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ar X iv :h ep -t h /9 8 1 2 0 3 7 v 2 1 1 D ec 1 9 9 8 CALT-68-2204 hep-th/9812037 Introduction to M Theory and AdS/CFT Duality1 John H. Schwarz California Institute of Technology, Pasadena, CA 91125, USA Abstract An introductory survey of some of the developments that have taken place in superstring theory in the past few years is presented. The main focus is on three particular dualities. The first one is the appearance of an 11th dimension in the strong coupling limit of the type IIA theory, which give rise to M theory. The second one is the duality between the type IIB theory compactified on a circle and M theory on a two-torus. The final topic is an introduction to the recently proposed duality between superstring theory or M theory on certain anti de Sitter space backgrounds and conformally invariant quantum field theories. To be published in the Proceedings of Quantum Aspects of Gauge Theories, Supersymmetry, and Unification Corfu, Greece – September 1998 1Work supported in part by the U.S. Dept. of Energy under Grant No. DE-FG03-92-ER40701. 1 Introduction It is a pleasure to speak in such a remarkable venue – the old fortress of Corfu City. It is not often that one gets to meet inside a tourist attraction. The last such occasion for me was at a conference that took place inside the Chateau de Blois. Many of the talks in this conference will cover recent developments in superstring theory and M theory – in particular, the recent AdS/CFT conjecture. Since the organizers have chosen to place me first in the schedule, and since not everyone here is an expert in these matters, I have decided (in consultation with the organizers) to give a rather general intro- duction to some of these recent developments. Hopefully, this will help to provide some of the background that is needed for the more specialized talks that will follow. It should also relieve those speakers of the need to give an extended review of the basics. As I’m sure most of you know, the term M theory was introduced by Witten to describe the 11-dimensional quantum theory whose low energy effective description is 11-dimensional supergravity. However, the usage of this term has been extended by many authors (including myself on occasion) to refer to the underlying theory that reduces to the five different 10- dimensional superstring theories in various special limits, as well as the flat 11-dimensional theory in a sixth special limit. This is somewhat confusing. Therefore, in a recent talk at the Vancouver conference, Sen proposed that the term M theory should be reserved for the 11- dimensional quantum theory and that the (largely unknown) underlying fundamental theory, which is not specific to any particular spacetime dimension, should be called U Theory [1]. He suggested that U could stand for either “unknown” or “unified”. This nomenclature makes a lot of sense to me, so I will try to adhere to it. In the text that follows, I will use the term M theory and not the term U theory. However, its usage will occur only in the sense that Witten originally intended, namely an 11-dimensional quantum theory. In the first half of this talk (in sections 2 – 5) I will survey some of the basic facts about type IIA and type IIB superstrings in 10 dimensions and the dualities that related them to M theory. Aside from some minor editing, these sections are copied from a review that I wrote earlier this year [2]. Then, in section 6 I will give an introduction to the remarkable duality that has been proposed between superstring theory or M theory in certain anti de Sitter spacetime backgrounds and conformally invariant field theories. To be specific, I will focus on the duality between type IIB superstring theory in an AdS5×S5 background and N = 4 supersymmetric gauge theory. As I have already indicated, in this talk I will only survey 1 some of the basics, and leave the discussion of more advanced aspects of this subject to other speakers. For a more detailed survey of superstring theory and M theory I recommend the review paper written by Sen [3]. Polchinski’s new textbook is also recommended [4]. 2 Perturbative Superstring Theory Superstring theory first achieved widespread acceptance during the first superstring revolu- tion in 1984-85. There were three main developments at this time. The first was the discovery of an anomaly cancellation mechanism [5], which showed that supersymmetric gauge theories can be consistent in ten dimensions provided they are coupled to supergravity (as in type I superstring theory) and the gauge group is either SO(32) or E8 × E8. Any other group necessarily would give uncanceled gauge anomalies and hence inconsistency at the quantum level. The second development was the discovery of two new superstring theories—called heterotic string theories—with precisely these gauge groups [6]. The third development was the realization that the E8 ×E8 heterotic string theory admits solutions in which six of the space dimensions form a Calabi–Yau space, and that this results in a 4d effective theory at low energies with many qualitatively realistic features [7]. Unfortunately, there are very many Calabi–Yau spaces and a whole range of additional choices that can be made (orb- ifolds, Wilson loops, etc.). Thus there is an enormous variety of possibilities, none of which stands out as particularly special. In any case, after the first superstring revolution subsided, we had five distinct superstring theories with consistent weak coupling perturbation expansions, each in ten dimensions. Three of them, the type I theory and the two heterotic theories, have N = 1 supersymmetry in the ten-dimensional sense. Since the minimal 10d spinor is simultaneously Majorana and Weyl, this corresponds to 16 conserved supercharges. The other two theories, called type IIA and type IIB, have N = 2 supersymmetry (32 supercharges) [8]. In the IIA case the two spinors have opposite handedness so that the spectrum is left-right symmetric (nonchiral). In the IIB case the two spinors have the same handedness and the spectrum is chiral. The understanding of these five superstring theories was developed in the ensuing years. In each case it became clear, and was largely proved, that there are consistent perturbation expansions of on-shell scattering amplitudes. In four of the five cases (heterotic and type II) the fundamental strings are oriented and unbreakable. As a result, these theories have particularly simple perturbation expansions. Specifically, there is a unique Feynman diagram 2 at each order of the loop expansion. The Feynman diagrams depict string world sheets, and therefore they are two-dimensional surfaces. For these four theories the unique L- loop diagram is a closed orientable genus-L Riemann surface, which can be visualized as a sphere with L handles. External (incoming or outgoing) particles are represented by N points (or “punctures”) on the Riemann surface. A given diagram represents a well- defined integral of dimension 6L+2N −6. This integral has no ultraviolet divergences, even though the spectrum contains states of arbitrarily high spin (including a massless graviton). From the viewpoint of point-particle contributions, string and supersymmetry properties are responsible for incredible cancellations. Type I superstrings are unoriented and breakable. As a result, the perturbation expansion is more complicated for this theory, and the various world-sheet diagrams at a given order (determined by the Euler number) have to be combined properly to cancel divergences and anomalies [9]. 2.1 T Duality An important discovery that was made between the two superstring revolutions is called T duality [10]. This is a property of string theories that can be understood within the context of perturbation theory. (The discoveries associated with the second superstring revolution are mostly nonperturbative.) T duality shows that spacetime geometry, as probed by strings, has some surprising properties (sometimes referred to as quantum geometry). The basic idea can be illustrated by the simplest example in which one spatial dimension forms a circle (denoted S1). Then the ten-dimensional geometry is R9×S1. T duality identifies this string compactification with one of a second string theory also on R9 × S1. However, if the radii of the circles in the two dual descriptions are denoted R1 and R2, then R1R2 = α ′. (1) Here α′ = ℓ2s is the universal Regge slope parameter, and ℓs is the fundamental string length scale (for both string theories). The tension of a fundamental string is given by T = 2πm2s = 1 2πα′ , where we have introduced a fundamental string mass scale ms = (2πℓs) −1. Note that T duality implies that shrinking the circle to zero in one theory corresponds to decompactification of the dual theory. Compactification on a circle of radius R implies 3 that momenta in that direction are quantized, p = n/R. (These are called Kaluza–Klein excitations.) These momenta appear as masses for states that are massless from the higher- dimensional viewpoint. String theories also have a second class of excitations, called winding modes. Namely, a string wound m times around the circle has energy E = 2πR ·m · T = mR/α′ . Equation (1) shows that the winding modes and Kaluza–Klein excitations are interchanged under T duality. What does T duality imply for our five superstring theories? The IIA and IIB theories are T dual [11]. So compactifying the nonchiral IIA theory on a circle of radius R and letting R → 0 gives the chiral IIB theory in ten dimensions! This means, in particular, that they should not be regarded as distinct theories. The radius R is actually a vev of a scalar field, which arises as an internal component of the 10d metric tensor. Thus the type IIA and type IIB theories in 10d are two limiting points in a continuous moduli space of quantum vacua. The two heterotic theories are also T dual, though there are technical details involving Wilson loops, which we will not explain here. T duality applied to the type I theory gives a dual description, which is sometimes called I′. The names IA and IB have also been introduced by some authors. For the remainder of this paper, we will restrict attention to theories with maximal supersymmetry (32 conserved supercharges). This is sufficient to describe the basic ideas of M theory. Of course, it suppresses many fascinating and important issues and discoveries. In this way we will keep the presentation from becoming too long or too technical. The main focus will be to ask what happens when we go beyond perturbation theory and allow the coupling strength to become large in the type II theories. The answer in the IIA case, as we will see, is that another spatial dimension appears. 3 M Theory In the 1970s and 1980s various supersymmetry and supergravity theories were constructed. (See [12], for example.) In particular, supersymmetry representation theory showed that ten is the largest spacetime dimension in which there can be a matter theory (with spins ≤ 1) in which supersymmetry is realized linearly. A realization of this is 10d super Yang–Mills theory, which has 16 supercharges [13]. This is a pretty (i.e., very symmetrical) classical 4 field theory, but at the quantum level it is both nonrenormalizable and anomalous for any nonabelian gauge group. However, as we indicated earlier, both problems can be overcome for suitable gauge groups (SO(32) or E8 × E8) when the Yang–Mills theory is embedded in a type I or heterotic string theory. The largest possible spacetime dimension for a supergravity theory (with spins ≤ 2), on the other hand, is eleven. Eleven-dimensional supergravity, which has 32 conserved supercharges, was constructed 20 years ago [14]. It has three kinds of fields—the graviton field (with 44 polarizations), the gravitino field (with 128 polarizations), and a three-index antisymmetric tensor gauge field Cµνρ (with 84 polarizations). These massless particles are referred to collectively as the supergraviton. 11d supergravity is also a pretty classical field theory, which has attracted a lot of attention over the years. It is not chiral, and therefore not subject to anomaly problems.2 It is also nonrenormalizable, and thus it cannot be a fundamental theory. (Though it is difficult to demonstrate explicitly that it is not finite as a result of “miraculous” cancellations, we now know that this is not the case.) However, we now believe that it is a low-energy effective description of M theory, which is a well- defined quantum theory [16]. This means, in particular, that higher dimension terms in the effective action for the supergravity fields have uniquely determined coefficients within the M theory setting, even though they are formally infinite (and hence undetermined) within the supergravity context. 3.1 Relation to Type IIA Superstring Theory Intriguing connections between type IIA string theory and 11d supergravity have been known for a long time. If one carries out dimensional reduction of 11d supergravity to 10d, one gets type IIA supergravity [17]. In this case dimensional reduction can be viewed as a compactification on a circle in which one drops all the Kaluza–Klein excitations. It is easy to show that this does not break any of the supersymmetries. The field equations of 11d supergravity admit a solution that describes a supermembrane. This solution has the property that the energy density is concentrated on a two-dimensional surface. A 3d world-volume description of the dynamics of this supermembrane, quite analo- gous to the 2d world volume actions of superstrings, has been constructed [18]. The authors suggested that a consistent 11d quantum theory might be defined in terms of this membrane, 2Unless the spacetime has boundaries. The anomaly associated to a 10d boundary can be canceled by introducing E8 supersymmetric gauge theory on the boundary [15]. 5 in analogy to string theories in ten dimensions.3 Another striking result was the discovery of double dimensional reduction [19]. This is a dimensional reduction in which one com- pactifies on a circle, wraps one dimension of the membrane around the circle and drops all Kaluza–Klein excitations for both the spacetime theory and the world-volume theory. The remarkable fact is that this gives the (previously known) type IIA superstring world-volume action [20]. For many years these facts remained unexplained curiosities until they were reconsidered by Townsend [21] and by Witten [16]. The conclusion is that type IIA superstring theory really does have a circular 11th dimension in addition to the previously known ten space- time dimensions. This fact was not recognized earlier because the appearance of the 11th dimension is a nonperturbative phenomenon, not visible in perturbation theory. To explain the relation between M theory and type IIA string theory, a good approach is to identify the parameters that characterize each of them and to explain how they are related. Eleven-dimensional supergravity (and hence M theory, too) has no dimensionless parameters. As we have seen, there are no massless scalar fields, whose vevs could give parameters. The only parameter is the 11d Newton constant, which raised to a suitable power (−1/9), gives the 11d Planck mass mp. When M theory is compactified on a circle (so that the spacetime geometry is R10 × S1) another parameter is the radius R of the circle. Now consider the parameters of type IIA superstring theory. They are the string mass scalems, introduced earlier, and the dimensionless string coupling constant gs. An important fact about all five superstring theories is that the coupling constant is not an arbitrary parameter. Rather, it is a dynamically determined vev of a scalar field, the dilaton, which is a supersymmetry partner of the graviton. With the usual conventions, one has gs = 〈eφ〉. We can identify compactified M theory with type IIA superstring theory by making the following correspondences: m2s = 2πRm 3 p (2) gs = 2πRms. (3) Using these one can derive other equivalent relations, such as gs = (2πRmp) 3/2 ms = g 1/3 s mp. 3It is now clear that this cannot be done in any straightforward manner, since there is no weak coupling limit in which the supermembrane describes all the finite-mass excitations. 6 The latter implies that the 11d Planck length is shorter than the string length scale at weak coupling by a factor of (gs) 1/3. Conventional string perturbation theory is an expansion in powers of gs at fixed ms. Equation (3) shows that this is equivalent to an expansion about R = 0. In particular, the strong coupling limit of type IIA superstring theory corresponds to decompactification of the eleventh dimension, so in a sense M theory is type IIA string theory at infinite coupling.4 This explains why the eleventh dimension was not discovered in studies of string perturbation theory. These relations encode some interesting facts. The fact relevant to eq. (2) concerns the interpretation of the fundamental type IIA string. Earlier we discussed the old notion of double dimensional reduction, which allowed one to derive the IIA superstring world-sheet action from the 11d supermembrane (or M2-brane) world-volume action. Now we can make a stronger statement: The fundamental IIA string actually is an M2-brane of M theory with one of its dimensions wrapped around the circular spatial dimension. No truncation to zero modes is required. Denoting the string and membrane tensions (energy per unit volume) by TF1 and TM2, one deduces that TF1 = 2πRTM2. (4) However, TF1 = 2πm 2 s and TM2 = 2πm 3 p. Combining these relations gives eq. (2). It should be emphasized that all the formulas in this section are exact, due to the large amount of unbroken supersymmetry. 3.2 p-Branes and D-Branes Type II superstring theories contain a variety of p-brane solutions that preserve half of the 32 supersymmetries. These are solutions in which the energy is concentrated on a p-dimensional spatial hypersurface. (Adding the time dimension, the world volume of a p-brane has p + 1 dimensions.) The corresponding solutions of supergravity theories were constructed by Horowitz and Strominger [22]. A large class of these p-brane excitations are called D-branes (or Dp-branes when we want to specify the dimension), whose tensions are given by [23] TDp = 2πm p+1 s /gs. (5) 4The E8 ×E8 heterotic string theory is also eleven-dimensional at strong coupling [15]. 7 This dependence on the coupling constant is one of the characteristic features of a D-brane. It is to be contrasted with the more familiar g−2 dependence of soliton masses (e.g., the ’t Hooft–Polyakov monopole). Another characteristic feature of D-branes is that they carry a charge that couples to a gauge field in the Ramond-Ramond (RR) sector of the theory. (Such fields can be described as bispinors.) The particular RR gauge fields that occur imply that even values of p occur in the IIA theory and odd values in the IIB theory. D-branes have a number of special properties, which make them especially interesting. By definition, they are branes on which strings can end—D stands for Dirichlet boundary conditions. The end of a string carries a charge, and the D-brane world-volume theory contains a U(1) gauge field that carries the associated flux. When n Dp-branes are coincident, or parallel and nearly coincident, the associated (p+ 1)-dimensional world-volume theory is a U(n) gauge theory. The n2 gauge bosons Aijµ and their supersymmetry partners arise as the ground states of oriented strings running from the ith Dp-brane to the jth Dp-brane. The diagonal elements, belonging to the Cartan subalgebra, are massless. The field Aijµ with i 6= j has a mass proportional to the separation of the ith and jth branes. This separation is described by the vev of a corresponding scalar field in the world-volume theory. In particular, the D2-brane of the type IIA theory corresponds to our friend the super- membrane of M theory, but now in a background geometry in which o