TASI 2003 lectures on AdS CFT

ar X iv :h ep -th /0 30 92 46 v5 7 Ja n 20 04 hep-th/0309246 TASI 2003 lectures on AdS/CFT Juan Maldacena Institute for Advanced Study Princeton, New Jersey 08540, USA We give a short introduction to AdS/CFT and its plane wave limit. September 2003 1. Introduction In these lecture notes we provide a short introduction to the ideas related to the correspondence between gauge theories and gravity theories. For other reviews of the subject, includding a more complete list of references, see [1,2,3,4,5]. We start by discussing the simplifications that ocurr in the large N limit of field theories. We discuss first the large N limit of vector theories, then the large N limit of theories where the fundamental fields are N ×N matrices and we show that these theories are expected to be described in terms of strings [6]. If we start with a four dimensional gauge theory, we might naively expect to find a strings moving in four dimensions. But strings are not consistent in four flat dimensions. As we try to proceed, we are forced to introduce at least one more dimension [7]. If the gauge theory is conformal, then the original flat dimensions, together with this new extra dimension are constrained by the symmetries to form an Anti-de-Sitter spacetime. We will describe some basic properties of Anti-de-Sitter spacetimes. Then we present the simplest example of the relationship between a four dimensional field theory and a gravity theory. Namely, we discuss the relationship between Yang Mills theory with four supersymmetries to type IIB superstring theory on AdS5 × S5 [8]. We later give the general prescription linking computations of correlation functions in the gauge theory to the computations of amplitudes in the gravity theory [9][10]. This is a general prescription that should hold for any field theory that has a gravity dual. Finally we discuss a particular limit of the relationship between N = 4 Yang Mills and AdS5 × S5 where we consider particles with large angular momentum on the sphere [11]. In this limit the relevant region of the geometry looks like a plane wave where we can quantize strings exactly. Through a simple gauge theory computation one can reproduce the string spectrum. 2. Large N There are theories that contain a large number of fields related by a symmetry such as SO(N) or U(N). These theories simplify when N is taken to infinity. For a more extended discussion of this subject see [12]. 1 2.1. Large N for vector theories Consider a theory with N fields ηi, where i = 1, · · ·N with O(N) symmetry. For example, S = 1 2g20 ∫ d2σ(∂~n)2 , ~n2 = 1 (2.1) First note that the effective coupling of the theory is g20N . The theory simplifies in the limit where we keep g20N fixed and we take N →∞. In this limit only a subset of Feynman diagrams survives. A very convenient way to proceed is to introduce a Lagrange multiplier, λ, that enforces the constraint in (2.1) and integrate out the fields ~n. S = 1 2g20 ∫ d2σ[(∂~n)2 + λ(~n2 − 1)] S = N 2 [ log det(−∂2 + λ)− 1 g20N ∫ λd2σ ] (2.2) We get a classical theory for λ in this large N limit, so we set ∂S ∂λ = 0. We get 1 = Ng20 ∫ d2p (2π)2 1 p2 + λ2 = Ng20 4π log Λ2/λ λ =Λ2e − 4pi Ng2 0 = µ2e − 4pi Ng2 (2.3) where g0 is the bare coupling, by absorbing the Λ dependence in g0 we define the renor- malized coupling g. Notice that the cutoff dependence of g0 is that of an asymptotically free theory. By looking again at (2.2) we find that the expectation value for λ in (2.3) gives a mass to the ~n fields. Moreover, the model has an unbroken O(N) symmetry. The fact the O(N) symmetry is restored is consistent with the fact that in two dimensions we cannot break continuous symmetries. Note that the g2N dependence of λ in (2.3) implies that the mass for ~n is non-perturbative in g2N . Notice that, even though the dependence of the mass in g2N looks non-perturbative, we have obtained this result by summing Feyn- man diagrams, in particular we obtained it through a one loop diagram contribution to the effective action and then balancing this term against a tree level term. Large N was crucial to ensure that no further diagrams contribute. This theory is similar to QCD4 since it is asymptotically free and has a mass gap. It has a large N expansion and the large N expansion contains the fact that the theory has a mass gap. This mass gap is non perturbative in g2N . 2 2.2. Matrix theories Consider theories where the basic field is a hermitian matrix M . This arises, for example, in a U(N) gauge theory, or a U(N) gauge theory with matter fields in the adjoint representation. The Lagrangian has a schematic form L = 1 g2 Tr[(∂M)2 +M2 +M3 + · · ·] = 1 g2 Tr[(∂M)2 + V (M)] (2.4) The action is U(N) invariant M → UMU†. It is convenient to introduce a double line notation to keep track of the matrix indices Mj i ij Fig. 1: Propagator . . . . Fig. 2: Vertices Each propagator, fig. 1, contributes a factor g2 in the Feynman diagrams. Each vertex, fig. 2, contributes a factor of 1/g2. Finally, each closed line will contain a sum over the gauge index and will contribute a factor of N , see Fig. 3. i Fig. 3: Closed line contributes a factor of N . Each diagram contributes with (g2)#Propagators−#verticesN#Closed lines (2.5) We can draw these diagrams on a two dimensional surface and think of it as a geometric figure. We see that (2.5) becomes N#Faces−#Edges+#vertices(g2N)Power = N2−2h(g2N)Power (2.6) where h is the genus of the two dimensional surface. Namely, a sphere has genus h = 0, a torus has genus h = 1, etc. 3 3N (g N2) ~ 2 (g2N) N (g N2) ~ 2 (g2N)2 24 Fig. 4: Planar diagrams ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� � � � � �� Torus N0(g2N) Fig. 5: A non-planar diagram A few examples of diagrams that can be drawn on a plane or a sphere are shown in fig. 4, and example of a diagram that cannot be drawn on a sphere but can be drawn on a torus is shown in fig. 5. The sum of all planar diagrams gives N2[c0 + c1(g 2N) + c2(g 2N)2 + · · ·] = N2f(g2N) (2.7) where the ci are numerical coefficients depending on the detailed evaluation of each Feyn- man graph. This detailed evaluation contains the momentum integrals. The full partition function has the form logZ = ∞∑ h=0 N2−2hfh(g2N) (2.8) The ’t Hooft limit is N →∞ , λ ≡ g2N = fixed (2.9) λ is the ’t Hooft coupling. In this limit only the planar diagrams contribute. As λ gets large a large number of diagrams contribute and they become dense on the sphere, so we might think that they describe a discretized worldsheet of some string theory. This worldsheet theory is defined to be whatever results from summing the planar diagrams. This argument is valid for any matrix theory. The argument does not give us a practical way of finding the worldsheet theory. In bosonic Yang Mills theory g2 runs. In fact, the beta function has a smooth large N limit λ˙ = β(λ) + o(1/N2) (2.10) 4 So we have λ(E). The string description will be appropriate where λ(E) becomes large. If we add matter in the fundamental, then we get diagrams with boundaries. These give rise to open strings which are mesons containing a quark and anti-quark. Some features of QCD with N = 3 are similar to those of N = ∞, like the fact that mesons contain a quark and anti-quark, and that they have small interactions. Strings are also suggested by the existence of Regge trajectories. Namely that particles with highest spin for a given mass obey α′m2 = J + const. Confinement is also closely associated to a string that forms between the quark and anti-quark. Though we will see later that the string description does not necessarily imply confinement. 2.3. Large N correlators Consider operators of the form O = Ntr[f(M)] (2.11) diagrammatically represented in Fig. 6. + . . . Fig. 6: Operator insertion If we add them to the action, they have the same scaling as an extra interaction vertex. In the large N limit their correlation functions factorize 〈tr[f1(M)]tr[f2(M)]〉 = 〈tr[f1(M)]〉〈tr[f2(M)]〉+ o(1/N2) (2.12) Notice that this implies that the leading contribution is a disconnected diagram. All connected correlation functions of operators normalized as in (2.11) go like N2. This means that 〈OO〉c ∼ N2 , 〈OOO〉c ∼ N2 , 〈OOO〉c〈OO〉3/2c ∼ 1 N (2.13) where the subscript indicates the connected part. In the string description the insertion of these operators corresponds to the insertion of a vertex operator on the string worldsheet. An interesting operator in gauge theories is the Wilson loop operator W (C) = NTr[Pe ∮ C A ] (2.14) For a contour of large area the expectation value of this operator should go like e−T (Area) for a confining theory. T is then the string tension. 5 3. Guessing the string theory Rather than summing all Feynman diagrams one would like to guess what the final string theory description is. Naively, for d = 4 Yang Mills we expect to get a bosonic string theory that lives in four dimensions. We know this is not correct. The bosonic string is not consistent quantum mechanically in d = 4. It is consistent in d = 26 flat dimensions, but this is not the theory we are interested in. The reason for this inconsistency is that the classical Polyakov action S ∼ ∫ √ ggab∂aX∂bX (3.1) has a Weyl symmetry gab → Ωgab which is not a symmetry quantum mechanically. In the quantum theory, under a change metric of the form gab = e φgˆab the partition function e−Seff (g) = ∫ DXD(bc)e−S[X,g]−S[b,c,g] (3.2) changes as Seff (g)− Seff (gˆ) = (26−D) 48π ∫ 1 2 (∇ˆφ)2 + Rˆ(2)φ+ µ2eφ (3.3) This action for φ is called “Liouville” action. Even though the initial classical action for the conformal factor in the metric was zero, a non-trivial action was generated quantum mechanically. Integrating over φ is like adding a new dimension. For D ≤ 1 this is the right answer. We start with a matrix integral or a matrix quantum mechanics and we get a string in one or two dimensions. Actually, it is necessary to do a particular scaling limit in the matrix quantum mechanics which involves N →∞ and a tuning of the potential that is analogous to taking the ’t Hooft coupling to a region where there is a large number of Feynman diagrams that contribute, see [13]. For D = 4 it is not known how to quantize the Liouville action. Nevertheless the lesson we extract is that we need to include at least one extra dimension. So we introduce an extra dimension and look for the most general string theory. If we are interested in four dimensional gauge theories we look for strings in five dimensions. We need to specify the space where string moves. It should have 4d Poincare symmetry, so the metric has the form ds2 = w(z)2(dx21+3 + dz 2) (3.4) we have used the reparametrization symmetry to set the coefficient of dz2 equal to that of dx2. 6 Now suppose that we were dealing with a scale invariant field theory. N = 4 Yang Mills is an example. Then x→ λx (3.5) should be a symmetry. But in string theory we have a scale, set by the string tension. So the only way that a string (with the usual Nambu action1) could be symmetric under (3.5) is that this scaling is an isometry of (3.4). This means that z → λz and that w = R/z. So we are dealing with 5 dimensional Anti-de-Sitter space ds2 = R2 dx2 + dz2 z2 (3.6) R4 Z 1 __ Boundary Z=0 Z Warp Factor W ~ Horizon Z=1 Fig. 7: A sketch of Anti-de-Sitter space. We emphasize the behavior of the warp factor. This is a spacetime with constant negative curvature and it is the most symmetric spacetime with negative curvature. The most symmetric spacetime with positive curvature is de-Sitter. In Euclidean space the most symmetric positive curvature space is a sphere and the most symmetric negative curvature one is hyperbolic space. These are the Euclidean continuation of de-Sitter and Anti-de-Sitter respectively. 3.1. Conformal symmetry A local field theory that is scale invariant is usually also conformal invariant. The change in the action due to a change in the metric is δS = ∫ Tµνδgµν (3.7) 1 It is possible to write a string action that is conformal invariant in four dimensions [14] but it is not know how to quantize it. 7 Under a coordinate transformation xµ → xµ + ζµ(x) the action changes by (3.7) with δgµν = ∇µζν +∇νζµ (3.8) If ζµ generates an isometry then the metric is left invariant, so that we have δgµν = 0 in (3.8). The scale transformation (3.5) gives δgµν = 2δλgµν . The action would be invariant if Tµµ = 0. In this case the action is also invariant under coordinate transformations such that δgµν = h(x)gµν in (3.8), and h is any function. Coordinate transformations of this type are called conformal transformations. In d=4 they form the group SO(2, 4). This group is obtained by adding to the Poincare group the scale transformation and the inversion ~x → −~x/x2. We see that the inversion maps the origin to infinity. It turns out that the conformal group acts more nicely if we compactify the space and we consider S3 × R in the Lorentzian case or S4 in the Euclidean case. Note also that if the trace of the stress tensor is zero, then the theory is also Weyl invariant, it is invariant under a rescaling of the metric g → Ω2g. In the quantum case this symmetry will have a calculable anomaly and one can find the change in physical quantities under such a rescaling. 3.2. Isometries of AdS In order to see clearly the AdS isometries we write AdS as a hypersurface in R2,4 −X2−1 −X20 +X21 +X22 +X23 +X24 = −R2 (3.9) Note that even though the ambient space has 2 time directions the surface contains only one time direction, the other is orthogonal to the surface. You should not be confused by these two times, AdS is an ordinary Lorentzian space with one time!. We recover (3.6) after writing X−1 +X4 = R/z, Xµ = Rxµ/z for µ = 0, · · ·3. By choosing an appropriate parameterization of (3.9) we can also write the metric ds2 = R2[− cosh2 ρdτ2 + dρ2 + sinh2 ρdΩ23] (3.10) These are called “global” coordinates. They cover the whole AdS space. In contrast the “Poincare” coordinates in (3.6) only cover a portion. Note that translations in τ correspond to rotations of X−1 and X0 in (3.9). So in the construction based on (3.9) we would get closed timelike curves. Fortunately we can go to the covering space and consider (3.10) 8 with τ non-compact. When we talk about AdS we are always going to think about this covering space. In the metric (3.10) we can take out a factor of cosh2 ρ, and define a new coordinate dx = dρ/ cosh ρ. We see that the range of x is finite. This allows us to understand the Penrose diagram of AdS. It is a solid cylinder whose boundary is S3 × R where R corresponds to the time direction. The field theory will be defined on this boundary. On this boundary the isometries of AdS act like the conformal group acts in four dimensions. The proper distance to the boundary along a surface of constant time is infinite. (a) (b) (c) Light RayTime Solid S3 ρ =∞ ρ = 0 Massive ParticleCylinder Fig. 8: (a) Penrose diagram of AdS. (b) Trajectory of a light ray. (c) Trajectory of massive geodesics. ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ��� ��� ��� ��� ��� ��� Horizon, Z =1 & t = + � 1 Poin are Pat h Fig. 9: The coordinates of (3.6) cover only the region of global AdS contained between the two shaded hyperplanes. These hyperplanes correspond to the horizons at z =∞ and t = ±∞. Finally note that Weyl transformations in the 4d theory correspond to picking different functions as conformal factors when we compute the Penrose diagram so that the boundary will have different metrics which differ by an overall function of the coordinates on the boundary. 3.3. Mapping of states and operators 9 Operator States of CFT on S 3 Operators of Eu lidean CFT on R 4 S 3 R = Fig. 10: We can map states of the field theory on S3 × R to operators on R4. In a CFT we have a correspondence between operators on R4 and states on the cylinder S3 × R. This can be seen as follows. We start with a state on the cylinder, we go to Euclidean time and then notice that the cylinder and the plane differ only by a Weyl transformation so that the two theories are related. The vacuum on the cylinder corresponds to the identity on the plane. The energy of the state in the cylinder corresponds to the conformal dimension of the operator on the plane, Ecyl = ∆. 3.4. N = 4 U(N) Yang Mills and strings on AdS5 × S5 Consider a theory with four supersymmetries in 4 dimensions, namely sixteen real supercharges. This theory has a unique field content, there is a unique supermultiplet. Our only freedom is the choice of gauge group and coupling constants. The field content is as follows. One vector field or gauge boson Aµ, six scalars φ I I = 1, ..., 6 and four fermions χαi, χα˙i¯, where α and α˙ are four dimensional chiral and anti-chiral spinor indices respectively and i = 1, 2, 3, 4 is an index in the 4 of SU(4) = SO(6) and i¯ in the 4¯. (The 4 is the spinor of SO(6)). The theory has a global S0(6) symmetry. This symmetry does not commute with the supercharges, since different components of the multiplet have different SO(6) quantum numbers. In fact, the supercharges are in the 4 and 4¯ of SU(4). A symmetry that does not commute with the supercharges is called an “R” symmetry. Note that SU(4) is a chiral symmetry. The Lagrangian is schematically of the form L = 1 g2 Tr [ F 2 + (Dφ)2 + χ¯ 6 Dχ+ ∑ IJ [φIφJ ]2 + χ¯ΓIφIχ ] + θTr[F ∧ F ] (3.11) It contains two parameters, the coupling constant and a theta angle. The theory is scale invariant quantum mechanically. Namely the beta function is zero to all orders. So it is also conformal invariant. The extra conformal symmetries commuted with the 16 ordinary 10 supersymmetries give 16 new supersymmetries. In any conformal theory we have this doubling of supersymmetries. The theory has an S-duality under which τYM = θ 2π + i 2π g2YM (3.12) transforms into −1/τ . This combines with shifts in the θ angle into SL(2, Z) acting on τ as it usually acts on the upper half plane. The ’t Hooft coupling is λ = g2YMN . 3.5. IIB strings on AdS5 × S5. Suppose that the radius is large. We will later find under which conditions this is true. Then we are looking for a solution of the type IIB supergravity equations of motion. These equations follow form the action2 S ∼ ∫ √ gR+ F 25 (3.13) plus the self duality constraint for the fiveform field strength, F5 = ∗F5, which has to be imposed by hand. Due to the existence of D3 branes the flux of F5 is quantized.∫ S5 F5 = N (3.14) Choosing a fiveform fieldstrength proportional to the volume form on S5 plus the volume form on AdS5 we find that AdS5 × S5 is a solution. The radius R of the sphere and the radius of AdS are R = (4πgsN) 1/4ls ∼ N1/4lpl (3.15) where gs is the string coupling and 2πl 2 s is the inverse of the string tension. It is clear from (3.13) that the radius in Planck units should have this form since the F 25 term in the action scales like N2, while the first term in (3.13) scales like R8. The equations of motion will balance these two terms giving (3.15). It is also amusing to understand the energetics that gives rise to the negative cosmolog- ical constant in AdS5. For this purpose consider a compactification of the ten dimensional theory on S5 to five dimensions with a fiveform fieldstrength on an S5 of radius r. Then the five dimensional action is schematically S = ∫ r5 √ g5R (5) −√g5N2r−5 +√g5r3 = ∫ √ gERE − V (r) with V (r) = r−25/3(N2r−5 − r3) (3.16) 2 We have suppressed the dependence of the action on the fields that are not important for our purposes. 11 From F 0 V(r) AdS 2 From Curvature Fig. 11: Effective potential after com