Superstring and Gauge Theory Correspondence

Transcription

1 Superstring and Gauge Theory Correspondence Ardian Nata Atmaja Supervisor: Prof. K.S. Narain The Abdus Salam International Centre for Theoretical Physics High Energy, Cosmology and Astroparticle Physics Group Diploma Programme Trieste, Italy August 2006

2 Acknowledgement I am grateful indebted to all my professors in the Diploma programme for always willing to listen; in particular Prof. K.S. Narain who has guided this work despite of his several occupations and willing to spend some of his time for discussion during the working of this thesis. I also would like to thank to Ms Patricia Passarela for her assistance and also to all diploma students specially in High Energy and all the postdoc for their assistance during this one year ICTP Diploma programme. I also thank Prof Randjbar Daemi the head of the High Energy Physics group and Prof Srenivasan, the Director of the abdus salam International Center for Theoretical Physics (ICTP), United Nations educational,unesco, the International Atomic Energy Agency (IAEA) and the Italian government for the hospitality at ICTP during the Diploma course studies. Above all, I thank Allah SWT for making all the things possible. Ardian Nata Atmaja August, 2006 i

3 Contents Acknowledgement i 1 Introduction 1 2 Anti de Sitter space Geometry of Minkowskian AdS Geometry of Euclidean AdS N = 4 Super Yang-Mills Superconformal N = 4 Super Yang-Mills Type IIB Supergravity Theory Spinors in general dimensions Supersymmetry in general dimensions Type IIB Supergravity Particle and Field Contents Type IIB Supergravity Actions Branes in Supergravity Brane Solution in Type IIB Supergravity D3-brane solution Maldacena AdS/CFT Correspondence Non-Abelian Gauge Symmetry on D3-branes The Maldacena Limit The AdS/CF T Conjecture Mapping Global Symmetries AdS/CF T Correlation Function Ansatz for effective action Computation massless scalar field bulk-boundary propagator. 21 ii

4 5.5.3 Computation massive scalar field bulk-boundary propagator Kaluza-Klein mode in AdS side and conformal mass dimension of CF T operator Conclusion 29 A SO(2,4) and SO(6) Algebra 30 A.1 Four dimensional representation of SO(2, 4) generators A.2 Four dimensional representation of SO(6) generators iii

5 Chapter 1 Introduction In the strongest form of the conjecture, the correspondence is to hold for all values of N and all regimes of coupling g s = gy 2 M. Certain limits of the conjecture are, however, also highly non-trivial. The t Hooft limit on the SYM-side [2], in which λ gy 2 M N is fixed as N corresponds to classical string theory on AdS 5 S 5 (no string loops) on the AdS-side. In this sense, classical string theory on AdS 5 S 5 provides with a classical Lagrangian formulation of the large N dynamics of N = 4 SYM theory, often referred to as the masterfield equations. A further limit λ reduces classical string theory to classical Type IIB supergravity on AdS 5 S 5. Thus, strong coupling dynamics in SYM theory (at least in the large N limit) is mapped onto classical low energy dynamics in supergravity and string theory, a problem that offers a reasonable chance for solution. The conjecture correspondence is between a 10-dimensional theory of gravity and a 4-dimensional theory without gravity at all. The fact that all the 10-dimensional dynamical degrees of freedom can somehow be encoded in a 4-dimensional theory living at the boundary of AdS 5 suggests that the gravity bulk dynamics results from a holographic image generated by the dynamics of the boundary theory. Therefore, the correspondence is also often referred to as holographic [3]. The main idea in Maldacena conjecture [1] was that a suitable theory on AdS d+1 would be equivalent to a conformal field theory in d dimensions. The most surprising statement in Maldacena conjecture was that to describe the N = 4 super Yang-Mills theory in four dimensions, one should not use only just low energy supergravity on AdS 5 but the whole infinite tower of massive Kaluza-Klein states on AdS 5 S 5. Chiral fields in the four dimensional N = 4 theory (this is fields in small representations of the supersymmetry algebra) correspond to Kaluza-Klein harmonics on AdS 5 S 5. 1

6 Chapter 2 Anti de Sitter space AdS d+1 space has many unusual properties such as a boundary at spatial infinity [4]. In fact, the boundary M d is a copy of d dimensional Minkowski space (with some points at infinity added). The symmetry group SO(2, d) of AdS d+1 acts on M d as the conformal group. This means that there are two ways to get a physical theory with SO(2, d) symmetry: in a relativistic field theory (with or without gravity) on AdS d+1 or in a conformal field theory on M d. There are many metric representations to describe AdS space-time. In this thesis we will describe two description of AdS space-time which are Minkowskian and Euclidean AdS. 2.1 Geometry of Minkowskian AdS In the Minkowskian AdS d+1 (of unit radius) may be defined in R d+1 with coordinates (X 1, X 0,, X d ) as the (d + 1)-dimensional connected hyperboloid with isometry SO(2, d) which is given by the equation X 2 1 X X X 2 d = 1 (2.1) with induced metric ds 2 = dx 1 2 dx0 2 + dx dxd 2. The topology of the manifold is that of the cylinder S 1 R times sphere S d 1, and is therefore not simply connected. The topology of boundary is consequently given by AdS d+1 = S 1 S d 1. 2

7 2.2 Geometry of Euclidean AdS The Euclidean AdS d+1 (of unit radius) may be defined in Minkowski flat space R d+1 with coordinates (X 1, X 0,, X d ) as the (d + 1)-dimensional disconnected hyperboloid with isometry SO(1, d + 1) is given by the equation X X X X 2 d = 1 (2.2) with induced metric ds 2 = dx dx0 2 + dx dxd 2. The topology of the manifold is that of R d+1. The topology of boundary is that of the d-sphere given by AdS d+1 = S d. Consider a Euclidean space coordinates (X 0,..., X d ), and let B d+1 be the open unit ball, d i=0 X2 i < 1. AdS d+1 can be identified as B d+1 with the metric [3] ds 2 = 4 d i=0 dx2 i (1 X 2 ) 2, (2.3) with X 2 = d i=0 X2 i. We can compactify B d+1 to get the closed unit ball Bd+1, defined by d i=0 X2 i 1. Its boundary is the sphere S d, defined by d i=0 X2 i = 1. S d is the Euclidean version of the conformal compactification of Minkowski space, and the fact that S d is the boundary of Bd+1 is the Euclidean version of the statement that Minkowski space is the boundary of AdS d+1. The metric (2.3) on B d+1 does not extend over B d+1, or define a metric on S d, because it is singular at X = 1. To get a metric which does extend over B d+1, one can pick a function f on B d+1 which is positive on B d+1 and has a first order zero on the boundary (for instance, one can take f = 1 X 2 ), and replace ds 2 by d s 2 = f 2 ds 2. (2.4) Then d s 2 restricts to a metric on S d. As there is no natural choice of f, this metric is only well-defined up to conformal transformations. One could, in other words, replace f by f fe w, (2.5) with w any real function on B d+1, and this would induce the conformal transformation d s 2 e 2w d s 2 (2.6) in the metric of S d. Thus, while AdS d+1 has (in its Euclidean version) a metric invariant under SO(1, d + 1), the boundary S d has only a conformal structure, which is preserved by the action of SO(1, d + 1). 3

8 Alternatively, with the substitution X = tanh(y/2), one can put the AdS d+1 metric (2.3) in the form ds 2 = dy 2 + sinh 2 y dω 2 d, (2.7) where dω 2 d is the metric on the d-dimensional unit sphere, and 0 y <. In this representation, the boundary is at y =. Introducing coordinates X 1 + X 0 = 1 z 0 and z i = z 0 X i for i = 1,, d, we map Euclidean AdS d+1 onto the upper half space H d+1, with z 0 > 0 and Poincaré metric ds 2 is defined by H d+1 = { (z 0, z), z 0 R +, z R d}, ds 2 = 1 z 2 0 ( dz d z 2). (2.8) In this representation, the boundary consists of a copy of R d, at z 0 = 0, together with a single point P at z 0 = (z 0 = consists of a single point since the metric in the x i direction vanishes as z 0 ). Thus, from this point of view, the boundary of AdS d+1 is a conformal compactification of R d obtained by adding in a point P at infinity; this of course gives a sphere S d. In this thesis we will use Euclidean AdS representation most of the time since its more easy to deal with the calculation. 4

9 Chapter 3 N = 4 Super Yang-Mills In this chapter we will discus briefly about super Yang-Mills side of the conjecture. The Lagrangian for the N = 4 super Yang-Mills theory is unique and given by [5] ( L = Tr 1 2g F µνf µν + θ I 2 8π F F 2 µν µν i λ a σ µ D µ λ a D µ X i D ν X i a i + [ ] gci ab λ a X i, λ b + g C [ ) iab λa X i, λ b] + g2 [X i, X j ] 2 (3.1) 2 a,b,i i,j a,b,i This super Yang-Mills Lagrangian consist of one vector(with field strenght F µν ), four Weyl fermions(λ a, a, b = 1, 2, 3, 4), and six real scalars(x i, i, j = 1, 2, 3, 4, 5, 6) with all in adjoint representation of gauge group SU(N). The constants Ci ab and C iab are related to the Clifford Dirac matrices for SO(6) R SU(4) R which is R-symmetry group of the supercharges. This is evident when considering N = 4 super Yang- Mills in four dimensional as the reduction on T 6 of ten dimensional super Yang-Mills theory(with 16 supercharges). The Lagrangian above is invariant under N = 4 Poincaré supersymmetry, whose transformation laws are given by δx i = [ Q a α, X i] = C iab λ ab δλ βb = {Q a α, λ βb = F µν + (σ µν ) α β δa b + [ X i, X j] ɛ αβ (C ij ) a b { } δ λ b β = Q a α, λ = C b β i ab σ µ α β D µx i δa µ = [Q a α, A µ ] = (σ µ ) β α λ ȧ β (3.2) 5

10 The constants (C ij ) a b are related to bilinears in Clifford Dirac matrices of SO(6) R. The lagrangian above is classically scale invariant (combined with Poincaré invariant to become a larger conformal group SO(2, 4) SU(2, 2)) and all terms in the Lagrangian have mass dimensions 4. The combination of N = 4 Poincaré supersymmetry and conformal invariance produces and even larger superconformal symmetry which is given by the supergroup SU(2, 2 4). The N = 4 super Yang-Mills has renormalization group β-function which is identically zero and it is believed that the theory is ultraviolet finite. The theory is exactly scale invariant at the quantum level, and the superconformal group SU(2, 2 4) is a fully quantum mechanical symmetry. 3.1 Superconformal N = 4 Super Yang-Mills In this section we will show that global continuous symmetry group of N = 4 super Yang-Mills is given by supergroup SU(2, 2 4) [6]. The ingredients are as follows. Conformal Symmetry Forming the group SO(2, 4) SU(2, 2) is generated by translations P µ, Lorentz transformations L µν, dilations D and special conformal transformations K µ. R-symmetry Forming the group SO(6) R SU(4) R and generated by T A, A = 1,..., 15. Poincaré Supersymmetry Generated by the supercharges Q a α and their complex conjugates Q αa, a = 1,..., 4. The presence of these charges results immediately from N = 4 Poincaré supersymmetry. Conformal supersymmetry Generated by the supercharges S αa and their complex conjugates S ā α. The presence of these symmetries results from the fact that the Poincaré supersymmetry and the special conformal transformations K µ do not commute. Since both are symmetries, their commutator must also be a symmetry, and these are the S generator. Detail algebra and spinor representation of SO(2,4) and SO(6) can be seen on Appendix A. The two bosonic subalgebras SO(2, 4) and SU(4) R are commute, and the 6

11 relations between the supercharges are given below. { } Q a α, Q b β = {Sαa, S βb } = { Q a α, S } β b = 0 { Q a α, Q } βb = 2σ µ α β P µδb a { S αa, S } ḃ = 2σ µ β α β K µδa b { } Q a α, S βb = ɛαβ (δb a D + Tb a ) δa b L µν σ µν αβ (3.3) 7

12 Chapter 4 Type IIB Supergravity Theory 4.1 Spinors in general dimensions Consider D-dimensional Minkowski space-time M D with flat metric η µν = diag ( + +) with µ, ν = 0, 1,..., D 1. The Lorentz group is SO(1, D 1) and the generators of the Lorentz algebra J µν obey the standard structure relations [J µν, J ρσ ] = iη µρ J νσ + iη νρ J µσ iη νσ J µρ + iη µσ J νρ. (4.1) The Dirac spinor representation, denoted S D, is defined in terms of the standard Clifford Dirac matrices Γ µ, J µν = i 4 [Γ µ, Γ ν ], {Γ µ, Γν} = 2η µν. (4.2) Its (complex) dimensions is given by dim C S D = 2 D/2. For D even, the Dirac spinor representation is always reducible because in that case there exist a chirality matrix Γ, with square Γ 2 = I, which anti-commutes with all Γ µ and therefore commutes with J µν, Γ i 1 2 D(D 1)+1 Γ 0 Γ 1 Γ d 1 { Γ, Γµ } = { Γ, Jµν } = 0. (4.3) As a result, the Dirac spinor is the direct sum of two Weyl spinors S D = S + S. The reality properties of the Weyl spinors depends on D(mod 8), and is given as follows, D 0, 4(mod 8) S = S + both complex D 2, 6(mod 8) S, S + self-conjugate. (4.4) 8

13 The charge conjugate ψ c of a Dirac spinor ψ is defined by ψ c CΓ 0 ψ, CΓ µ C 1 = (Γ µ ) T. (4.5) The proper Lorentz invariant condition for reality is that a spinor be its own charge conjugate ψ c = ψ, such a spinor is called Majorana spinor. In dimensions D 0, 4 (mod 8), a Majorana spinor is equivalent to a Weyl spinor, while in dimension D 2 (mod 8) it is possible to impose the Majorana and Weyl conditions at the same time resulting Majorana-Weyl spinors [7]. 4.2 Supersymmetry in general dimensions The basic Poincaré supersymmetry algebra in M D is obtained by supplementing the Poincaré algebra with N supercharges Q I α, I = 1,..., N. Here Q transform in the spinor representation S with α = 1,..., dim C S is the spinor index. We will restrict on a class of supersymmetry representation in which we have a massless graviton, such as in supergravity and superstring theory. Therefore, we may ignore the central charges. Considering theories with a massless graviton and assuming that supersymmetry is realized linearly, the massless graviton must be part of a massless supermultiplet of states and fields. This multiplet cannot contain fields and states of spin > 2. With vanishing central charges, we shall consider the Poincaré supersymmetry algebras of the form [8] { Q I α, ( ) } Q J β = 2δJ I (Γ µ ) β α P µ { } Q I α, Q J β = 0. (4.6) To analyze massless representations, choose P µ = (E, 0,..., 0, E), E > 0, so that the supersymmetry algebra in this representation simplifies and becomes { Q I α, ( ) } ( Q J 4E 0 β = 2δJ I 0 0 ) β. (4.7) α On this unitary massless representation, half of the supercharges effectively vanish. Half of the remaining supercharges may be viewed as lowering operators for the Clifford algebra, while the other half may be viewed as raising operators. Thus, the total number of raising operator is 1 4.N.dim RS. Each operator raising helicity by 1/2, 9

14 and total helicity ranging at most from -2 to +2, we should have at most 8 raising operator and this produces an important bound N.dim R S 32. (4.8) In other words, the maximum number of Poincaré supercharges is always 32. For N = 4 super Yang-Mills in four dimensions we will have 16 Poincaré supercharges and 16 Conformal supercharges in irreducible representation, so total we have 32 supercharges. 4.3 Type IIB Supergravity Particle and Field Contents In this thesis we will restrict our discussion on the list of the field and particle contents for N = 2, D = 10 Type IIB theory, which is chiral and has two Majorana-Weyl gravitini of the same chirality. The N = 2, D = 10 Type IIB theory, which is obtained by looking at low energy effective action of Type IIB superstring, has the following field and particle contents G µν SO(8) 35 B metric - graviton C + iφ 2 B axion - dilaton B T ypeiib µν + ia 2µν 56 B rank 2 antisymmetric A + (4.9) 4µνρσ 35 B antisymmetric rank 4 ψ µα, I I = 1, F Majorana - Weyl gravitinos λ I α, I = 1, 2 16 F Majorana - Weyl dilatinos The rank 4 antisymmetric tensor A + µνρσ has self-dual field strength, a fact that is indicated with the + superscript. The gravitinos satisfies Γ-tracelessness condition (Γ µ ) βα ψ µα = 0. The two gravitinos ψµα I have the same chirality, while the two dilatinos λ I α also have the same chirality but opposite to that of the gravitinos. 4.4 Type IIB Supergravity Actions There exist but not completely satisfactory action for the Type IIB theory, since it involves an antisymmetric field A + 4 with self-dual field strength. However, one may write an action involving both dualities of A 4 1/4!A 4µνρλ dx µ dx ν dx ρ dx λ and 10

15 then impose the self-duality as a supplementary field equation. In that case one will obtain 1 [9] [7] S IIB = 1 Ge ( 2Φ 2R 4κ 2 G + 8 µ Φ µ Φ H 3 2) B 1 [ G ( F F ) ] A +4 H 3 F 3 4κ 2 B +fermions (4.10) with 2κ 2 B = (2π)7 g 2 sα 4, where g s is the string coupling contant and the field strengths are defined by F 1 = dc, H 3 = db, F 3 = da 2, F 5 = da + 4 F 3 = F 3 CH 3, F5 = F A 2 H B F 3 (4.11) and we have to impose the self-duality condition F 5 = F 5. The above form of the action naturally arises from the string low energy approximation. The firs line in (4.10) originates from the NS-NS sector while the second line originates from R-R sector. 4.5 Branes in Supergravity A rank p+1 antisymmetric tensor field A µ1 µ p+1 can be indentified with a (p+1)-form, A p+1 1 (p + 1)! A µ 1 µ p+1 dx µ 1 dx µ p+1. (4.12) A (p + 1)-form naturally couples to geometrical objects Σ p+1 of spacetime dimension p + 1, because a diffeomorphism invariant action may be constructed as follows S p+1 = T p+1 Σ p+1 A p+1. (4.13) The action is invariant under Abelian gauge transformations ρ p (x) of rank p A p+1 A p+1 + dρ p (4.14) 1 We use the notation G detg µν and G F p 2 1 p! GG µ1ν1 G µpνp Fµ1 µp F ν1 νp where F denotes the complex conjugate of F. For real fields, this definition coincides with that of [7] 11

16 because S p+1 transform with total derivative. The field A p+1 has a gauge invariant field strength F p+2 which is a (p + 2)-form whose flux is conserved. Solutions to supergravity with non-trivial A p+1 charge are referred to as p-brane, after the spacedimension of their geometry. Each A p+1 gauge field has a magnetic dual A magn D 3 p which is a differential form field of rank D 3 p, whose field strength is related to that of A p+1 by Poincaré duality da magn D 3 p da p+1. (4.15) This also means that each p-brane has a magnetic dual, which is a (D 4 p)-brane and which now couples to the field A magn D 3 p. The possible branes in Type IIB supergravity are distinguished as follow. When the antisymmetric field which the charge they carry is in R-R sector, the brane is referred to as a D-brane. On the other hand, the 1-brane that couples to NS-NS field B µν is the fundamental string denoted by F1, whose magnetic dual is NS5 [10] Brane Solution in Type IIB Supergravity Each brane is realized as 1/2 BPS solution in supergravity. A p-brane has a (p + 1)- dimensional flat hypersurface, with Poincaré invariance group R p+1 SO(1, p). The transverse space is then of dimension D p 1 and solutions may always be found with maximal rotational symmetry SO(D p 1) in this transverse space. So, p-branes in supergravity may be thought of as solutions with symmetry groups D = 10, R p+1 SO(1, p) SO(9 p). (4.16) So in D3-brane case then this becomes symmetry group of R 4 SO(1, 3) SO(6). We also denote the coordinates as follows Coordinates to brane x µ µ = 0, 1,, p Coordinates to brane y u = x p+u u = 1, 2,, D p 1. (4.17) Poincaré invariance in (p + 1)-dimensions forces the metric in those directions to be a rescaling of the Minkowski flat metric, while rotation invariance in the transverse directions forces the metric in those directions to be a rescaling of the Euclidean metric in those dimensions. Furthermore, the metric rescaling functions should be independent of x µ, µ = 0, 1,, p. Substituting an Ansatz with the above restriction into the field equation, the metric solution for the branes can be expressed in term of single function H as follows [11] Dp-brane, ds 2 = H( y) 1 2 dx µ dx µ + H( y) 1 2 d y 2, e Φ = H( y) 3 p 4 (4.18) 12

17 Here, the Dp-brane metric is expressed in the string frame. The single function H must be harmonic with respect to y. Assuming maximal rotational symmetry by SO(D p 1) in the transversal dimensions, and using the fact that the metric should tend to flat space-time as y, the most general solution is parametrized by a single scale factor L and is given by H(y) = 1 + LD p 3 y D p 3 (4.19) Since α is the only dimensionful parameter of the theory, L must be a numerical constant (possibly dependent on the dimensionless string couplings) times the above α dependence. Of particular interest will be the solution of N coincident branes, for which we have L D p 3 = Nρ p. For Dp branes, we have ρ p = g s (4π) (5 p)/2 Γ((7 p)/2)(α ) (D p 3)/ D3-brane solution Now, lets take a look at one of D-branes in Type IIB supergravity which is mainly discussed in this thesis. The D3-brane solution is of special interest for many reasons : 1. Its worldbrane has 4-dimensional Poincaré invariance. 2. It has constant axion and dilaton fields. 3. it is regular at y = it is self-dual. This D3-brane solution is characterized by g s = e φ, C constant B µν = A 2µν = 0 ds 2 = H(y) 1/2 dx µ dx µ + H(y) 1/2 (dy 2 + y 2 dω 2 5) F + 5µνρστ = ɛ µνρστυ υ H (4.20) Here, ɛ µνρστυ is the volume element transverse to the 4-dimensional Minkowski D3- brane in D = 10. The N-brane solution with general locations of N I parallel D3- branes located at transverse position y i is given by H( y) = 1 + N I=1 13 4πg s N I (α ) 2 y y I 4 (4.21)

18 where the total number of D3-branes is N = I N I. It is useful to compare the scales involved in the D3 brane solution and their relations with the coupling constant. 2 The radius L of the D3 brane solution to string theory is a scale that is not necessarily of the same order of magnitude as the Planck length l P, which is defined by l 2 P = α. Their ratio is given instead by L 4 = 4πg s Nl 4 P. For g s N 1, the radius L is much smaller than the string length l P, and thus the supergravity approximation is not expected to be a reliable approximation to the full string solution. In this regime we have g s 1, so that string perturbation theory is expected to be reliable and the D3 brane may be treated using conformal field theory techniques. For g s N 1, the radius L is much larger than the string length l P, and thus the supergravity approximation is expected to be a good approximation to the full string solution. 2 The discussion given here may be extended to Dp-branes to some extent. However, when p 3, the dilaton is not constant and the strength of the coupling will depend upon the distance to the brane. 14

19 Chapter 5 Maldacena AdS/CFT Correspondence 5.1 Non-Abelian Gauge Symmetry on D3-branes In this chapter we will discuss more detail about how Maldacena conjecture can be reached by considering both point of views of D-brane from open and close string and taking certain limits. The open strings whose both end points are attached to a single brane can have arbitrarily short length and therefore must be massless. This excitation mode induces a massless U(1) gauge theory on the worldbrane which is effectively 4-dimensional flat spacetime [12]. Since the brane breaks half of total number of supersymmetries (it is 1/2 BPS), the U(1) gauge theory must have N = 4 Poincaré supersymmetry. In the low energy approximation (which has at most two derivative on bosons and one derivative on fermions) the N = 4 supersymmetric U(1) gauge theory is free. In the case of N > 1 parallel separated D3-branes, the end points of an open string may be attached to the same brane. For each brane, these strings have small length and therefore must be massless. These excitation modes induce a massless U(1) N gauge theory with N = 4 supersymmetry in the low energy limit. In the limit where the N branes all are coincident, all string states would be massless and U(1) N gauge symmetry is enhanced to a full U(N) gauge symmetry. The overall factor U(1) = U(N)/SU(N) actually corresponds to the overall position of the branes and may be ignored when considering dynamics on the branes, such that leaving only a SU(N) gauge symmetry [13]. So, in the low energy limit, N coincident branes support an N = 4 super Yang-Mills theory in 4-dimensional with gauge group SU(N). 15

20 Figure 5.1: D-branes : (a) single, (b) well-separated, (c) (almost) coincident 5.2 The Maldacena Limit Now, we consider for N D3-branes solution in order to implement the limit needed for the conjecture. The spacetime metric of N coincident D3-branes can be written as 1 ) 1 ) 1 ds 2 = (1 + L4 2 ηij dx i dx j + (1 + L4 2 (dy 2 + y 2 dω5) 2 (5.1) where the radius L of the D3-brane is given by y 4 y 4 L 4 = 4πg s Nα 2. (5.2) Consider two regimes of metric (5.1) on its limits. For y L the result will be in flat spacetime R 10. When y < L, the geometry is often referred to as the throat and would at first appear to be singular as y L. But as we redefine the coordinate into u L2 y (5.3) 1 In this section, we shall denote 10-dimensional indices by M, N,, 5-dimensional indices by µ, ν, and 4-dimensional Minkowski indices by i, j,, and the Minkowski metric by η ij = diag( + ++). 16

21 and the large u limit, this will transform into the following asymptotic form ( 1 ds 2 = L 2 u η ijdx i dx j + 1 ) 2 u 2 du2 + dω 2 5 (5.4) which corresponds to a product geometry of the sphere S 5, with metric L 2 dω 2 5, and the hyperbolic space AdS 5, with constant negative curvature metric L 2 u 2 (η ij dx i dx j + du 2 ). So, the geometry close to the brane (y 0 or u ) is regular and highly symmetrical, and may be summarized as AdS 5 S 5 where both components have identical radius L. Figure 5.2: Minkowski region and throat region of D3-brane The Maldacena limit [1] is implemented by keeping fixed g s and N as well as all physical length scales(such as mass), while α 0. In this limit, only the AdS 5 S 5 region of the D3-brane geometry survives the limit and contributes to the string dynamics of physical process, while the asymptotically flat region decouples from the theory. We can see in more precise way of this decoupling is by taking the Maldacena limit directly on the string theory non-linear sigma model in the D3-brane background. We shall concentrate here on the metric part, thereby ignoring the contributions from the tensor field F + 5. We denote the D = 10 coordinates by x M, M = 0, 1,, 9, and the metric by G MN (x). The first 4 coordinates coincide with x µ of the Poincaré invariant 17

22 D3 worldvolume, while the coordinates on the 5-sphere are x M for M = 5,, 9 and x 4 = u. The full D3-brane metric of (5.1) takes the form ds 2 = G MN dx M dx N = L 2 Ḡ MN (x; L)dx M dx N, where the rescaled metric ḠMN is given by ) 1 Ḡ MN (x; L)dx M dx N = (1 + L4 2 du 2 ( u 4 u + 2 dω2 5) + (1 + L4 u 4 ) u 2 η ijdx i dx j (5.5) Inserting this metric into the non-linear sigma model, we obtain S G = 1 γγ mn G 4πα MN (x) m x M n x N = L2 γγ mn Ḡ Σ 4πα MN (x; L) m x M n x N Σ (5.6) The overall coupling constant for the sigma model dynamics is given by L 2 λ = λ g 4πα s N (5.7) 4π Keeping g s and N fixed but letting α 0 implies that L 0. Under this limit the sigma model action admits a smooth limit, given by λ S G = γγ mn Ḡ MN (x; 0) m x M n x N (5.8) 4π Σ where the metric ḠMN(x; 0) is the metric on AdS 5 S 5, Ḡ MN (x; L)dx M dx N = 1 u 2 η ijdx i dx j + du2 u 2 + dω2 5 (5.9) rescaled to unit radius. Manifestly, the coupling 1/ λ has taken over the role of α as the non-linear sigma model coupling constant and the radius L has cancelled out. 5.3 The AdS/CF T Conjecture The conjecture that appears in Maldacena paper [1] related the two theory: Type IIB superstring theory on AdS 5 S 5 where both AdS 5 and S 5 have the same radius L, where the 5-form F + 5 has integer flux N = S 5 F + 5 and where the string coupling is g s. N = 4 super Yang-Mills theory in 4-dimensional with gauge group SU(N) and Yang-Mills coupling g Y M in its (super)conformal phase. 18

23 With the identifications between the parameters of both theories, g s = gy 2 M, L 4 = 4πg s Nα 2 (5.10) and the axion expectation value is equal with the super Yang-Mills instanton angle C = θ I. This statement of conjecture is referred to as the strong form, as it is to hold for all values of N and of g s = gy 2 M. Actually we have three forms of the AdS/CF T conjecture in order of decreasing strength as below N = 4 conformal SYM all N, g Y M g s = g 2 Y M t Hooft limit of N = 4 SYM λ = g 2 Y M N fixed, N 1/N expansion Large λ limit of N = 4 SYM (for N ) λ 1/2 expansion Full Quantum Type IIB string theory on AdS 5 S 5 L 4 = 4πg s Nα 2 Classical Type IIB string theory on AdS 5 S 5 g s string loop expansion Classical Type IIB supergravity on AdS 5 S 5 α expansion Table 5.1: AdS/CFT correspondnce 5.4 Mapping Global Symmetries The main requirement for the AdS/CF T correspondence is that the global unbroken symmetries of the two theories be identical. The continuous global symmetry 19

24 of N = 4 super-yang-mills theory in its conformal phase was previously shown to be the superconformal group SU(2, 2 4), whose maximal bosonic subgroup is SU(2, 2) SU(4) R SO(2, 4) SO(6) R. Recall that the bosonic subgroup arises as the product of the conformal group SO(2, 4) in 4-dimensions by the SU(4) R automorphism group of the N = 4 Poincaré supersymmetry algebra. This bosonic group is immediately recognized on the AdS side as the isometry group of the AdS 5 S 5 background. The completion into the full supergroup SU(2, 2 4) and arises on the AdS side because 16 of the 32 Poincaré supersymmetries are preserved by the array of N parallel D3-branes, and in the AdS limit, are supplemented by another 16 conformal supersymmetries (which are broken in the full D3-brane geometry). Thus, the global symmetry SU(2, 2 4) matches on both sides of the AdS/CF T correspondence. 5.5 AdS/CF T Correlation Function Ansatz for effective action In this section, we will consider in general AdS d+1 S d+1 space-time and derive relation formula of Kaluza-Klein mode and conformal dimension of operator in the boundary of AdS. Now, consider a massless scalar field φ on AdS d+1 S d+1, satisfying the simple Laplace equation D i D i φ = 0, with no mass term or curvature coupling, or some other equation with the same basic property: the existence of a unique solution on B d+1 with any given boundary values. We define φ 0 be the restriction of φ to the boundary of AdS d+1 (from now on, we will use AdS d+1 rather then AdS d+1 S d+1 ). We will assume that in the correspondence between AdS d+1 and conformal field theory on the boundary, φ 0 should be considered to couple to a conformal field O, via a coupling φ S d 0 O. This assumption is natural given the relation of the conjectured AdS/CF T correspondence to analyses [14] of interactions of fields with branes. The φ 0 is conformally invariant, with conformal weight zero, since the use of a function f as in (2.4) to define a metric on S d does not enter at all in the definition of φ 0. So conformal invariance dictates that O must have conformal dimension d. We will show later in more precise formula. Now, consider Z S (φ 0 ) be the supergravity (or string) partition function on B d+1 computed with the boundary condition that at infinity φ approaches a given function φ 0. For example, in the approximation of classical supergravity, one computes Z S (φ 0 ) by simply extending φ 0 over B d+1 as a solution φ of the classical supergravity equations, and then writing Z S (φ 0 ) = exp( I S (φ)), (5.11) 20

25 where I S is the classical supergravity action. If classical supergravity is not an adequate approximation, then we must include string theory corrections to I S (and the equations for φ), or include quantum loops (computed in an expansion around the solution φ) rather than just evaluating the classical action. Criteria under which stringy and quantum corrections are small are given in Maldacena limit [1]. The ansatz for the precise relation of conformal field theory on the boundary to AdS space is that [3] exp φ 0 O S d CF T = Z S (φ 0 ). (5.12) For a preliminary check, note that in the case of N = 4 supergravity in four dimensions, this has the expected scaling of the t Hooft large N limit, since the supergravity action I S that appears in (5.11) is of order N Computation massless scalar field bulk-boundary propagator First we consider an AdS theory that contains a massless scalar φ with action I(φ) = 1 d d+1 x g dφ 2. (5.13) 2 B d+1 We assume that the boundary value φ 0 of φ is the source for a field O and that to compute the two point function of O, we must evaluate I(φ) for a classical solution with boundary value φ 0. For this, we must solve for φ in terms of φ 0, and then evaluate the classical action (5.13) for the field φ. In order to solve for φ in terms of φ 0, we first look for a Green s function, a solution K of the Laplace equation on B d+1 whose boundary value is a delta function at a point P on the boundary. To find this function, it is convenient to use the representation of B d+1 as the upper half space with metric (2.8) and take P to be the point at z 0 =. The boundary conditions and metric are invariant under translations of the z i, so K will have this symmetry and is a function only of z 0. The Laplace equation reads d z d+1 d 0 K(z 0 ) = 0. (5.14) dz 0 dz 0 2 The action contains an Einstein term, of order 1/g 2 s. This is of order N 2 as g st g 2 Y M 1/N. It also contains a term F 2 with F the Ramond-Ramond five-form field strength; this has no power of g st but is of order N 2 as, with N units of RR flux, one has F N. 21

26 We will get solution K(z 0 ) = contant which is not vanish at z 0 = 0 and the solution that vanishes at z 0 = 0 is K(z 0 ) = c z d 0 (5.15) with c a constant. We take the last solution since we require that in the boundary should be vanish except at on point(here, we already picked up the point P at z 0 = ). Since this grows at infinity, there is some sort of singularity at the boundary point P. To show that this singularity is a delta function, it helps to make an SO(1, d + 1) transformation that maps P to a finite point. The transformation z i z i z0 2 +, i = 0,..., d (5.16) d j=1 z2 j maps P to the origin, x i = 0, i = 0,..., d, and transforms K to K(z) = c z0 d (z (5.17) d j=1 z2 j )d Using scaling argument we can show that dz 1... dz d K(z) is independent of x 0 and also, as z 0 0, K vanishes except at z 1 =... = z d = 0. So for z 0 0, K becomes a delta function supported at z i = 0, with unit coefficient if c is chosen correctly. Henceforth, we write z for the d-tuple z 1, z 2,... z d, and z 2 = d j=1 z2 j. Using this Green s function, the solution of the Laplace equation on the upper half space with boundary values φ 0 is φ(z 0, z i ) = c dz z0 d (z0 2 + z z 2 ) φ 0(z i). (5.18) d (dz is an abbreviation for dz 1dz 2... dz d ). It follows that for z 0 0, where we can ignore z 0 in the denumerator such that (z0 2 + z z 2 ) d z z 2d φ dc z0 d 1 dz φ 0 (z ) + z 0 z z O(zd+1 2d 0 ). (5.19) By integrating by parts, one can express I(φ) as a surface integral. Since φ φ 0 for z 0 0, and φ/ z 0 behaves as in (5.19),then (5.13) can be evaluated to give I(φ) = cd dz dz φ 0(z)φ 0 (z ). (5.20) 2 z z 2d So the two point function of the operator O which is O(z)O(z ) z z 2d, as expected for a field O of conformal dimension d. 22

27 5.5.3 Computation massive scalar field bulk-boundary propagator In the ideas of [1], we need to consider massive excitations as well. The reason for this is that in, for instance, the AdS 5 S 5 example, the radius of the S 5 is comparable to the radius of curvature of the AdS 5, so that the inverse radius of curvature (which behaves in AdS space roughly as the smallest wavelength of any excitation, as seen in [15]) is comparable to the masses of the Kaluza-Klein excitations. Lets now, we consider a scalar with mass m in AdS d+1. The action is given by I(φ) = 1 d d+1 x g ( dφ 2 + m 2 φ 2). (5.21) 2 B d+1 By using metric (2.7), the wave equation receives an extra contribution from the mass term, and is now ( 1 d (sinh y) d dy (sinh y)d d dy + ) L2 sinh 2 y + m2 φ = 0. (5.22) The L 2 term is still irrelevant at large y such that we can get the approximation for y, sinh y e y so (5.22) becomes 1 ( d e y d ) e y dy dy φ + m 2 φ = 0. (5.23) We can take φ = e λy and substitute to equation (5.23), then λ(λ + d)e (λ+d)y + m 2 e (λ+d)y = 0, (5.24) so we get equation m 2 = λ(λ + d). This will give two solutions λ ± = d 2 ± 1 2 d2 + 4m 2 (5.25) with m 2 is limited to the region in which this quadratic equation has real roots. One linear combination of the two solutions extends smoothly over the interior of AdS d+1 and this solution behaves at infinity as e λ +y. This means that we cannot find a solution of the massive equation of motion (5.21) that approaches a constant at infinity. The closest that we can do is the following. Pick any positive function f on B d+1 that has a simple zero on the boundary. For instance, f could be e y (which has a simple zero on the boundary, as one can see 23

28 by mapping back to the unit ball with X = tanh(y/2)). Then one can look for a solution of the equation of motion that behaves as φ f λ + φ 0, (5.26) with an arbitrary function φ 0 on the boundary. The definition of φ 0 as a function depends on the choice of a particular f, which was the same choice used in (2.4) to define a metric (and not just a conformal structure) on the boundary of AdS d+1. If we transform f e w f, the metric will transform by d s 2 e 2w d s 2. At the same time, (5.26) becomes φ f λ + e λ +w φ 0 and shows that φ 0 will transform by φ 0 e wλ + φ 0. This transformation under conformal rescalings of the metric shows that φ 0 must be understood as a conformal density of length dimension λ +, or mass dimension λ + (what we mean by dimension is the mass dimension). If, therefore, in a conformal field theory on the boundary of AdS d+1, there is a coupling φ 0 O for some operator O, then O must have conformal dimension d + λ +. We will verify this by computing the two point function of the field O just like we did in the massless case before. In that case, we need to find the explicit form of a function φ that obeys the massive wave equation and behaves as f λ + φ 0 at infinity. We represent AdS space as the half-space z 0 0 with metric (2.8). As before, the first step is to find a Green s function, that is a solution of ( D i D i + m 2 )K = 0 that vanishes on the boundary except at one point. Again taking this point to be at z 0 =, K should be a function of z 0 only, and the equation reduces to ( ) By taking K(z 0 ) = c z 0 The solutions are z d+1 0 d z d+1 d 0 + m 2 dz 0 dz 0 K(z 0 ) = 0. (5.27) and substituting to (5.27) then we will get ( d)z 0 + m 2 z 0 = 0 m 2 = ( d). (5.28) = d 2 ± 1 2 d2 + 4m 2, (5.29) since we have restricted that K(z 0 ) = 0 at z 0 = 0 so we need > 0 which mean we choose = d d2 + 4m 2 = d + λ +. (5.30) 24

29 So the solution that vanishes for z 0 = 0 is K(z 0 ) = c z d+λ + 0, = d + λ +. After the inversion z i z i /(z z 2 ), we get the solution K = c z d+λ + 0 (z z 2 ) d+λ +, (5.31) which behaves as z d+λ + 0 for z 0 0 except for a singularity at x = 0. Now, c z d+2λ + 0 lim z 0 0 (z0 2 + z 2 ) d+λ + is a multiple of δ(z), as proved by using a scaling argument to show that c z d+2λ + 0 dz (z0 2 + z 2 ) d+λ + (5.32) (5.33) is independent of z 0, and observing that for z 0 0, the function in (5.32) is supported near z = 0. So if we use the Green s function K to construct the solution φ(z) = c dz z d+λ + 0 (z0 2 + z z 2 ) φ 0(z ) (5.34) d+λ + of the massive wave equation, then φ behaves for z 0 0 like z λ + 0 φ 0 (z), as expected. Moreover, given the solution (5.34), one can evaluate the action (5.21) by the same arguments (integration by parts and reduction to a surface term) that we used in the massless case, with the result I(φ) = c (d + λ + ) 2 dz dz φ 0 (z)φ 0 (z ) z z 2(λ ++d). (5.35) This is the expected two point function of a conformal field O of dimension = λ + +d. In sum, the dimension of a conformal field on the boundary that is related to a field of mass m on AdS space is = d + λ +, and is the larger root of ( d) = m 2. (5.36) Thus = 1 2 (d + d 2 + 4m 2 ). (5.37) 25

30 5.5.4 Kaluza-Klein mode in AdS side and conformal mass dimension of CF T operator For the limit in which classical supergravity is valid, the states with masses of order one are precisely the Kaluza-Klein modes. Indeed, for example the AdS 5 S 5 model, the radius of the S 5 is comparable to the radius of curvature of the AdS 5 factor, so the Kaluza-Klein harmonics have masses of order one, in units of the AdS 5 length scale. Stringy excitations are much heavier; for instance, in this model they have masses of order (gy 2 M N)1/4 (which is large in the limit in which classical supergravity is valid), as shown in [1]. Since the dimension of a scalar operator in the four-dimensional conformal field theory is = d + λ + = d + d 2 + 4m 2, (5.38) 2 the particles of very large mass in AdS space correspond to operators of very high dimension in the conformal field theory. The conformal fields with dimensions of order one correspond therefore precisely to the Kaluza-Klein excitations. Since the Kaluza-Klein excitations are in small representations of supersymmetry, their masses are protected against quantum and stringy corrections. The conformal fields that correspond to these excitations are similarly in small representations, with dimensions that are protected against quantum corrections. It is therefore possible to test the conjectured AdS/CF T correspondence by comparing the Kaluza-Klein harmonics on AdS 5 S 5 to the operators in the N = 4 super Yang- Mills theory that are in small representations of supersymmetry; those operators are discussed in [16]. The Kaluza-Klein harmonics have been completely worked out in [17]. The operators of spin zero are in five infinite families. We will make the comparison to the N = 4 theory for only one of families that contain relevant or marginal operator which is related with a scalar in type IIB supergravity(in this case we consider the dilaton field). We look at the following facts about the N = 4 theory super Yang-Mills. This theory has an R-symmetry group SU(4), which is a cover of SO(6). It has six real scalars X a in the 6 or vector of SO(6) and adjoint represention of the gauge group, and four fermions λ A α, also in the adjoint representation of the gauge group, in the 4 of SU(4) or positive chirality spinor of SO(6)(Here a = 1,..., 6 is a vector index of SO(6), A = 1,..., 4 is a positive chirality SO(6) spinor index, and α is a positive chirality Lorentz spinor index.) From the point of view of an N = 1 subalgebra of the supersymmetry algebra, 26

31 this theory has three chiral multiplet Φ z, z = 1,..., 3 in the adjoint representation, and a gauge multiplet W α, also in the adjoint representation, that contains the gauge field strength. The superpotential is W = ɛ z1 z 2 z 3 Tr (Φ z 1 [Φ z 2, Φ z 3 ]). Viewed as an N = 2 theory, this theory has a vector multiplet and a hypermultiplet in the adjoint representation. The conformal field theory also contains a special marginal operator, the derivative of the Lagrangian density with respect to the Yang-Mills coupling. This operator is in a small representation, since the Lagrangian can not be written as an integral over a superspace with 16 fermionic coordinates. It transforms in the singlet representation of SO(6), and, being a marginal operator, is related to a supergravity mode with m 2 = 0, in fact, the dilaton. To exhibit this operator as the first in an infinite series, view the N = 4 theory as an N = 2 theory with a vector multiplet and an adjoint hypermultiplet. Let a be the complex scalar in the vector multiplet. Possible Lagrangians for the vector multiplets, with N = 2 supersymmetry and the smallest possible number of derivatives, are determined by a prepotential, which is a holomorphic, gauge-invariant function of a. A term Tr (a r ) in the prepotential leads to an operator Q r = Tr (a r 2 F ij F ij ) +... which could be added to the Lagrangian density while preserving N = 2 supersymmetry. These operators are all in small representations as the couplings coming from the prepotential can not be obtained by integration over all of N = 2 superspace. Q r has dimension 2 + r in free field theory, and this dimension is protected by supersymmetry. Since a is part of a vector of SO(6), Q r is part of a set of operators transforming in the (r 2) th symmetric tensor representation of SO(6). (In fact, Q r can be viewed as a highest weight vector for this representation). In the AdS 5 S 5 Type IIB supergravity, we can write the dilaton equation in two part of space-time ( AdS5 + S 5) Φ = 0 (5.39) where means Laplacian operator. By writing the dilaton Φ in expansion of spherical harmonic functions such that Φ = k Z φ k Y k (5.40) with Y k is a spherical harmonic function of rank k in S 5, so that φ k is a scalar field on AdS 5 which transform in the [0, k, 0] irreducible of SO(6). With this expansion, the equation (5.39) becomes ( AdS5 + k(k + 4)) φ k = 0, k 0 ( AdS5 + m 2) φ k = 0. (5.41) 27

32 This will give the mass for the field φ k as Kaluza-Klein scalar and if we set k = r 2(from previous discussion), we have for k 0 an operator of dimension k + 4 in the k th symmetric tensor representation, corresponding to Kaluza-Klein harmonics in that representation with mass m 2 = k(k + 4). (5.42) 28

33 Chapter 6 Conclusion The Anti-de Sitter/Conformal Field Theory (AdS/CF T ) correspondence, as originally conjectured by Maldacena [1], describes remarkable equivalence between two unrelated theories. On one side (the AdS-side) of the correspondence, we have 10- dimensional Type IIB string theory on the product space AdS 5 S 5, where the radius L of AdS 5 and S 5 are given by L 4 = 4πg s Nα 2, where g s is the string coupling. On the other side (the super Yang-Mills) of the correspondence, we have 4-dimensional super Yang-Mills theory with maximal N = 4 supersymmetry, gauge group SU(N), Yang- Mills coupling g 2 Y M = g s in the conformal phase. The AdS/CF T conjecture states that these two theories, including operator observables, states, correlation functions and full dynamics, are equivalent to one another [1] where (in this thesis) we show the relation for scalar field(in this case is the dilaton) with operator in super Yang-Mills by showing relation between Kaluza-Klein mode and conformal dimension. 29

34 Appendix A SO(2,4) and SO(6) Algebra The symmetries of AdS 5 S 5 form the supergroup P SU(2, 2 4), the bosonic component of this group is given by SO(2, 4) SO(6). The SO(2, 4) corresponds to the conformal group in the N = 4 super Yang-Mills in four dimensions, while SO(6) to its R-symmetry. The conformal group in four dimensions SO(2, 4) is generated by the Lorentz generator M µν, the four momentum P µ, the generator of special conformal transformations K µ, (µν = 0, 1, 2, 3) and the Dilatation generator D. The algebra of the generators is given by [M µν, M ρσ ] = i (η νρ M µσ η µρ M νσ η νσ M µρ + η µσ M νρ ), [P µ, M ρσ ] = i (η µρ P σ η µσ P ρ ), [K µ, M ρσ ] = i (η µρ K σ η µσ K ρ ), [D, M µν ] = [P µ, P ν ] = [K µ, K ν ] = 0, [P µ, D] = ip µ, [K µ, D] = ik µ, [P µ, K nu] = 2i (η µν D M µν ). (A.1) where η µν = diag(, +, +, +). The correspondence with the SO(2, 4) generators is made with the identifications M µ5 = 1 2 (P µ K µ ), M µ6 = 1 2 (P µ + K µ ), M 56 = D, (A.2) the these generators satisfy the SO(2, 4) algebra given by [M AB, M CD ] = i (η BC M A D η AC M B D η BD M A C + η AD M B C). (A.3) Here A, B,... = 0, 1,..., 6 and η AB = diag(, +, +, +, +, ). 30

35 A.1 Four dimensional representation of SO(2, 4) generators The four dimensional Weyl representation of SO(2, 4) generators is as follows M µν = i 4 [γµ, γ ν ], M µ5 = i 2 γµ γ ν, M µ6 = 1 2 γµ, M 56 = 1 2 γ5. (A.4) with the γ matrices are 4 4, SO(1, 3)gamma matrices in the Dirac representation which obey the algebra γ µ, γ ν = 2η µν, η µν = diag( 1, 1, 1, 1), and µ, ν = 0, 1, 2, 3, 4 as spacetime indices. These matrices are given by ( ) 1 0 γ 0 =, 0 1 ( ) 0 σ γ m m = σ m, 0 ( ) 0 1 γ 5 = i (A.5) 1 0 where σ m refers to the Pauli matrices with index m = 1, 2, 3 as space index. A.2 Four dimensional representation of SO(6) generators The conventions for the four dimensional Weyl representation of SO(6) generators is as follow. M ij = i 4 [ γ i, γ j], M i5 = i 2 γi γ 5, M i6 = 1 2 γi, M 56 = 1 2 γ5. (A.6) Where γ i and 4 4 SO(4) gamma matrices in the Weyl representation are given by ( ) 0 σ γ i i = σ i (A.7) 0 31

36 by using σ i = (1, i σ) and σ i = (1, i σ). These gamma matrices obey the algebra {γ i, γ j } = 2δ ij. Finally γ 5 is given by ( ) 1 0 γ 5 = γ 1 γ 2 γ 3 γ 4 = (A.8) 0 1 with i, j = 1, 2, 3, 4 as indices of SO(4) SO(6). 32

37 Bibliography [1] J.Maldacena, The Large N Limit Of Superconformal Field Theory And Supergravity, hep-th/ [2] G. t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl.Phys.B 72, 461(1974). [3] E.Witten, Anti-de Sitter Space and Holography, Adv.Theor.Math.Phys. 2, 253(1998)[arXiv:hep-th/ ]. [4] S.W.Hawking and G.F.R.Ellis, The Large Scale Structure Of Spacetime, Cambridge University Press(1973). [5] R.Grimm,M.Sohnius and J.Wess, Extended Supersymmetry and Gauge Theories, Nucl.Phys.B 133, 275(1978). [6] S.Minwalla, Restrictions Imposed by Superconformal Invariance on Quantum Field Theories, Adv.Theor.Math.Phys.2, 781(1998). [7] J.Polchinski, String Theory, Vol. II, Cambrige University Press(1998). [8] W.Nahm, Supersymmetries and Their Representations, Nucl.Phys.B 135, 149(1978). [9] P.S.Howe and P.C.West, The Complete N=2, D=10 Supergravity, Nucl.Phys.B 238, 181(1984). [10] C.G.Callan, J.A.Harvey and A.Strominger, Worldbrane Actions for String Solitons, Nucl.Phys.B 367, 60(1991). [11] G.Horowitz and A.Strominger, Nucl.Phys.B 360, 197(1991). 33