flat space physics, a procedure which is in principle (but not in detail) similar to the one used in matrix theory [24].

The fact that the field theory lives in a lower-dimensional space blends in perfectly with some previous speculations about quantum gravity. It was suggested [25], [26] that quantum gravity theories should be holographic, in the sense that physics in some region can be described by a theory at the boundary with no more than one degree of freedom per Planck area. This “holographic” principle comes from thinking about the Bekenstein bound which states that the maximum amount of entropy in some region is given by the area of the region in Planck units [27]. The reason for this bound is that otherwise black hole formation could violate the second law of thermodynamics. We will see that the correspondence between field theories and string theory on AdS space (including gravity) is a concrete realization of this holographic principle.

The review is organized as follows.

In the rest of the introductory chapter, we present background material. In Section 1.2, we present the 't Hooft large N limit and its indication that gauge theories may be dual to string theories. In Section 1.3, we review the p-brane supergravity solutions. We discuss D-branes, their worldvolume theory and their relation to the p-branes. We discuss greybody factors and their calculation for black holes built out of D-branes.

In Section 2, we review conformal field theories and AdS spaces. In Section 2.1, we give a brief description of conformal field theories. In Section 2.2, we summarize the geometry of AdS spaces and gauged supergravities.

In Section 3, we “derive” the correspondence between supersymmetric Yang–Mills theory and string theory on AdS₅×S⁵ from the physics of D3-branes in string theory. We define, in Section 3.1, the correspondence between fields in the string theory and operators of the conformal field theory and the prescription for the computation of correlation functions. We also point out that the correspondence gives an explicit holographic description of gravity. In Section 3.2, we review the direct tests of the duality, including matching the spectrum of chiral primary operators and some correlation functions and anomalies. Computation of correlation functions is reviewed in Section 3.3. The isomorphism of the Hilbert spaces of string theory on AdS spaces and of CFTs is described in Section 3.4. We describe how to introduce Wilson loop operators in Section 3.5. In Section 3.6, we analyze finite temperature theories and the thermal phase transition.

In Section 4, we review other topics involving AdS₅. In Section 4.1, we consider some other gauge theories that arise from D-branes at orbifolds, orientifolds, or conifold points. In Section 4.2, we review how baryons and instantons arise in the string theory description. In Section 4.3, we study some deformations of the CFT and how they arise in the string theory description.

In Section 5, we describe a similar correspondence involving (1+1)-dimensional CFTs and AdS₃ spaces. We also describe the relation of these results to black holes in five dimensions.

In Section 6, we consider other examples of the AdS/CFT correspondence as well as non conformal and non supersymmetric cases. In Section 6.1, we analyze the M2 and M5 branes theories, and go on to describe situations that are not conformal, realized on the worldvolume of various Dp-branes, and the “little string theories” on the worldvolume of NS 5-branes. In Section 6.2, we describe an approach to studying theories that are confining and have a behavior similar to QCD in three and four dimensions. We discuss confinement, θ-vacua, the mass spectrum and other dynamical aspects of these theories.

Finally, the last section is devoted to a summary and discussion.

Other reviews of this subject are [28], [29], [30], [31].

The relation between gauge theories and string theories has been an interesting topic of research for over three decades. String theory was originally developed as a theory for the strong interactions, due to various string-like aspects of the strong interactions, such as confinement and Regge behavior. It was later realized that there is another description of the strong interactions, in terms of an SU(3) gauge theory (QCD), which is consistent with all experimental data to date. However, while the gauge theory description is very useful for studying the high-energy behavior of the strong interactions, it is very difficult to use it to study low-energy issues such as confinement and chiral symmetry breaking (the only current method for addressing these issues in the full non-Abelian gauge theory is by numerical simulations). In the last few years many examples of the phenomenon generally known as “duality” have been discovered, in which a single theory has (at least) two different descriptions, such that when one description is weakly coupled the other is strongly coupled and vice versa (examples of this phenomenon in two-dimensional field theories have been known for many years). One could hope that a similar phenomenon would apply in the theory of the strong interactions, and that a “dual” description of QCD exists which would be more appropriate for studying the low-energy regime where the gauge theory description is strongly coupled.

There are several indications that this “dual” description could be a string theory. QCD has in it string-like objects which are the flux tubes or Wilson lines. If we try to separate a quark from an anti-quark, a flux tube forms between them (if ψ is a quark field, the operator $\text{ψ}\text{̄}\text{(0)ψ(x)}$ is not gauge-invariant but the operator $\text{ψ}\text{̄}\text{(0)P}\text{exp}\text{(}\text{i}\text{∫}\text{0}\text{x}\text{A}{}_{\mathrm{\mu }}\text{d}\text{x}{}^{\mathrm{\mu }}\text{)ψ(x)}$ is gauge-invariant). In many ways these flux tubes behave like strings, and there have been many attempts to write down a string theory describing the strong interactions in which the flux tubes are the basic objects. It is clear that such a stringy description would have many desirable phenomenological attributes since, after all, this is how string theory was originally discovered. The most direct indication from the gauge theory that it could be described in terms of a string theory comes from the 't Hooft large N limit [3], which we will now describe in detail.

Yang–Mills (YM) theories in four dimensions have no dimensionless parameters, since the gauge coupling is dimensionally transmuted into the QCD scale Λ_QCD (which is the only mass scale in these theories). Thus, there is no obvious perturbation expansion that can be performed to learn about the physics near the scale Λ_QCD. However, an additional parameter of SU(N) gauge theories is the integer number N, and one may hope that the gauge theories may simplify at large N (despite the larger number of degrees of freedom), and have a perturbation expansion in terms of the parameter 1/N. This turns out to be true, as shown by 't Hooft based on the following analysis (reviews of large N QCD may be found in [32], [33]).

First, we need to understand how to scale the coupling g_YM as we take N→∞. In an asymptotically free theory, like pure YM theory, it is natural to scale g_YM so that Λ_QCD remains constant in the large N limit. The beta function equation for pure SU(N) YM theory is

$$\text{μ}\text{d}\text{g}{}_{\mathrm{<mtext>YM</mtext>}}\text{d}\text{μ}\text{=−}\text{11}\text{3}\text{N}\text{g}{}_{\mathrm{<mtext>YM</mtext>}}{}^3\text{16}\text{π}{}^2\text{+}\text{O}\text{(g}{}_{\mathrm{<mtext>YM</mtext>}}{}^5\text{)}\text{,}$$

so the leading terms are of the same order for large N if we take N→∞ while keeping λ≡g_YM²N fixed (one can show that the higher order terms are also of the same order in this limit). This is known as the ’t Hooft limit. The same behavior is valid if we include also matter fields (fermions or scalars) in the adjoint representation, as long as the theory is still asymptotically free. If the theory is conformal, such as the $\text{N}\text{=4}$ SYM theory which we will discuss in detail below, it is not obvious that the limit of constant λ is the only one that makes sense, and indeed we will see that other limits, in which λ→∞, are also possible. However, the limit of constant λ is still a particularly interesting limit and we will focus on it in the remainder of this section.

Instead of focusing just on the YM theory, let us describe a general theory which has some fields Φ_i^a, where a is an index in the adjoint representation of SU(N), and i is some label of the field (a spin index, a flavor index, etc.). Some of these fields can be ghost fields (as will be the case in gauge theory). We will assume that as in the YM theory (and in the $\text{N}\text{=4}$ SYM theory), the 3-point vertices of all these fields are proportional to g_YM, and the 4-point functions to g_YM², so the Lagrangian is of the schematic form

$$\text{L}\text{∼}\text{Tr}\text{(dΦ}{}_{\mathrm{i}}\text{dΦ}{}_{\mathrm{i}}\text{)+g}{}_{\mathrm{<mtext>YM</mtext>}}\text{c}{}^{\mathrm{ijk}}\text{Tr}\text{(Φ}{}_{\mathrm{i}}\text{Φ}{}_{\mathrm{j}}\text{Φ}{}_{\mathrm{k}}\text{)+g}{}_{\mathrm{<mtext>YM</mtext>}}{}^2\text{d}{}^{\mathrm{ijkl}}\text{Tr}\text{(Φ}{}_{\mathrm{i}}\text{Φ}{}_{\mathrm{j}}\text{Φ}{}_{\mathrm{k}}\text{Φ}{}_{\mathrm{l}}\text{)}\text{,}$$

for some constants c^ijk and d^ijkl (where we have assumed that the interactions are SU(N)-invariant; mass terms can also be added and do not change the analysis). Rescaling the fields by $\text{Φ}\text{̃}{}_{\mathrm{i}}\text{≡g}{}_{\mathrm{<mtext>YM</mtext>}}\text{Φ}{}_{\mathrm{i}}$, the Lagrangian becomes

$$\text{L}\text{∼}\text{1}\text{g}{}_{\mathrm{<mtext>YM</mtext>}}{}^2\text{[}\text{Tr}\text{(d}\text{Φ}\text{̃}{}_{\mathrm{i}}\text{d}\text{Φ}\text{̃}{}_{\mathrm{i}}\text{)+c}{}^{\mathrm{ijk}}\text{Tr}\text{(}\text{Φ}\text{̃}{}_{\mathrm{i}}\text{Φ}\text{̃}{}_{\mathrm{j}}\text{Φ}\text{̃}{}_{\mathrm{k}}\text{)+d}{}^{\mathrm{ijkl}}\text{Tr}\text{(}\text{Φ}\text{̃}{}_{\mathrm{i}}\text{Φ}\text{̃}{}_{\mathrm{j}}\text{Φ}\text{̃}{}_{\mathrm{k}}\text{Φ}\text{̃}{}_{\mathrm{l}}\text{)]}\text{,}$$

with a coefficient of 1/g_YM²=N/λ in front of the whole Lagrangian.

Now, we can ask what happens to correlation functions in the limit of large N with constant λ. Naively, this is a classical limit since the coefficient in front of the Lagrangian diverges, but in fact this is not true since the number of components in the fields also goes to infinity in this limit. We can write the Feynman diagrams of the theory (1.3) in a double-line notation, in which an adjoint field Φ^a is represented as a direct product of a fundamental and an anti-fundamental field, Φⁱ_j, as in Fig. (1.1). The interaction vertices we wrote are all consistent with this sort of notation. The propagators are also consistent with it in a U(N) theory; in an SU(N) theory there is a small mixing term

$$\text{〈Φ}{}^{\mathrm{i}}{}_{\mathrm{j}}\text{Φ}{}^{\mathrm{k}}{}_{\mathrm{l}}\text{〉∝}\text{δ}{}^{\mathrm{i}}{}_{\mathrm{l}}\text{δ}{}^{\mathrm{j}}{}_{\mathrm{k}}\text{−}\text{1}\text{N}\text{δ}{}^{\mathrm{i}}{}_{\mathrm{j}}\text{δ}{}^{\mathrm{k}}{}_{\mathrm{l}}\text{,}$$

which makes the expansion slightly more complicated, but this involves only subleading terms in the large N limit so we will neglect this difference here. Ignoring the second term the propagator for the adjoint field is (in terms of the index structure) like that of a fundamental–anti-fundamental pair. Thus, any Feynman diagram of adjoint fields may be viewed as a network of double lines. Let us begin by analyzing vacuum diagrams (the generalization to adding external fields is simple and will be discussed below). In such a diagram we can view these double lines as forming the edges in a simplicial decomposition (for example, it could be a triangulation) of a surface, if we view each single-line loop as the perimeter of a face of the simplicial decomposition. The resulting surface will be oriented since the lines have an orientation (in one direction for a fundamental index and in the opposite direction for an anti-fundamental index). When we compactify space by adding a point at infinity, each diagram thus corresponds to a compact, closed, oriented surface.

What is the power of N and λ associated with such a diagram? From the form of (1.3) it is clear that each vertex carries a coefficient proportional to N/λ, while propagators are proportional to λ/N. Additional powers of N come from the sum over the indices in the loops, which gives a factor of N for each loop in the diagram (since each index has N possible values). Thus, we find that a diagram with V vertices, E propagators (=edges in the simplicial decomposition) and F loops (=faces in the simplicial decomposition) comes with a coefficient proportional to

$$\text{N}{}^{\mathrm{V-E+F}}\text{λ}{}^{\mathrm{E-V}}\text{=N}{}^{\mathrm{\chi }}\text{λ}{}^{\mathrm{E-V}}\text{,}$$

where χ≡V−E+F is the Euler character of the surface corresponding to the diagram. For closed oriented surfaces, χ=2−2g where g is the genus (the number of handles) of the surface.² Thus, the perturbative expansion of any diagram in the field theory may be written as a double expansion of the form

$$\text{∑}\text{g=0}\text{∞}\text{N}{}^{\mathrm{2-2g}}\text{∑}\text{i=0}\text{∞}\text{c}{}_{\mathrm{g,i}}\text{λ}{}^{\mathrm{i}}\text{=}\text{∑}\text{g=0}\text{∞}\text{N}{}^{\mathrm{2-2g}}\text{f}{}_{\mathrm{g}}\text{(λ),}$$

where f_g is some polynomial in λ (in an asymptotically free theory the λ-dependence will turn into some Λ_QCD-dependence but the general form is similar; infrared divergences could also lead to the appearance of terms which are not integer powers of λ). In the large N limit we see that any computation will be dominated by the surfaces of maximal χ or minimal genus, which are surfaces with the topology of a sphere (or equivalently a plane). All these planar diagrams will give a contribution of order N², while all other diagrams will be suppressed by powers of 1/N². For example, the first diagram in Fig. 1.1 is planar and proportional to N²⁻³⁺³=N², while the second one is not and is proportional to N⁴⁻⁶⁺²=N⁰. We presented our analysis for a general theory, but in particular it is true for any gauge theory coupled to adjoint matter fields, like the $\text{N}\text{=4}$ SYM theory. The rest of our discussion will be limited mostly to gauge theories, where only gauge-invariant (SU(N)-invariant) objects are usually of interest.

The form of the expansion (1.6) is the same as one finds in a perturbative theory with closed oriented strings, if we identify 1/N as the string coupling constant.³ Of course, we do not really see any strings in the expansion, but just diagrams with holes in them; however, one can hope that in a full non-perturbative description of the field theory the holes will “close” and the surfaces of the Feynman diagrams will become actual closed surfaces. The analogy of (1.6) with perturbative string theory is one of the strongest motivations for believing that field theories and string theories are related, and it suggests that this relation would be more visible in the large N limit where the dual string theory may be weakly coupled. However, since the analysis was based on perturbation theory which generally does not converge, it is far from a rigorous derivation of such a relation, but rather an indication that it might apply, at least for some field theories (there are certainly also effects like instantons which are non-perturbative in the 1/N expansion, and an exact matching with string theory would require a matching of such effects with non-perturbative effects in string theory).

The fact that 1/N behaves as a coupling constant in the large N limit can also be seen directly in the field theory analysis of the 't Hooft limit. While we have derived the behavior (1.6) only for vacuum diagrams, it actually holds for any correlation function of a product of gauge-invariant fields 〈∏_j=1ⁿG_j〉 such that each G_j cannot be written as a product of two gauge-invariant fields (for instance, G_j can be of the form $\text{(1/N)}\text{Tr}\text{(}\text{∏}\text{i}\text{Φ}{}_{\mathrm{i}}\text{)}$). We can study such a correlation function by adding to the action S→S+N∑g_jG_j, and then, if W is the sum of connected vacuum diagrams we discussed above (but now computed with the new action),

Our analysis of the vacuum diagrams above holds also for these diagrams, since we put in additional vertices with a factor of N, and, in the double-line representation, each of the operators we inserted becomes a vertex of the simplicial decomposition of the surface (this would not be true for operators which are themselves products, and which would correspond to more than one vertex). Thus, the leading contribution to 〈∏_j=1ⁿG_j〉 will come from planar diagrams with n additional operator insertions, leading to

$$\text{∏}\text{j=1}\text{n}\text{G}{}_{\mathrm{j}}\text{∝N}{}^{\mathrm{2-n}}$$

in the 't Hooft limit. We see that (in terms of powers of N) the 2-point functions of the G_j's come out to be canonically normalized, while 3-point functions are proportional to 1/N, so indeed 1/N is the coupling constant in this limit (higher genus diagrams do not affect this conclusion since they just add higher-order terms in 1/N). In the string theory analogy, the operators G_j would become vertex operators inserted on the string world-sheet. For asymptotically free confining theories (like QCD) one can show that in the large N limit they have an infinite spectrum of stable particles with rising masses (as expected in a free string theory). Many additional properties of the large N limit are discussed in [34], [32] and other references.

The analysis we did of the 't Hooft limit for SU(N) theories with adjoint fields can easily be generalized to other cases. Matter in the fundamental representation appears as single-line propagators in the diagrams, which correspond to boundaries of the corresponding surfaces. Thus, if we have such matter we need to sum also over surfaces with boundaries, as in open string theories. For SO(N) or USp(N) gauge theories we can represent the adjoint representation as a product of two fundamental representations (instead of a fundamental and an anti-fundamental representation), and the fundamental representation is real, so no arrows appear on the propagators in the diagram, and the resulting surfaces may be non-orientable. Thus, these theories seem to be related to non-orientable string theories [35]. We will not discuss these cases in detail here, some of the relevant aspects will be discussed in Section 4.1.2 below.

Our analysis thus far indicates that gauge theories may be dual to string theories with a coupling proportional to 1/N in the 't Hooft limit, but it gives no indication as to precisely which string theory is dual to a particular gauge theory. For two-dimensional gauge theories much progress has been made in formulating the appropriate string theories [36], [37], [38], [39], [40], [41], [42], [43], but for four dimensional gauge theories there was no concrete construction of a corresponding string theory before the results reported below, since the planar diagram expansion (which corresponds to the free string theory) is very complicated. Various direct approaches towards constructing the relevant string theory were attempted, many of which were based on the loop equations [44] for the Wilson loop observables in the field theory, which are directly connected to a string-type description.

Attempts to directly construct a string theory equivalent to a four dimensional gauge theory are plagued with the well-known problems of string theory in four dimensions (or generally below the critical dimension). In particular, additional fields must be added on the worldsheet beyond the four embedding coordinates of the string to ensure consistency of the theory. In the standard quantization of four-dimensional string theory an additional field called the Liouville field arises [4], which may be interpreted as a fifth space–time dimension. Polyakov has suggested [45], [46] that such a five dimensional string theory could be related to four-dimensional gauge theories if the couplings of the Liouville field to the other fields take some specific forms. As we will see, the AdS/CFT correspondence realizes this idea, but with five additional dimensions (in addition to the radial coordinate on AdS which can be thought of as a generalization of the Liouville field), leading to a standard (critical) ten-dimensional string theory.

The recent insight into the connection between large N field theories and string theory has emerged from the study of p-branes in string theory. The p-branes were originally found as classical solutions to supergravity, which is the low-energy limit of string theory. Later it was pointed out by Polchinski that D-branes give their full string theoretical description. Various comparisons of the two descriptions led to the discovery of the AdS/CFT correspondence.

String theory has a variety of classical solutions corresponding to extended black holes [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57]. Complete descriptions of all possible black hole solutions would be beyond the scope of this review, and we will discuss here only illustrative examples corresponding to parallel Dp branes. For a more extensive review of extended objects in string theory, see [58], [59].

Let us consider type II string theory in ten dimensions, and look for a black hole solution carrying electric charge with respect to the Ramond–Ramond (R–R) (p+1)-form A_p+1 [48], [53], [56]. In type IIA (IIB) theory, p is even (odd). The theory contains also magnetically charged (6−p)-branes, which are electrically charged under the dual $\text{dA}{}_{\mathrm{7-p}}\text{=∗dA}{}_{\mathrm{p+1}}$ potential. Therefore, R–R charges have to be quantized according to the Dirac quantization condition. To find the solution, we start with the low-energy effective action in the string frame,

$$\text{S=}\text{1}\text{(2}\text{π}\text{)}{}^7\text{l}{}_{\mathrm{<mtext>s</mtext>}}{}^8\text{∫}\text{d}{}^{10}\text{x}\text{−g}\text{e}{}^{\mathrm{-2\phi }}\text{(}\text{R}\text{+4(}\text{∇}\text{φ)}{}^2\text{)−}\text{2}\text{(8−p)!}\text{F}{}_{\mathrm{p+2}}{}^2\text{,}$$

where l_s is the string length, related to the string tension (2πα′)⁻¹ as α′=l_s², and F_p+2 is the field strength of the (p+1)-form potential, F_p+2=dA_p+1. In the self-dual case of p=3, we work directly with the equations of motion. We then look for a solution corresponding to a p-dimensional electric source of charge N for A_p+1, by requiring the Euclidean symmetry ISO(p) in p-dimensions:

$$\text{d}\text{s}{}^2\text{=}\text{d}\text{s}{}_{\mathrm{10-p}}{}^2\text{+}\text{e}{}^{\mathrm{\alpha }}\text{∑}\text{i=1}\text{p}\text{d}\text{x}{}^{\mathrm{i}}\text{d}\text{x}{}^{\mathrm{i}}\text{.}$$

Here ds_10−p² is a Lorentzian-signature metric in (10−p)-dimensions. We also assume that the metric is spherically symmetric in (10−p) dimensions with the R–R source at the origin,

$$\text{∫}\text{S}{}^{\mathrm{8-p}}{}^{\mathrm{\ast }}\text{F}{}_{\mathrm{p+2}}\text{=N}\text{,}$$

where S^8−p is the (8−p)-sphere surrounding the source. By using the Euclidean symmetry ISO(p), we can reduce the problem to the one of finding a spherically symmetric charged black hole solution in (10−p) dimensions [48], [53], [56]. The resulting metric, in the string frame, is given by

$$\text{d}\text{s}{}^2\text{=−}\text{f}{}_{\mathrm{+}}\text{(ρ)}\text{f}{}_{\mathrm{-}}\text{(ρ)}\text{d}\text{t}{}^2\text{+}\text{f}{}_{\mathrm{-}}\text{(ρ)}\text{∑}\text{i=1}\text{p}\text{d}\text{x}{}^{\mathrm{i}}\text{d}\text{x}{}^{\mathrm{i}}\text{+}\text{f}{}_{\mathrm{-}}\text{(ρ)}{}^{\mathrm{-1/2-(5-p)/(7-p)}}\text{f}{}_{\mathrm{+}}\text{(ρ)}\text{d}\text{ρ}{}^2\text{+r}{}^2\text{f}{}_{\mathrm{-}}\text{(ρ)}{}^{\mathrm{<mspace\; sp="0.16"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>1</mtext><mtext>2</mtext><mtext>-(5-p)/(7-p)</mtext>}}\text{dΩ}{}_{\mathrm{8-p}}{}^2\text{,}$$

with the dilaton field,

$$\text{e}{}^{\mathrm{-2\phi }}\text{=g}{}_{\mathrm{<mtext>s</mtext>}}{}^{\mathrm{-2}}\text{f}{}_{\mathrm{-}}\text{(ρ)}{}^{\mathrm{-(p-3)/2}}\text{,}$$

where

$$\text{f}{}_{\mathrm{\pm }}\text{(ρ)=1−}\text{r}{}_{\mathrm{\pm }}\text{ρ}{}^{\mathrm{7-p}}\text{,}$$

and g_s is the asymptotic string coupling constant. The parameters r₊ and r₋ are related to the mass M (per unit volume) and the RR charge N of the solution by

$$\text{M=}\text{1}\text{(7−p)(2}\text{π}\text{)}{}^7\text{d}{}_{\mathrm{p}}\text{l}{}_{\mathrm{<mtext>P</mtext>}}{}^8\text{((8−p)r}{}_{\mathrm{+}}{}^{\mathrm{7-p}}\text{−r}{}_{\mathrm{-}}{}^{\mathrm{7-p}}\text{),}\text{N=}\text{1}\text{d}{}_{\mathrm{p}}\text{g}{}_{\mathrm{<mtext>s</mtext>}}\text{l}{}_{\mathrm{<mtext>s</mtext>}}{}^{\mathrm{7-p}}\text{(r}{}_{\mathrm{+}}\text{r}{}_{\mathrm{-}}\text{)}{}^{\mathrm{(7-p)/2}}\text{,}$$

where l_P=g_s^1/4l_s is the 10-dimensional Planck length and d_p is a numerical factor,

$$\text{d}{}_{\mathrm{p}}\text{=2}{}^{\mathrm{5-p}}\text{π}{}^{\mathrm{(5-p)/2}}\text{Γ}\text{7−p}\text{2}\text{.}$$

The metric in the Einstein frame, $\text{(g}{}_{\mathrm{<mtext>E</mtext>}}\text{)}{}_{\mathrm{\mu \nu }}$, is defined by multiplying the string frame metric g_μν by $\text{g}{}_{\mathrm{<mtext>s</mtext>}}\text{e}{}^{\mathrm{-\phi }}$ in (1.9), so that the action takes the standard Einstein–Hilbert form,

$$\text{S=}\text{1}\text{(2}\text{π}\text{)}{}^7\text{l}{}_{\mathrm{<mtext>P</mtext>}}{}^8\text{∫}\text{d}{}^{10}\text{x}\text{−g}{}_{\mathrm{<mtext>E</mtext>}}\text{(}\text{R}{}_{\mathrm{<mtext>E</mtext>}}\text{−}\text{1}\text{2}\text{(}\text{∇}\text{φ)}{}^2\text{+⋯)}\text{.}$$

The Einstein frame metric has a horizon at ρ=r₊. For p≤6, there is also a curvature singularity at ρ=r₋. When r₊>r₋, the singularity is covered by the horizon and the solution can be regarded as a black hole. When r₊<r₋, there is a time like naked singularity and the Cauchy problem is not well-posed.

The situation is subtle in the critical case r₊=r₋. If p≠3, the horizon and the singularity coincide and there is a “null” singularity.⁴ Moreover, the dilaton either diverges or vanishes at ρ=r₊. This singularity, however, is milder than in the case of r₊<r₋, and the supergravity description is still valid up to a certain distance from the singularity. The situation is much better for p=3. In this case, the dilaton is constant. Moreover, the ρ=r₊ surface is regular even when r₊=r₋, allowing a smooth analytic extension beyond ρ=r₊ [60].

According to (1.15), for a fixed value of N, the mass M is an increasing function of r₊. The condition r₊≥r₋ for the absence of the time like naked singularity therefore translates into an inequality between the mass M and the R–R charge N, of the form

$$\text{M≥}\text{N}\text{(2}\text{π}\text{)}{}^{\mathrm{p}}\text{g}{}_{\mathrm{<mtext>s</mtext>}}\text{l}{}_{\mathrm{<mtext>s</mtext>}}{}^{\mathrm{p+1}}\text{.}$$

The solution whose mass M is at the lower bound of this inequality is called an extremal p-brane. On the other hand, when M is strictly greater than that, we have a non-extremal black p-brane. It is called black since there is an event horizon for r₊>r₋. The area of the black hole horizon goes to zero in the extremal limit r₊=r₋. Since the extremal solution with p≠3 has a singularity, the supergravity description breaks down near ρ=r₊ and we need to use the full string theory. The D-brane construction discussed below will give exactly such a description. The inequality (1.18) is also the BPS bound with respect to the 10-dimensional supersymmetry, and the extremal solution r₊=r₋ preserves one half of the supersymmetry in the regime where we can trust the supergravity description. This suggests that the extremal p-brane is a ground state of the black p-brane for a given charge N.

The extremal limit r₊=r₋ of the solution (1.12) is given by

$$\text{d}\text{s}{}^2\text{=}\text{f}{}_{\mathrm{+}}\text{(ρ)}\text{−}\text{d}\text{t}{}^2\text{+}\text{∑}\text{i=1}\text{p}\text{d}\text{x}{}^{\mathrm{i}}\text{d}\text{x}{}^{\mathrm{i}}\text{+f}{}_{\mathrm{+}}\text{(ρ)}{}^{\mathrm{-3/2-(5-p)/(7-p)}}\text{d}\text{ρ}{}^2\text{+ρ}{}^2\text{f}{}_{\mathrm{+}}\text{(ρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>-(5-p)/(7-p)</mtext>}}\text{dΩ}{}_{\mathrm{8-p}}{}^2\text{.}$$

In this limit, the symmetry of the metric is enhanced from the Euclidean group ISO(p) to the Poincaré group $\text{ISO(p,}\text{1)}$. This fits well with the interpretation that the extremal solution corresponds to the ground state of the black p-brane. To describe the geometry of the extremal solution outside of the horizon, it is often useful to define a new coordinate r by

$$\text{r}{}^{\mathrm{7-p}}\text{≡ρ}{}^{\mathrm{7-p}}\text{−r}{}_{\mathrm{+}}{}^{\mathrm{7-p}}\text{,}$$

and introduce the isotropic coordinates, r^a=rθ^a ($\text{a=1,…,}\text{9−p;}\text{∑}{}_{\mathrm{a}}\text{(θ}{}^{\mathrm{a}}\text{)}{}^2\text{=1}$). The metric and the dilaton for the extremal p-brane are then written as

$$\text{d}\text{s}{}^2\text{=}\text{1}\text{H(r)}\text{−}\text{d}\text{t}{}^2\text{+}\text{∑}\text{i=1}\text{p}\text{d}\text{x}{}^{\mathrm{i}}\text{d}\text{x}{}^{\mathrm{i}}\text{+}\text{H(r)}\text{∑}\text{a=1}\text{9−p}\text{d}\text{r}{}^{\mathrm{a}}\text{d}\text{r}{}^{\mathrm{a}}\text{,}$$

$$\text{e}{}^{\mathrm{\phi }}\text{=g}{}_{\mathrm{<mtext>s</mtext>}}\text{H(r)}{}^{\mathrm{(3-p)/4}}\text{,}$$

where

$$\text{H(r)=}\text{1}\text{f}{}_{\mathrm{+}}\text{(ρ)}\text{=1+}\text{r}{}_{\mathrm{+}}{}^{\mathrm{7-p}}\text{r}{}^{\mathrm{7-p}}\text{,}\text{r}{}_{\mathrm{+}}{}^{\mathrm{7-p}}\text{=d}{}_{\mathrm{p}}\text{g}{}_{\mathrm{<mtext>s</mtext>}}\text{Nl}{}_{\mathrm{<mtext>s</mtext>}}{}^{\mathrm{7-p}}\text{.}$$

The horizon is now located at r=0.

In general, , give a solution to the supergravity equations of motion for any function H(r) which is a harmonic function in the (9−p) dimensions which are transverse to the p-brane. For example, we may consider a more general solution, of the form

$$\text{H(}\text{r}\text{)=1+}\text{∑}\text{i=1}\text{k}\text{r}{}_{\mathrm{(i)+}}{}^{\mathrm{7-p}}\text{|}\text{r}\text{−}\text{r}{}_{\mathrm{i}}\text{|}{}^{\mathrm{7-p}}\text{,}\text{r}{}_{\mathrm{(i)+}}{}^{\mathrm{7-p}}\text{=d}{}_{\mathrm{p}}\text{g}{}_{\mathrm{<mtext>s</mtext>}}\text{N}{}_{\mathrm{i}}\text{l}{}_{\mathrm{<mtext>s</mtext>}}{}^{\mathrm{7-p}}\text{.}$$

This is called a multi-centered solution and represents parallel extremal p-branes located at k different locations, $\text{r}\text{=}\text{r}{}_{\mathrm{i}}\text{(i=1,…,}\text{k}$), each of which carries N_i units of the R–R charge.

So far we have discussed the black p-brane using the classical supergravity. This description is appropriate when the curvature of the p-brane geometry is small compared to the string scale, so that stringy corrections are negligible. Since the strength of the curvature is characterized by r₊, this requires r₊⪢l_s. To suppress string loop corrections, the effective string coupling e^φ also needs to be kept small. When p=3, the dilaton is constant and we can make it small everywhere in the 3-brane geometry by setting g_s<1, namely l_P<l_s. If g_s>1 we might need to do an S-duality, g_s→1/g_s, first. Moreover, in this case it is known that the metric (1.21) can be analytically extended beyond the horizon r=0, and that the maximally extended metric is geodesically complete and without a singularity [60]. The strength of the curvature is then uniformly bounded by r₊⁻². To summarize, for p=3, the supergravity approximation is valid when

$$\text{l}{}_{\mathrm{<mtext>P</mtext>}}\text{<l}{}_{\mathrm{<mtext>s</mtext>}}\text{⪡r}{}_{\mathrm{+}}\text{.}$$

Since r₊ is related to the R–R charge N as

$$\text{r}{}_{\mathrm{+}}{}^{\mathrm{7-p}}\text{=d}{}_{\mathrm{<mtext>p</mtext>}}\text{g}{}_{\mathrm{<mtext>s</mtext>}}\text{Nl}{}_{\mathrm{<mtext>s</mtext>}}{}^{\mathrm{7-p}}\text{,}$$

this can also be expressed as

$$\text{1⪡g}{}_{\mathrm{<mtext>s</mtext>}}\text{N<N}\text{.}$$

For p≠3, the metric is singular at r=0, and the supergravity description is valid only in a limited region of the space time.

Alternatively, the extremal p-brane can be described as a D-brane. For a review of D-branes, see [61]. The Dp-brane is a (p+1)-dimensional hyperplane in space time where an open string can end. By the worldsheet duality, this means that the D-brane is also a source of closed strings (see Fig. 1.2). In particular, it can carry the R–R charges. It was shown in [6] that, if we put N Dp-branes on top of each other, the resulting (p+1)-dimensional hyperplane carries exactly N units of the (p+1)-form charge. On the worldsheet of a type II string, the left-moving degrees of freedom and the right-moving degrees of freedom carry separate space time supercharges. Since the open string boundary condition identifies the left and right movers, the D-brane breaks at least one-half of the space time supercharges. In type IIA (IIB) string theory, precisely one-half of the supersymmetry is preserved if p is even (odd). This is consistent with the types of R–R charges that appear in the theory. Thus, the Dp-brane is a BPS object in string theory which carries exactly the same charge as the black p-brane solution in supergravity.

It is believed that the extremal p-brane in supergravity and the Dp-brane are two different descriptions of the same object. The D-brane uses the string worldsheet and, therefore, is a good description in string perturbation theory. When there are N D-branes on top of each other, the effective loop expansion parameter for the open strings is g_sN rather than g_s, since each open string boundary loop ending on the D-branes comes with the Chan–Paton factor N as well as the string coupling g_s. Thus, the D-brane description is good when g_sN⪡1. This is complementary to the regime (1.27) where the supergravity description is appropriate.

The low-energy effective theory of open strings on the Dp-brane is the U(N) gauge theory in (p+1) dimensions with 16 supercharges [9]. The theory has (9−p) scalar fields Φ in the adjoint representation of U(N). If the vacuum expectation value 〈Φ〉 has k distinct eigenvalues,⁵ with N₁ identical eigenvalues $\text{φ}{}_1\text{,}\text{N}{}_2$ identical eigenvalues φ₂ and so on, the gauge group U(N) is broken to U(N₁)×⋯×U(N_k). This corresponds to the situation when N₁ D-branes are at $\text{r}{}_1\text{=}\text{φ}{}_1\text{l}{}_{\mathrm{<mtext>s</mtext>}}{}^2\text{,}\text{N}{}_2$ Dp-branes are at r₂=φ₂l_s², and so on. In this case, there are massive W-bosons for the broken gauge groups. The W-boson in the bi-fundamental representation of U(N_i)×U(N_j) comes from the open string stretching between the D-branes at r_i and r_j, and the mass of the W-boson is proportional to the Euclidean distance |r_i−r_j| between the D-branes. It is important to note that the same result is obtained if we use the supergravity solution for the multi-centered p-brane (1.24) and compute the mass of the string going from r_i to r_j, since the factor H(r)^1/4 from the metric in the r-space (1.21) is cancelled by the redshift factor H(r)^−1/4 when converting the string tension into energy. Both the D-brane description and the supergravity solution give the same value of the W-boson mass, since it is determined by the BPS condition.

An important precursor to the AdS/CFT correspondence was the calculation of greybody factors for black holes built out of D-branes. It was noted in [14] that Hawking radiation could be mimicked by processes where two open strings collide on a D-brane and form a closed string which propagates into the bulk. The classic computation of Hawking (see, for example, [62] for details) shows in a semi-classical approximation that the differential rate of spontaneous emission of particles of energy ω from a black hole is

where v is the velocity of the emitted particle in the transverse directions, and the sign in the denominator is minus for bosons and plus for fermions. We use n to denote the number of spatial dimensions around the black hole (or if we are dealing with a black brane, it is the number of spatial dimensions perpendicular to the world-volume of the brane). T_H is the Hawking temperature, and σ_absorb is the cross-section for a particle coming in from infinity to be absorbed by the black hole. In the differential emission rate, the emitted particle is required to have a momentum in a small region dⁿk, and ω is a function of k. To obtain a total emission rate we would integrate (1.28) over all k.

If σ_absorb were a constant, then (1.28) tells us that the emission spectrum is the same as that of a blackbody. Typically, σ_absorb is not constant, but varies appreciably over the range of finite ω/T_H. The consequent deviations from the pure blackbody spectrum have earned σ_absorb the name “greybody factor”. A successful microscopic account of black hole thermodynamics should be able to predict these greybody factors. In [16] and its many successors, it was shown that the D-branes provided an account of black hole microstates which was successful in this respect.

Our first goal will be to see how greybody factors are computed in the context of quantum fields in curved space-time. The literature on this subject is immense. We refer the reader to [63] for an overview of the General Relativity literature, and to [18], [11], [59] and references therein for a first look at the string theory additions.

In studying scattering of particles off of a black hole (or any fixed target), it is convenient to make a partial wave expansion. For simplicity, let us restrict the discussion to scalar fields. Assuming that the black hole is spherically symmetric, one can write the asymptotic behavior at infinity of the time-independent scattering solution as

$$\text{φ(}\text{r}\text{)∼}\text{e}{}^{\mathrm{<mtext>i</mtext><mtext>kx</mtext>}}\text{+f}\text{(θ)}\text{e}{}^{\mathrm{<mtext>i</mtext><mtext>kr</mtext>}}\text{r}{}^{\mathrm{n/2}}\text{∼}\text{∑}\text{ℓ=0}\text{∞}\text{1}\text{2}\text{P}\text{̃}{}_{\mathrm{\ell }}\text{(}\text{cos}\text{θ)}\text{S}{}_{\mathrm{\ell }}\text{e}{}^{\mathrm{<mtext>i</mtext><mtext>kr</mtext>}}\text{+(−1)}{}^{\mathrm{\ell }}\text{i}{}^{\mathrm{n}}\text{e}{}^{\mathrm{<mtext>-</mtext><mtext>i</mtext><mtext>kr</mtext>}}\text{(}\text{i}\text{kr)}{}^{\mathrm{n/2}}\text{,}$$

where $\text{x=r}\text{cos}\text{θ}$. The term e^ikx represents the incident wave, and the second term in the first line represents the scattered wave. The $\text{P}\text{̃}{}_{\mathrm{\ell }}\text{(}\text{cos}\text{θ)}$ are generalizations of Legendre polynomials. The absorption probability for a given partial wave is given by P_ℓ=1−|S_ℓ|². An application of the Optical Theorem leads to the absorption cross section [64]

$$\text{σ}{}_{\mathrm{<mtext>abs</mtext>}}{}^{\mathrm{\ell }}\text{=}\text{2}{}^{\mathrm{n-1}}\text{π}{}^{\mathrm{(n-1)/2}}\text{k}{}^{\mathrm{n}}\text{Γ}\text{n−1}\text{2}\text{ℓ+}\text{n−1}\text{2}\text{ℓ+n−2}\text{ℓ}\text{P}{}_{\mathrm{\ell }}\text{.}$$

Sometimes the absorption probability P_ℓ is called the greybody factor.

The strategy of absorption calculations in supergravity is to solve a linearized wave equation, most often the Klein–Gordon equation □φ=0, using separation of variables, $\text{φ=}\text{e}{}^{\mathrm{<mtext>-i</mtext><mtext>\omega t</mtext>}}\text{P}{}_{\mathrm{\ell }}\text{(}\text{cos}\text{θ)R(r)}$. Typically, the radial function cannot be expressed in terms of known functions, so some approximation scheme is used, as we will explain in more detail below. Boundary conditions are imposed at the black hole horizon corresponding to infalling matter. Once the solution is obtained, one can either use the asymptotics (1.29) to obtain S_ℓ and from it P_ℓ and σ_abs^ℓ, or compute the particle flux at infinity and at the horizon and note that particle number conservation implies that P_ℓ is their ratio.

One of the few known universal results is that for $\text{ω/T}{}_{\mathrm{<mtext>H</mtext>}}\text{⪡1,}\text{σ}{}_{\mathrm{<mtext>abs</mtext>}}$ for an s-wave massless scalar approaches the horizon area of the black hole [65]. This result holds for any spherically symmetric black hole in any dimension. For ω much larger than any characteristic curvature scale of the geometry, one can use the geometric optics approximation to find σ_abs.

We will be interested in the particular black hole geometries for which string theory provides a candidate description of the microstates. Let us start with N coincident D3-branes, where the low-energy world-volume theory is $\text{d=4}\text{N}\text{=4}\text{U(N)}$ gauge theory. The equation of motion for the dilaton is □φ=0 where □ is the Laplacian for the metric

$$\text{d}\text{s}{}^2\text{=}\text{1+}\text{R}{}^4\text{r}{}^4{}^{\mathrm{-1/2}}\text{(−}\text{d}\text{t}{}^2\text{+}\text{d}\text{x}{}_1{}^2\text{+}\text{d}\text{x}{}_2{}^2\text{+}\text{d}\text{x}{}_3{}^2\text{)+}\text{1+}\text{R}{}^4\text{r}{}^4{}^{\mathrm{1/2}}\text{(}\text{d}\text{r}{}^2\text{+r}{}^2\text{d}\text{Ω}{}_5{}^2\text{)}\text{.}$$

It is convenient to change radial variables: $\text{r=R}\text{e}{}^{\mathrm{-z}}\text{,}\text{φ=}\text{e}{}^{\mathrm{2z}}\text{ψ}$. The radial equation for the ℓth partial wave is

$$\text{[}\text{∂}{}_{\mathrm{z}}{}^2\text{+2ω}{}^2\text{R}{}^2\text{cosh}\text{2z−(ℓ+2)}{}^2\text{]ψ}{}_{\mathrm{\ell }}\text{(z)=0}\text{,}$$

which is precisely Schrodinger's equation with a potential $\text{V(z)=−2ω}{}^2\text{R}{}^2\text{cosh}\text{2z}$. The absorption probability is precisely the tunneling probability for the barrier V(z): the transmitted wave at large positive z represents particles falling into the D3-branes. At leading order in small ωR, the absorption probability for the ℓth partial wave is

$$\text{P}{}_{\mathrm{\ell }}\text{=}\text{4}\text{π}{}^2\text{(ℓ+1)!}{}^4\text{(ℓ+2)}{}^2\text{ωR}\text{2}{}^{\mathrm{8+4\ell }}\text{.}$$

This result, together with a recursive algorithm for computing all corrections as a series in ωR, was obtained in [66] from properties of associated Mathieu functions, which are the solutions of (1.32). An exact solution of a radial equation in terms of known special functions is rare. We will therefore present a standard approximation technique (developed in [67] and applied to the problem at hand in [10]) which is sufficient to obtain the leading term of (1.33). Besides, for comparison with string theory predictions we are generally interested only in this leading term.

The idea is to find limiting forms of the radial equation which can be solved exactly, and then to match the limiting solutions together to approximate the full solution. Usually, a uniformly good approximation can be found in the limit of small energy. The reason, intuitively speaking, is that on a compact range of radii excluding asymptotic infinity and the horizon, the zero energy solution is nearly equal to solutions with very small energy; and outside this region the wave equation usually has a simple limiting form. So one solves the equation in various regions and then matches together a global solution.

It is elementary to show that this can be done for (1.32) using two regions:

$$\text{far}\text{region:}\text{z⪢}\text{log}\text{ωR}\text{[}\text{∂}{}_{\mathrm{z}}{}^2\text{+ω}{}^2\text{R}{}^2\text{e}{}^{\mathrm{2z}}\text{−(ℓ+2)}{}^2\text{]ψ=0}\text{ψ(z)=H}{}^{\mathrm{(1)}}{}_{\mathrm{\ell +2}}\text{(ωR}\text{e}{}^{\mathrm{z}}\text{)}\text{near}\text{region:}\text{z⪡−}\text{log}\text{ωR}\text{[}\text{∂}{}_{\mathrm{z}}{}^2\text{+ω}{}^2\text{R}{}^2\text{e}{}^{\mathrm{-2z}}\text{−(ℓ+2)}{}^2\text{]ψ=0}\text{ψ(z)=aJ}{}_{\mathrm{\ell +2}}\text{(ωR}\text{e}{}^{\mathrm{-z}}\text{)}$$

It is amusing to note the $\text{Z}{}_2$ symmetry, z→−z, which exchanges the far region, where the first equation in (1.34) is just free particle propagation in flat space, and the near region, where the second equation in (1.34) describes a free particle in AdS₅. This peculiar symmetry was first pointed out in [10]. It follows from the fact that the full D3-brane metric comes back to itself, up to a conformal rescaling, if one sends r→R²/r. A similar duality exists between six-dimensional flat space and AdS₃×S³ in the (D1–D5)-brane solution, where the Laplace equation again can be solved in terms of Mathieu functions [68], [69]. To our knowledge there is no deep understanding of this “inversion duality”.

For low energies ωR⪡1, the near and far regions overlap in a large domain, $\text{log}\text{ωR⪡z⪡−}\text{log}\text{ωR}$, and by comparing the solutions in this overlap region one can fix a and reproduce the leading term in (1.33). It is possible but tedious to obtain the leading correction by treating the small terms which were dropped from the potential to obtain the limiting forms in (1.34) as perturbations. This strategy was pursued in [70], [71] before the exact solution was known, and in cases where there is no exact solution. The validity of the matching technique is discussed in [63], but we know of no rigorous proof that it holds in all the circumstances in which it has been applied.

The successful comparison of the s-wave dilaton cross-section in [10] with a perturbative calculation on the D3-brane world-volume was the first hint that Green's functions of $\text{N}\text{=4}$ super-Yang–Mills theory could be computed from supergravity. In summarizing the calculation, we will follow more closely the conventions in [11], and give an indication of the first application of non-renormalization arguments [12] to understand why the agreement between supergravity and perturbative gauge theory existed despite their applicability in opposite limits of the 't Hooft coupling.

Setting normalization conventions so that the pole in the propagator of the gauge bosons has residue one at tree level, we have the following action for the dilaton plus the fields on the brane:

$$\text{S=}\text{1}\text{2κ}{}^2\text{∫}\text{d}{}^{10}\text{x}\text{g}\text{[}\text{R}\text{−}\text{1}\text{2}\text{(}\text{∂}\text{φ)}{}^2\text{+⋯]+∫}\text{d}{}^4\text{x}\text{[−}\text{1}\text{4}\text{e}{}^{\mathrm{-\phi }}\text{Tr}\text{F}{}_{\mathrm{\mu \nu }}{}^2\text{+⋯]}\text{,}$$

where we have omitted other supergravity fields, their interactions with one another, and also terms with the lower spin fields in the gauge theory action. A plane wave of dilatons with energy ω and momentum perpendicular to the brane is kinematically equivalent on the world-volume to a massive particle which can decay into two gauge bosons through the coupling $\text{1}\text{4}\text{φ}\text{Tr}\text{F}{}_{\mathrm{\mu \nu }}{}^2$. In fact, the absorption cross-section is given precisely by the usual expression for the decay rate into k particles:

$$\text{σ}{}_{\mathrm{<mtext>abs</mtext>}}\text{=}\text{1}\text{S}{}_{\mathrm{<mtext>f</mtext>}}\text{1}\text{2ω}\text{∫}\text{d}{}^3\text{p}{}_1\text{(2}\text{π}\text{)}{}^3\text{2ω}{}_1\text{⋯}\text{d}{}^3\text{p}{}_{\mathrm{k}}\text{(2}\text{π}\text{)}{}^3\text{2ω}{}_{\mathrm{k}}\text{(2}\text{π}\text{)}{}^4\text{δ}{}^4\text{(P}{}_{\mathrm{<mtext>f</mtext>}}\text{−P}{}_{\mathrm{<mtext>i</mtext>}}\text{)}\text{|}\text{M}\text{|}{}^2\text{.}$$

In the Feynman rules for $\text{M}$, a factor of $\text{2κ}{}^2$ attaches to an external dilaton line on account of the non-standard normalization of its kinetic term in (1.35). This factor gives σ_abs the correct dimensions: it is a length to the fifth power, as appropriate for six transverse spatial dimensions. In (1.36), $\text{|}\text{M}\text{|}{}^2$ indicates summation over distinguishable processes: in the case of the s-wave dilaton there are N² such processes because of the number of gauge bosons. One easily verifies that $\text{|}\text{M}\text{|}{}^2\text{=N}{}^2\text{κ}{}^2\text{ω}{}^4$. S_f is a symmetry factor for identical particles in the final state: in the case of the s-wave dilaton, S_f=2 because the outgoing gauge bosons are identical.

Carrying out the ℓ=0 calculation explicitly, one finds

$$\text{σ}{}_{\mathrm{<mtext>abs</mtext>}}\text{=N}{}^2\text{κ}{}^2\text{ω}{}^3\text{/32}\text{π}\text{,}$$

which, using (1.30) and the relation between R and N, can be shown to be in precise agreement with the leading term of P₀ in (1.33). This is now understood to be due to a non-renormalization theorem for the two-point function of the operator $\text{O}{}_4\text{=}\text{1}\text{4}\text{Tr}\text{F}{}^2$.

To understand the connection with two-point functions, note that an absorption calculation is insensitive to the final state on the D-brane world-volume. The absorption cross-section is therefore related to the discontinuity in the cut of the two-point function of the operator to which the external field couples. To state a result of some generality, let us suppose that a scalar field φ in ten dimensions couples to a gauge theory operator through the action

where we use x to denote the four coordinates parallel to the world-volume and y_i to denote the other six. An example where this would be the right sort of coupling is the ℓth partial wave of the dilaton [11]. The ℓth partial wave absorption cross-section for a particle with initial momentum $\text{p=ω(}\text{t}\text{̂}\text{+}\text{y}\text{̂}{}_1\text{)}$ is obtained by summing over all final states that could be created by the operator $\text{O}{}^{\mathrm{1\dots 1}}$:⁶

$$\text{σ}{}_{\mathrm{<mtext>abs</mtext>}}\text{=}\text{1}\text{2ω}\text{∑}\text{n}\text{∏}\text{i=1}\text{n}\text{1}\text{S}{}_{\mathrm{<mtext>f</mtext>}}\text{d}{}^3\text{p}{}_{\mathrm{i}}\text{(2}\text{π}\text{)}{}^3\text{2ω}{}_{\mathrm{i}}\text{(2}\text{π}\text{)}{}^4\text{δ}{}^4\text{(P}{}_{\mathrm{f}}\text{−P}{}_{\mathrm{<mtext>i</mtext>}}\text{)}\text{|}\text{M}\text{|}{}^2\text{=}\text{2κ}{}^2\text{ω}{}^{\mathrm{\ell }}\text{2}\text{i}\text{ω}\text{Disc}\text{∫}\text{d}{}^4\text{x}\text{e}{}^{\mathrm{<mtext>i</mtext><mtext>p·x</mtext>}}\text{〈}\text{O}{}^{\mathrm{1\dots 1}}\text{(x)}\text{O}{}^{\mathrm{1\dots 1}}\text{(0)〉}{}_{\mathrm{p=(\omega ,0,0,0)}}\text{.}$$

In the second equality we have applied the Optical Theorem (see Fig. 1.3). The factor of 2κ² is the square of the external leg factor for the incoming closed string state, which was included in the invariant amplitude $\text{M}$. The factor of ω^ℓ arises from acting with the ℓ derivatives in (1.38) on the incoming plane wave state. The symbol Disc indicates that one is looking at the unitarity cut in the two-point function in the s plane, where s=p². The two-point function can be reconstructed uniquely from this cut, together with some weak conditions on regularity for large momenta. Results analogous to (1.39) can be stated for incoming particles with spin, only it becomes more complicated because a polarization must be specified, and the two-point function in momentum space includes a polynomial in p which depends on the polarization.

Expressing absorption cross-sections in terms of two-point functions helps illustrate why there is ever agreement between the tree-level calculation indicated in (1.36) and the supergravity result, which one would a priori expect to pick up all sorts of radiative corrections. Indeed, it was observed in [12] that the s-wave graviton cross-section agreed between supergravity and tree-level gauge theory because the correlator 〈T_αβT_γδ〉 enjoys a non-renormalization theorem. One way to see that there must be such a non-renormalization theorem is to note that conformal Ward identities relate this two-point function to 〈T^μ_μT_αβT_γδ〉 (see for example [72] for the details), and supersymmetry in turn relates this anomalous three-point function to the anomalous VEV's of the divergence of R-currents in the presence of external sources for them. The Adler-Bardeen theorem [73] protects these anomalies, hence the conclusion.

Another case which has been studied extensively is a system consisting of several D1 and D5 branes. The D1-branes are delocalized on the four extra dimensions of the D5-brane, which are taken to be small, so that the total system is effectively (1+1)-dimensional. We will discuss the physics of this system more extensively in chapter 5, and the reader can also find background material in [59]. For now our purpose is to show how supergravity absorption calculations relate to finite-temperature Green's functions in the (1+1)-dimensional theory.

Introducing momentum along the spatial world-volume (carried by open strings attached to the branes), one obtains the following ten-dimensional metric and dilaton:

$$\text{d}\text{s}{}_{\mathrm{<mtext>10,</mtext><mtext>str</mtext>}}{}^2\text{=H}{}_1{}^{\mathrm{-1/2}}\text{H}{}_5{}^{\mathrm{-1/2}}\text{−}\text{d}\text{t}{}^2\text{+}\text{d}\text{x}{}_5{}^2\text{+}\text{r}{}_0{}^2\text{r}{}^2\text{(}\text{cosh}\text{σ}\text{d}\text{t+}\text{sinh}\text{σ}\text{d}\text{x}{}_5\text{)}{}^2\text{+H}{}_1{}^{\mathrm{1/2}}\text{H}{}_5{}^{\mathrm{-1/2}}\text{∑}\text{i=1}\text{4}\text{d}\text{y}{}_{\mathrm{i}}{}^2\text{+H}{}_1{}^{\mathrm{1/2}}\text{H}{}_5{}^{\mathrm{1/2}}\text{1−}\text{r}{}_0{}^2\text{r}{}^2{}^{\mathrm{-1}}\text{d}\text{r}{}^2\text{+r}{}^2\text{d}\text{Ω}{}_3{}^2$$

$$\text{H}{}_1\text{=1+}\text{r}{}_1{}^2\text{r}{}^2\text{,}\text{H}{}_5\text{=1+}\text{r}{}_5{}^2\text{r}{}^2\text{.}$$

The quantities r₁², r₅², and $\text{r}{}_{\mathrm{K}}{}^2\text{=r}{}_0{}^2\text{sinh}{}^2\text{σ}$ are related to the number of D1-branes, the number of D5-branes, and the net number of units of momentum in the x₅ direction, respectively. The horizon radius, r₀, is related to the non-extremality. For details, see for example [18]. If r₀=0 then there are only left-moving open strings on the world-volume; if r₀≠0 then there are both left-movers and right-movers. The Hawking temperature can be expressed as 2/T_H=1/T_L+1/T_R, where

$$\text{T}{}_{\mathrm{<mtext>L</mtext>}}\text{=}\text{1}\text{π}\text{r}{}_0\text{e}{}^{\mathrm{\sigma }}\text{2r}{}_1\text{r}{}_5\text{,}\text{T}{}_{\mathrm{<mtext>R</mtext>}}\text{=}\text{1}\text{π}\text{r}{}_0\text{e}{}^{\mathrm{-\sigma }}\text{2r}{}_1\text{r}{}_5\text{.}$$

T_L and T_R have the interpretation of temperatures of the left-moving and right-moving sectors of the (1+1)-dimensional world-volume theory. There is a detailed and remarkably successful account of the Bekenstein–Hawking entropy using statistical mechanics in the world-volume theory. It was initiated in [13], developed in a number of subsequent papers, and has been reviewed in [59].

The region of parameter space which we will be interested in is

$$\text{r}{}_0\text{,}\text{r}{}_{\mathrm{K}}\text{⪡r}{}_1\text{,}\text{r}{}_5$$

This is known as the dilute gas regime. The total energy of the open strings on the branes is much less than the constituent mass of either the D1-branes or the D5-branes. We are also interested in low energies $\text{ωr}{}_1\text{,}\text{ωr}{}_5\text{⪡1}$, but ω/T_L,R can be arbitrary owing to , . The (D1–D5)-brane system is not stable because left-moving open strings can run into right-moving open string and form a closed string: indeed, this is exactly the process we aim to quantify. Since we have collisions of left and right moving excitations we expect that the answer will contain the left and right moving occupation factors, so that the emission rate is

where g_eff is independent of the temperature and measures the coupling of the open strings to the closed strings. The functional form of (1.43) seems, at first sight, to be different from (1.28). But in order to compare them we should calculate the absorption cross section appearing in (1.28).

Off-diagonal gravitons h_y1y2 (with y_1,2 in the compact directions) reduce to scalars in six dimensions which obey the massless Klein–Gordon equation. These so-called minimal scalars have been the subject of the most detailed study. We will consider only the s-wave and we take the momentum along the string to be zero. Separation of variables leads to the radial equation

$$\text{h}\text{r}{}^3\text{∂}{}_{\mathrm{r}}\text{hr}{}^2\text{∂}{}_{\mathrm{r}}\text{+ω}{}^2\text{f}\text{R=0}\text{,}\text{h=1−r}{}_0{}^2\text{/r}{}^2\text{,}\text{f=(1+r}{}_1{}^2\text{/r}{}^2\text{)(1+r}{}_5{}^2\text{/r}{}^2\text{)(1+r}{}_0{}^2\text{sinh}{}^2\text{σ/r}{}^2\text{)}\text{.}$$

Close to the horizon, a convenient radial variable is z=h=1−r₀²/r². The matching procedure can be summarized as follows:

$$\text{far}\text{region:}\text{1}\text{r}{}^3\text{∂}{}_{\mathrm{r}}\text{r}{}^3\text{∂}{}_{\mathrm{r}}\text{+ω}{}^2\text{R=0}\text{,}\text{R=A}\text{J}{}_1\text{(ωr)}\text{r}{}^{\mathrm{3/2}}\text{,}$$

After matching the near and far regions together and comparing the infalling flux at infinity and at the horizon, one arrives at

This has precisely the right form to ensure the matching of (1.43) with (1.28) (note that for a massless particle with no momentum along the black string v=1 in (1.28)). It is possible to be more precise and calculate the coefficient in (1.43) and actually check that the matching is precise [16]. We leave this to Section 5.

Both in the D3-brane analysis and in the (D1–D5)-brane analysis, one can see that all the interesting physics is resulting from the near-horizon region: the far region wave-function describes free particle propagation. For quanta whose Compton wavelength is much larger than the size of the black hole, the effect of the far region is merely to set the boundary conditions in the near region. See Fig. 1.4. This provides a motivation for the prescription for computing Green's functions, to be described in Section 3.3: as the calculations of this section demonstrate, cutting out the near-horizon region of the supergravity geometry and replacing it with the D-branes leads to an identical response to low-energy external probes.