Introduction to String Theory and Gauge/Gravity duality for students in QCD and QGP phenomenology
aa r X i v : . [ h e p - ph ] A p r Introduction to String Theory and Gauge/Gravity dualityfor students in QCD and QGP phenomenology ∗ Institut de Physique Th´eorique;URA 2306, unit´e de recherche associ´ee au CNRS,IPhT, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France
Robi Peschanski † String theory has been initially derived from motivations coming fromstrong interaction phenomenology, but its application faced deep concep-tual and practical difficulties. The strong interactions found their the-oretical fundation elsewhere, namely on QCD, the quantum gauge fieldtheory of quarks and gluons. Recently, the Gauge/Gravity correspondenceallowed to initiate a reformulation of the connection between strings andgauge field theories, avoiding some of the initial drawbacks and openingthe way to new insights on the gauge theory at strong coupling and even-tually QCD. Among others, the recent applications of the Gauge/Gravitycorrespondence to the formation of the QGP, the quark-gluon plasma, inheavy-ion reactions seem to provide a physically interesting insight on phe-nomenological features of the reactions. In these lectures we will give asimplified introduction to those aspects of string theory which, at the ori-gin and in the recent developments, are connected to strong interactions,for those students which are starting to learn QCD and QGP physics froman experimental or phenomenological point of view.PACS numbers: 11.15.-q,11.25.Tq ∗ Presented at the School on QCD, Low-x Physics and Diffraction,Copanello, Calabria, Italy, July 2007. † email:[email protected] (1) proceedings printed on October 25, 2018 CONTENTSLecture I: String Theory via
Strong Interactions1.
The origin of String Theory The Veneziano Formula and Dual Resonance Models From Dual Amplitudes to Strings String Symmetries Problem: Why 26 (or 10) Dimensions?
Lecture II: Gauge/Gravity correspondence6.
An Open-Closed String Connection the AdS/CFT Correspondence Wilson loops, Minimal Surfaces and Confinement Application: A Dual Model for Dipole Amplitudes
Lecture III: Quark Gluon Plasma/Black Hole Duality10.
QGP Formation and Hydrodynamics
AdS/CFT and Holographic Hydrodynamics
QGP and Black Holes: From Statics to Dynamics
Thermalization and Isotropization
Outlook roceedings printed on October 25, 2018 Lecture I: String Theory via
Strong Interactions1.
The origin of String Theory
There is an intimate but rather controversial relationship between stronginteractions and string theory. As well-known, the birth of string theorycomes from the observation of many puzzling features of strong interactionscattering amplitudes from the phenomenological point of view. In a modernlanguage, we call them “soft” reactions since they involve small- p T hadrons,and thus a strong coupling constant α s ( p T ) = O (1) or more preventing onefrom using known perturbative techniques of field theory.It took more or less six years, from 1968 to 1974 starting from the formu-lation of the Veneziano amplitude, to obtain a first consistent formulationof the underlying string theoretical framework. Strangely enough, it is atthe very same time, in 1974, that Quantum Field Theory in the form ofQuantum Chromodynamics (QCD), started to be identified as the correctmicroscopic theoretical fundation of strong interactions in terms of quarksand gluons. In fact, it has already been realized that the construction ofstring theory in the physical 3+1-dimensional Minkowski space has led to nu-merous difficulties and inconsistencies with the observed features of stronginteractions.It is well-known that starting from that period, string theory and QCDstudies followed divergent paths, the former being promoted after 1983 toa serious candidate for the unification of fundamental forces and quantumgravity and the second showing more and more ability to describe the fea-tures of quark and gluon interactions at high energy with unprecedentingaccuracy.Now, the divorce could have been complete and definitive, when in 1997appeared a new historical twist with the conjecture named “AdS/CFT cor-respondence” and its various generalizations and developments involvinga new duality relation between gauge theories and gravitational interac-tions in an higher-dimensional space. Interestingly, some of the majordrawbacks found previously for applying string theory to strongly inter-acting gauge fields have been avoided and a new formulation of gauge fieldtheory at strong coupling emerged. Since 1997, the developments of theGauge/Gravity correspondence are numerous.Many of these new developments are not directly connected to QCD,which indeed does not admit for the moment a correct dual formulation.However, they open the way for new tools for computing amplitudes andother observables of gauge field theories in terms of their gravity dual. Onevery promising aspect of this connection concerns the formation of a Quark-Gluon Plasma (QGP) in heavy-ion reactions. Indeed, the phenomenologicalfeatures coming from the experiments at RHIC point to the formation of proceedings printed on October 25, 2018 a strongly coupled plasma of deconfined gauge fields. In this case, onemay expect that features of the AdS/CFT correspondence may be rele-vant. Hence this problem appears to give a stimulating testing ground forthe Gauge/Gravity correspondence and its physical relevance for QCD andparticle physics.Our aim in these lectures is to provide one possible introduction tothose aspects of the construction of string theory and its applications,mainly the AdS/CFT correspondence, which could be of interest for the stu-dents in QCD and QGP phenomenology. The presentation is thus “strong-interaction oriented”, with both reasons that it uses as much as possible theparticle language, and that the speaker is more appropriately considered asa particle physicist than a string theorist. In this respect he is deeply grate-ful to his string theorists friends and collaborators, in first place RomualdJanik, for their help in many subtle and often technical aspects of stringtheory. In this respect, it is quite stimulating to take part in casting a newbridge between “particles and strings”. The Veneziano Formula and Dual Resonance Models
Shapiro-Virasoro AmplitudeVeneziano Amplitude { q^2 A (s,q^2) A (s,q^2)
R P s{ Fig. 1. “Duality Diagrams” representing the Veneziano and Shapiro-Virasoro am-plitudes . The hatched surface gives a representation of the string worldsheet of theprocess. s (resp. q ) are the square of the c.o.m. energy (resp. momentum trans-fer) of the 2-body scattering amplitude. A R ( s, q ) is, at high energy, the Reggeonamplitude; A P ( s, q ) is the Pomeron amplitude, see text. The Veneziano amplitude was the effective starting point of string the-ory, even if it took some years to fully realize the connection. At first theVeneziano amplitude was proposed as a way to formulate mathematically roceedings printed on October 25, 2018 an amplitude which could describe a troubling phenomenological feature oftwo-body hadron-hadron reactions; the “Resonance-Reggeon” duality. In-deed, the prominent feature in the low-energy domain of two-body hadron-hadron reactions is the presence of numerous hadronic resonances, such thatin some channels one can even describe the whole amplitude as a superposi-tion of resonances. At high-energy, the two-body hadron-hadron reactionscan also be described by the combination of amplitudes corresponding tothe exchange in the crossed channel ( q channel in Fig. 1) of particles andresonances, under the form of Regge poles which correspond to an analyticcontinuation in the spin variable.The term “duality” has been introduced to characterize the fact thatone should not describe the amplitude by adding the two kinds of descrip-tion. On the contrary, one expects an equivalent description in terms of asuperposition of resonance states and as a superposition of Reggeon contri-butions. In order to represent this feature, “duality diagrams” have beenproposed, as shown in Fig. 1, where the 2-dimensional surface is drawn todescribe the summation over states in the direct channel ( s -channel reso-nances, whre s is the square of the total energy) as well in the exchangedone ( t -channel reggeons where t = − q is the analytic continuation of thesquare of the total energy in the crossed channel). In terms of strings, itwill correspond to the string worldsheet associated to the amplitudes.The phenomenological constructions which were proposed to formulatethis property are called the “Dual Resonance Models”. As we shall see fur-ther, this qualitative representation will be promoted to a rigorous meaningin terms of string propagation and interaction. Note already that a topo-logical feature emerges from the diagrams of Fig. 1. Indeed, if one closesthe quark lines (in red in Fig. 1) corresponding to the ingoing and outgoingstates, they are characterized by a planar topology (Reggeon exchange, leftdiagram in Fig. 1) or a sphere topology with two holes (Pomeron exchange,right diagram in Fig. 1). This topological characteristics are indeed a ba-sic feature of string theory, corresponding to the invariance of the stringamplitudes w.r.t. the parametrization of the surface spanned by the string. [Exercise 2.1: Show that the “Shapiro-Virasoro diagram” istopologically equivalent to sticking two “Veneziano diagrams” to-gether in a specific way, i.e. with a “twist”] The first and pioneering step in the theoretical approach to dual reso-nance models is the proposal by Veneziano of a mathematical realizationof the dual amplitude corresponding to the planar topology (Reggeon ex-change) the well-known “Veneziano Amplitude”. In its simplest version it proceedings printed on October 25, 2018 reads: A R ( s, t ) = Γ( − α ( s )) Γ( − α ( t ))Γ( − α ( s ) − α ( t )) (1)with t ≡ − q and linear “Regge trajectories” α ( m ) = α (0) + α ′ m . (2)The Veneziano amplitude has quite remarkable features, thanks to the prop-erties of the Gamma function, which give an explicit realization of the du-ality properties. Indeed, in the s-channel as well as in the t-channel, itcorresponds to an infinite series of poles (and thus of states), but with afinite number of spins for each value of positive integer “level” α ( m ) = n ,since α ( s or t ) → n ∈ N ⇒ Γ( − α ( s ))Γ( − α ( t ))Γ( − α ( s ) − α ( t )) ≈≈ P olynomial { degree ≤ n } ( t or s )( n − α ( s or t )) . (3) [Exercise 2.2: prove formula (3) from Gamma function properties] Concerning the high-energy behaviour, one obtains s → ∞ ⇒ A R ( s, t ) = Γ( − α ( s )) Γ( − α ( t ))Γ( − α ( s ) − α ( t )) ≈ s α ( t ) Γ( − α ( t )) , (4)which is the typical dominant “Regge behaviour”, phenomenologically ob-served in hadron-hadron reactions, where the high-energy amplitude in thes-channel corresponds to the dominant Regge trajectory (higher spin fora given mass) in the crossed channels. Subdominant terms correspond tosecondary linear regge trajectories. A similar approach was proposed forthe sphere topology (the “Pomeron exchange”), resulting in the Shapiro-Virasoro amplitude A P ( s, t ) . [Exercise 2.3: prove formula (4) from Gamma function prop-erties] For phenomenological purpose, despite its remarkable properties, theVeneziano amplitude is not the full answer. Among other problems, allpoles are on the real s or t axis, and thus they correspond to unphysicalstable states and not resonances. In the following we shall focus on thetheoretical meaning of the Veneziano amplitude as the seed for string theory.As we shall see, a rigorous connection between the Veneziano amplitude andstrong interaction physics which was its initial motivation, required a moresophisticated framework. roceedings printed on October 25, 2018 From Dual Amplitudes to Strings (σ,τ) X d−Space σ τ Fig. 2.
String apparatus . Left: the 2-dimensional ( σ, τ ) string worldsheet. Right:the string embedding in the target space (here: flat d-dimensional space). X µ ( σ, τ )is the string position operator, see text. As clear from formula (3), the Veneziano amplitude corresponds to aninfinitely growing number of states as a function of the level ( n = α ( m n )).Such a spectrum is reminiscent of the classical oscillatory modes of a string.However, the construction of a quantum theory of strings and the identifica-tion of the Veneziano amplitude as a particular string interaction amplitudetook some time. In the following we will give the structure of the quantumposition operator for a (bosonic) string and sketch the derivation of theVeneziano amplitude as the tree-level string interaction amplitude.In order to describe the degrees of freedom of a relativistic string, it isuseful to introduce the following set of bosonic operators:[ a n,µ , a − m,ν ] = η µν δ nm ; [ˆ q µ , ˆ p ν ] = iη µν , (5)where one considers for the target space, see Fig. 2, the d -dimensional flatmetrics µ, ν ⇒ η µν = n , − ; − ∗ [ d − ] o . (6)In (5), the operators ˆ q, ˆ p describe the momentum and position of the stringcenter of mass, while a, a † are the bosonic annihilation and creation opera-tors describing the oscillator modes of the string. Using these definition, one proceedings printed on October 25, 2018 builds the string position operator at the boundary X µ ( τ, σ = ) as follows X µ ( σ = 0 , τ ) ≡ Q µ ( z ) = Q (+) µ ( z ) + Q (0) µ ( z ) + Q ( − ) µ ( z ) ; z = e iτ , (7)with Q (+) = i √ α ′ ∞ X n =1 a n √ n z − n ; Q ( − ) = − i √ α ′ ∞ X n =1 a − n √ n z n ; Q (0) = ˆ q − iα ′ ˆ p log z . (8)The σ -dependence is restored, specifying the boundary conditions of theopen string, by multiplying each term a ± n / √ n z ∓ n in (8) by cos nσ. The calculation of the amplitude is made by integration over the world-sheet variables of an overlap over plane wave operators A ∝ DQ j e ip j X j E σ,τ .Introducing the normal ordered vertex operators V ( p ; z ) ≡ : e ip · Q ( z ) := e ip · Q ( − ) e ip · ˆ q e α ′ p · ˆ p e ip · Q (+) , (9)the Veneziano amplitude B ( p + p → p + p ) in terms of string vertexoperators reads:(2 π ) d δ ( d ) ( X i =1 p i ) B = Z dz h , p | V ( p ; z = 1) V ( p ; z ) | , p i , (10)where the external states are h , p | ∝ h , | V ( p ; z → ∞ ) | , p i ∝ V ( p ; z → | , i (11)and where | , i denotes the vacuum state. The harmonic oscillators actingon this state build the Hilbert space of string states. The fact that three overthe four z i coordinates can be fixed at will comes from the string symmetrieswhich will be discussed in the next section.Using the definition (9) together with the relation V ( p i ; z i ) V ( p j ; z j ) =: V ( p i ; z i ) V ( p j ; z j ) : ( z i − z j ) α ′ p i · .p j (12)one finally finds B = Z dz z − − α ( s ) (1 − z ) − − α ( t ) = Γ( − α ( s ))Γ( − α ( t ))Γ( − α ( s ) − α ( t )) (13)which is nothing else han the Veneziano amplitude. An important steptowards the construction of string theory was made when the suitable gen-eralization to arbitrary number of legs B → B N has been performed. Theoperator formalism was then found and fully confirmed. roceedings printed on October 25, 2018 [Exercise 3.1: prove formula (12) from relations (5,8), usingthe Baker-Hausdorff formula e A e B = e B e A e [ A,B ] if [ A, B ] is scalar][Exercise 3.2: prove formula (13) from (10-12)]4. String Symmetries
The symmetries play a crucial role in the properties of string theory.Let us discuss the main features of string symmetries. There exists anexact symmetry group on the string worldsheet. It contains, respectively,dilatation, translation and inversion in the worldsheet variable z ≡ e iτ , withgenerators z ddz , ddz , − z ddz , respectively. These transformations correspond to the infinitesimal genera-tors (the algebra ) of the Projective (conformal) invariance group SU (1 , SU (1 ,
1) invariance with 3 generators, which allows one to arbitrarily fix 3among N values of the worldsheet variables in the expression of the ampli-tude B N , e.g. leaving one interation for the Veneziano amplitude (13).By extension, one also introduces the generalized conformal transforms z n +1 ddz , for all n, which however will not form an exact symmetry algebraat the quantum level, as we will discuss now. They will give rise to thefamous V irasoro Algebra with a “central extension” or quantum anomaly.As usual in the formulation of symmetries, a key point is to find anappropriate representation of the algebra in terms of physically meaningfulobjects, here the annihilation and creation operators describing the string.For this sake one forms the following operators L n = √ α ′ n ˆ p · a n + ∞ X m =1 p m ( n + m ) a n + m · a m + 12 n − X m =1 p m ( n − m ) a m − n · a m (14)which possess nice algebraic properties, when acting on the string positionoperator (7).Let us first consider the set of operators ( L , L − , L ) . One can provethat[ L , Q ( z )] = − z dQdz ; [ L − , Q ( z )] = − dQdz ; [ L , Q ( z )] = − z dQdz , (15)which demonstrate that they form an adequate representation of the algebra proceedings printed on October 25, 2018 of the projective symmetry group SU (1 , . More generally one finds[ L n , Q ( z )] = − z n +1 dQdz , (16)and thus a representation of the generalized projective transformations onthe string position operator and thus on the string states. [Exercise 4.1: prove formulae (15,16) using (14) and the com-mutation relations (5)] Now the symmetry properties will come from the commutation relationsbetween the L n generators, i.e. the Virasoro Algebra. One finds[ L n , L m ] = ( n − m ) L n + m + d n ( n − δ n + m, . (17) [Exercise 4.2: prove formula (17), starting with the simplercases when n = 0 , ± . ] The formula (17) calls for comments. From the algebra, it is easy tonote that, restricting (17) to n = 0 , ± , one finds[ L ± , L ] = ± L ± ; [ L , L − ] = 2 L (18)which is the algebra of generators of the SU (1 ,
1) group (analoguous to SU (2) and its generators L ± , L z , but with a change of sign in the [ L − , L ]relation which is related to the non-compactness of the group). Hence thissubalgebra indicates the exact symmetry under projective transforms.For higher | n | > , one notes the extra contribution d n ( n − δ n + m, , which is proportional to the target space dimension d. The “central charge”( d with the conventional normalization) is a fundamental contribution,showing that the generalized projective group is not an exact symmetry(unless other contributions cancel the central charge due to the dimension,which is precisely the condition for the existence of a consistent string the-ory). It will in fact be crucial in the striking feature of string theory toimply a constraint on the target space, i.e. the space in which it moves! Why 26 (or 10) Dimensions?
Now we have the tools to understand the old puzzle which has jeop-ardized the initial strong-interactions/strings connection. The question iswhether one can construct 4-dimensional string amplitudes in Minkowskispace and the answer is in fact “no”. Let us list the problems when facingthe construction of 4-d strings in a theoretically consistent way. One findsth following problems roceedings printed on October 25, 2018 i) Open (resp. closed) strings ⇒ Gauge (resp. Gravity) at lower energyii) Zero-mass asymptotic states: gauge bosons, gravitonsiii) Hadron spectrum not compatibleiv) QCD not obtainedv) Problem of dimensions: The Minkowskian string (resp. superstring)target-space is 26 (resp. 10) dimensional [Exercise 5.1: Given the ρ (spin 1, Mass 770 MeV) and f (spin2, mass 1270 MeV) mesons which are on the dominant hadronicReggeon trajectory and the fact that total hadronic cross-sectionsare constant with energy (up to logarithms) illustrate the thirdpoint of the list] Let us consider the problem of dimensions as a major illustration of thedeep implications of quantum consistency and symmetries of string theorybased on the Virasoro Algebra.The problem can be viewed in different ways. Here we shall take thepoint of view of the construction of an Hilbert space made of positivenorm particle states. Let us first remind the well-known construction ofthe Hilbert space for QED.If one considers the oscillator construction of the QED Hilbert space oneis led to satisfy, choosing the covariant gauge, the condition q µ a † µ | i = 0 , where q µ a † µ denotes the QED analogue (indeed ancestor) of the creationL-operator L . As is known from QED quantification, one may classify thefour vector states a † µ | i within three categories, namely a † T | i = | φ T i T ransverse photon states X | φ , i| = 1 a † − a † | i = | l i Longitudinal photon state h l | l i = 0 a † + a † | i = | s i Spurious photon state q µ a † µ = 0 . In a similar way for strings, and now for the whole hierarchy of operators L n , one considers the following (covariant) gauge conditions L n | φ string i = 0 f or n > , (19)which allows a similar generalized classification of states L n | φ string i = 0 On − shell states positive N orm h l string | l string i = 0 Of − shell states Zero − N ormL n | s string i 6 = 0 Spurious states U nphysical States . proceedings printed on October 25, 2018 Now, the key point for the construction of a consistent Hilbert space ofstring states is that spurious states decouple from the other ones. Buildinga simple example we shall prove that it implies a necessary condition overthe target space dimension. For simplicity, we shall not enter in the proofthat this is a sufficient condition for eliminating all spurious states from thespectrum.Let us consider the following spurious state: | s string i = L − | φ i + L − | φ i . (20) [Exercise 5.2: prove that the state defined by formula (20) isindeed spurious if the states φ , are physical on-shell states] Then acting on | s string i with an appropriate combination of L operators,one finds (cid:26) L + 32 L L (cid:27) | s string i = X i | s string , i i + d − | φ string i (21) [Exercise 5.3: prove Equation (21) by inserting the state (20)using the Virasoro algebra relation [ L , L − ] = 4 L + d and classi-fying the obtained states using (20)] The decoupling of spurious states requires that the subspace of spuriousstates should remain orthogonal from the physical spaces. Hence one getsthe condition d = 24 , characteristic of the bosonic string consistency. Asimilar condition applies to the supersymmetric versions of string theory inMinkowskian space, leading to d = 10 . The decoupling of non-physical statesis thus directly a consequence of the Virasoro Algebra and more specificallyof its central charge properties. roceedings printed on October 25, 2018 Lecture II: Gauge/Gravity correspondence6.
An Open-Closed String connection
We have discussed in section 5 the drawbacks of the initial attemptsto obtain strong interaction physics from string theory. Indeed, on the string-theoretical point of view, the dimensionality of a Minkowskian targetspace (26 or 10), the existence of zero mass states and their connection togauge field theory and gravity, among other features, seemed to invalidatea string description of hadronic interactions. On the field-theoretical pointof view, based on the existence of a satisfactory theoretical understandingof quark and gluon interactions at weak coupling in terms of QuantumChromodynamics, the challenge of a correct description of interactions atlong distance relies on the still unsolved problem of computing observablesat strong coupling. As we shall see now the Gauge/Gravity duality, a deep“geometrical” property of string amplitudes, and its precise formulation inthe case of the so-called AdS/CFT correspondence, seem to overcome atleast in principle, the difficulties from both string and field theory sides andopens a new way for the string approach to strong interactions.Let us first give a quite general argument, giving a qualitative explana-tion of this new way of approaching the problem. It relies on a connectionbetween open and closed strings which is displayed on Fig. 3. We considerthe configuration of two stacks of D-branes in the 10-dimensional targetspace of a superstring theory. The D-branes are kind of “solitonic” objectswhich form a consistent background in the string-theoretical framework. Inparticular, they are the locus of open string endpoints. In Fig. 3 they areunderstood as stacks of D branes which are two sets of copies of the 3 + 1Minkowski space, separated by a distance r in a fifth dimension, which willplay a special role in the following.One can geometrically interpret the cylinder shape of the interaction intwo equivalent ways: i) It may be seen as the propagation of a closed loop,starting on one D brane-stack and reaching the second one; ii) It may beseen alternatively as a one-loop contribution from open-strings since openstrings may have end-points on D branes. This equivalence, once given aprecise formulation in terms of a specific string theory, has quite intringuingand far-reaching consequences.Let us list some of them: • Gauge/Gravity duality. As we have alluded to in section , the mass-less modes of the string states are gauge fields for the open strings andgravitons for the closed strings. Hence the interaction amplitude depictedin Fig. 3 potentially identifies a tree-level gravitational interaction with agauge one at one-loop. • Short/Long distance relationship. When one consider a large 5th di- proceedings printed on October 25, 2018 Fig. 3.
Open ⇔ Closed duality and D-Branes.
Left: Cylinder topology describinga string interaction between two stacks of D-branes; Right: the interaction can bedescribed either by the exchange of a closed string propagating between the twostacks of branes or by the one-loop contribution of an open string attached to thetwo stacks (from reference [10]). mensional distance r , the closed string exchange is expected to be describedby a classical, weakly coupled, gravitational interaction. By contrast, atsmall distance, the open string interaction is well-described by the exchangeof its zero-mode states, that is the gauge vector fields. This is theoreticallyjustified, since the exchange of open strings with multiple combinations ofopen-string end-points between stacks of near-by D branes, are related atweak string coupling to generic SU ( N ) gauge field theories (see Fig. 4). • Weak/Strong coupling relationship. At short distance r, the SU ( N )gauge coupling is weak (due to asymptotic freedom for N ≥ D branes are some kind of very massive objets and are expected to generatea strong gravitational field in their neighbourhood. On the other end of thecomparison, at long distance r, the gravity is weak, while the open stringinteraction is expected to become strongly coupled, and moreover, all theexcitations of the open strings which correspond to the massive oscillatormodes are expected to contribute.From that comes the main feature of the Gauge/Gravity duality; Itmakes a deep connection between weak coupling on one side of the cor-respondence to the strong coupling regime of the other side. It is thusintrinsically a weak/strong coupling duality .In the present series of lectures, we are interested in the weak grav-ity/strong gauge coupling combination (the investigation of the weak gauge vs. strong gravity duality is yet another fascinating challenge). Obtainingvaluable new tools of investigation of gauge theories at strong coupling fromtheir gravity duals at weak coupling, and thus accessible to a quantitativeapproach. roceedings printed on October 25, 2018 The AdS/CFT correspondence
Fig. 4. SU ( N ) Gauge theory from D branes . The D branes are considered to bepractically at the same location in 10-dimensional space. The (short) open stringcombination of end-points leads to the adjoint representation of SU ( N ). The AdS/CFT correspondence has many interesting both formal andphysical facets. Concerning the aspects which are of interest for our prob-lem, it allows one to find relations between gauge field theories at strongcoupling and string gravity at weak coupling in the limit of large numberof colours ( N c → ∞ ). It can be examined quite precisely in the AdS /CFT case where CFT is the 4-dimensional conformal field theory correspondingto the SU ( N ) gauge theory with N = 4 supersymmetries. [Exercise 6.1: How many gauge bosons are expected in Fig. 4?] Some existing extensions to other gauge theories with broken conformalsymmetry and less or no supersymmetries will be valuable for our approach,since they lead to confining gauge theories which are more similar to QCD.Note that the appropriate string gravity dual of QCD has not yet beenidentified, and thus we are forced to restrict for the moment our use of proceedings printed on October 25, 2018 AdS/CFT correspondence to generic features of confining theories duals,see a discussion further on in this section.Let us recall the canonical derivation leading to the AdS backgroundsee Fig. 5. One starts from the (super)gravity classical solution of a systemof N D -branes in a 10 − D space of the (type IIB) superstrings. The metricssolution of the (super)Einstein equations read ds = f − / ( − dt + X − dx i )+ f / ( dr + r d Ω ) , (22)where the first four coordinates are on the brane and r corresponds to thecoordinate along the normal to the branes. In formula (22), one writes f = 1 + R r ; R = 4 πg Y M α ′ N , (23)where g Y M N is the so-called ‘t Hooft-Yang-Mills coupling equal to the stringcoupling g s and α ′ the string tension.One considers the limiting behaviour considered by Maldacena, whereone zooms on the neighbourhood of the branes while in the same time goingto the limit of weak string slope α ′ . The near-by space-time is thus distorteddue to the (super) gravitational field of the branes. One goes to the limitwhere
R f ixed ; α ′ ( → r ( → → z f ixed . (24)This, from the second equation of (23) obviously implies α ′ → , g Y M N ∼ R πα ′ → ∞ , (25) i.e. both a weak coupling limit for the string theory and a strong couplinglimit for the dual gauge field theory. By reorganizing the two parts of themetrics one obtains ds = 1 z ( − dt + X dx i + dz ) + R d Ω , (26)which corresponds to the AdS × S background structure. In (26) z ( − dt + P − dx i + dz ) , describes a Anti de Sitter geometrical space which is a5-dimensional hyperbolo¨ıd of equation − x − x + P i =2 x i = − R in a 6-dimensional flat Minkowski space S is the 5-sphere of metric R d Ω where the 5-d version of de Sitter geometrical space, whose 4-d version appears in generalrelativity, has a plus sign, i.e. obeys the equation − x + x + P i =2 x i = R . . roceedings printed on October 25, 2018 Weak Gravity
Strong Gravity
Strong Coupling
Weak Coupling
Macroscopic
AdS CFT
Microscopic } Gravity Source } N-Branes AdS x S
Superstring
SU(N) Gauge Theory on the N-Branes
Duality
Fig. 5.
Sketch of the AdS /CFT correspondence. Left: At long distance, thegravity source of the branes generate an Anti de Sitter background metric; Right:At short distance, the open strings on the branes boil down to a non-abelian SU ( N )gauge field theory with N = 4 supersymmetries. Ω is the 5-dimensional solid angle. More detailed analysis shows that theisometry group of the 5-sphere may be considered as the “gravity dual” ofthe N = 4 supersymmetries (for completion, the quantum number dual to N c , the number of colours, is the invariant charge carried by a Ramond-Ramond form field in (type IIB) superstrings.In the case of confining backgrounds, an intrinsic scale breaks conformalinvariance and is brought in the dual theory through e.g. a geometricalconstraint. For instance, a confining gauge theory is expected to be dual tostring theory in an AdS BH Black Hole (BH) background ds BH = 169 1 f ( z ) dz z + η µν dx µ dx ν z + . . . (27) proceedings printed on October 25, 2018 where f ( z ) = z / [1 − ( z/R ) ] and R is the position of the BH horizon.One may use this type of background to study the interplay between theconfining nature of gauge theory and its reggeization properties. Actuallythe qualitative arguments and approximations should be generic for mostconfining backgrounds. For instance, other geometries for (supersymmet-ric) confining theories have been discussed in this respect. They have theproperty that for small z , the geometry looks like AdS × S (in up to log-arithmic corrections related to asymptotic freedom) giving a coulombic q ¯ q potential. For large z the geometry is effectively flat. In all cases there isa scale, similar to R above, which marks a transition between the small z and large z regimes. Wilson Loops, Minimal Surfaces and Confinement
We will find appropriate in the next section to formulate the scatteringamplitudes in terms of Wilson loops, since the Gauge/Gravity “dictionnary”for Wilson loops has been proved to be well suitable for dual properties ingeneral. Let us thus introduce this dictionnary in the following. Let usintroduce the general framework within which Wilson loops in the “bound-ary” gauge field theory are in correspondence with minimal surfaces in the“bulk”.In a framework suitable for performing the AdS/CFT correspondence,quarks (resp. antiquarks) can be represented by colour sources in thefundamental (resp. anti fundamental) reps. of SU(N). In order to illustratethe way how one may formulate in practice the AdS/CFT correspondencein a context similar to QCD, let us consider first the example of the vacuumexpectation value ( vev ) of Wilson loops in a configuration parallel to the timedirection of the branes. We consider the large time limit and thus the loopsclose “near” infinity in the time direction (see Fig. 6). This configurationallows a determination of the potential between colour charges. The rˆoleof colour charges is played by open string states elongated between a stackof N c D branes on one side and one D brane near the boundary of AdSspace.The correspondence can be formulated as follows h e iP R C ~A · ~dl i = Z Σ e − Area (Σ) α ′ , (28)where C is the Wilson loop contour near the D branes and Σ any surfacein AdS -space with C as the boundary, see Fig. 6. In order to get correct quark degrees of freedom, e.g. flavor, a more complete geo-metrical set-up including D branes is used. For simplicity, an extra singlet term in the left-hand exponent, allowing to canceldivergences, is not written here. roceedings printed on October 25, 2018 HORIZON
Boundary
Fig. 6.
Sketch of minimal surfaces with Wilson loop boundaries: the potential con-figuration.
The Wilson loops correspond here to the potential configuration withtransverse boundaries at distance L and parallel time-like boundaries going to in-finity at the limit, see text. Left: Minimal surface in the presence of a confiningbackground such as (27); Right: Minimal surface corresponding to an AdS -likebackground such as (26). NB: In the figure, it is represented in an approximate casewhen the ratio of the Wilson loop transverse size to the horizon is small. Withouthorizon, the minimal surface at large transverse size would extend without limit. In the semi-classical approximation for the right-hand (gravity) side ofthe relation (28) where the Gauge/Gravity correspondence would give thestrong coupling value of the left-hand (gauge) side, the integration oversurfaces Σ boils down to h e iP R C ~A · ~dl i ≈ e − Areaminα ′ × F luct. , (29)where
Area min is the minimal surface whose boundary is the gauge-theoryWilson loop. The factor denoted
F luct. refers to the fluctuation determinantaround the minimal surface, corresponding to the first quantum correctionbeyond the classical approximation. It gives an interesting calculable semi-classical correction, as we shall see on the amplitude exemple.In Fig. 6, we have sketched the form of minimal surface solutions for the“confining”
AdS BH case, (see above (27)). For large separation of Wilsonlines, the minimal surface is bounded near the horizon and is consequentlycurved. At smaller separation, the solution becomes again similar to theconformal case, since the horizon cut-off does not play a big rˆole.In gauge theory, the quark-quark potential is known to be obtained froma suitable time-like infinite limit of the rectangular Wilson loop vev . One proceedings printed on October 25, 2018 has V ( L ) = lim T →∞ T × log h W ilson Loop i (30)Thanks to the Gauge/Gravity correspondence (28) and the classical approx-imation (29), one is able to get the strong coupling limit of the interquarkpotential from the large time limit of the Wilson loop v.e.v.: AdS : h W ilson Loop i = e T V ( L ) ∼ e √ g Y M N TL AdS BH : h W ilson Loop i = e T V ( L ) ∼ e TLR , where , are constants. The potential behaviour obeys the nonconfiningCoulomb law V ( L ) ∝ /L for the AdS case and the confining law V ( L ) ∝ L for the AdS BH case. An interesting nonanalytic dependence over the square-root coupling appears. Note again that, even in the case of a confininggeometry with an horizon at R , Wilson lines separated by a distance
L < Now we will come back to our original problem of describing in a consis-tent way the strong interaction two-body amplitudes which correspond, e.g. to the processes depicted in fig.1. There are different approaches to scat-tering amplitudes using gravity duals, including recently the formulation ofgluon amplitudes at strong coupling in the SU ( N ) gauge theory with N = 4supersymmetries. However, since we are interested in the present lecture inthe approach to hadronic scattering amplitudes, we search for both a field-theoretical formulation based on QCD and the determination of the gravityduals of the corresponding amplitudes. Concerning the nature of the dualtheory, the gravity dual theory of QCD has not yet been identified. Moregenerally, the problem of deriving a correspondence for a confining theorywith asymptotic freedom is not yet achieved. In the following we shall usean approach where only generic features of confining backgrounds allow todetermine some properties of the amplitudes. The price we pay is that wewill only be able to discuss the high-energy behaviour of the amplitudes, i.e. the high-energy regime, which was discussed in section 2 for instancein Eq.(4). Other properties of the amplitudes will not be discussed, andprobably are more difficult to derive in the absence of a better determineddual background to QCD. Using the AdS/CFT correspondence, we will find roceedings printed on October 25, 2018 a aL θ t xy Fig. 7. Wilson loops for Dipole-Dipole scattering in configuration space . The figureis drawn in the physical boundary configuration space ( t, x, y, z ) . Left: the two Wil-son loops corresponding to the elastic dipole-dipole amplitude A d − d P ( s, q ) . L is theimpact parameter distance between the colliding dipoles, a is the (small) q ¯ q distancein the dipoles. All q ¯ q trajectories are straight lines in the eikonal approximationRight: the Wilson loop (1 → → ′ → ′ → → → ′ → ′ → 1) in configurationspace corresponds to the inelastic dipole-dipole amplitude A d − d R ( s, q ) . The Wilsonlines 1 → → that two-body high-energy amplitudes in gauge field theories can be relatedto specific configurations of minimal surfaces.Using the Wilson loop properties, it is now possible to formulate theGauge/Gravity correspondence for the elastic and inelastic scattering am-plitude of massive colorless q ¯ q states in the space of QCD color dipoles. InFig. 7, one displayed the elastic and inelastic amplitudes of two dipoles inconfiguration space, corresponding respectively to A d − d P ( s, q ), and A d − d R ( s, q )appearing in Lecture I. We will here consider the amplitudes at high energy,i.e. the problem of “Reggeization”. Indeed, at high energy, fast movingcolour sources propagate along linear trajectories in coordinate space thanksto the eikonal approximation. This important property of high energy prop-agation of color sources will be helpful for the evaluation of the amplitudesthrough Gauge/Gravity duality.Let us first consider the elastic dipole amplitude, i.e. the left diagram ofFig. 7. In the gauge field theory, one may write it in terms of a correlatorbetween two Wilson lines in configuration space, namely A d − d P ( s, q ) = − is Z d x ⊥ e iqx ⊥ (cid:28) W W h W i h W i − (cid:29) (31) proceedings printed on October 25, 2018 where the Wilson loops W , W corresponding to the two colliding dipolesfollow classical straight lines for quark(antiquark) trajectories and close atinfinite time, as for the potential. The normalization {h W i h W i} of thecorrelator ensures that the amplitude vanishes when the Wilson loops getdecorrelated at large distances.One useful technique is to formulate the duality property in Euclidean R space where it takes the form of a well-defined geometrical interpretationin terms of a minimal surface problem. Then the analytic continuation fromEuclidean to Minkowski space allows one to find the physical solution.The Wilson line vev can be expressed as a minimal surface problemwith (approximately) two copies (for dipole size a ∼ 0) of a minimal surfacewhose boundaries are straight lines in a 3-dimensional coordinate space,placed at an impact parameter distance L and rotated one with respect tothe other by an angle θ, see Fig. 7. then the amplitude will be obtainedthrough the analytic continuation θ ↔ − iχ ; t Eucl ↔ it Mink , (32)where χ = log s/m is the total rapidity interval.In flat space, with the same boundary conditions, the minimal surfaceis the helicoid . One thus realizes that the problem can be formulated asa minimal surface problem whose mathematically well-defined solution isa generalized helicoidal manifold embedded in curved background spaces,such as Euclidean AdS Spaces. Unfortunately, this problem is rather dif-ficult to solve analytically, even in flat space. It is known as the Plateauproblem, namely the determination of minimal surfaces for given boundaryconditions.In fact, the definition of the minimal surface geometry in the conditionsof a confined AdS BH metrics (27) appears to be simpler, at least for theleading contribution. Indeed, in the configuration of Wilson lines of Fig. 6in the context of a confining theory, the AdS BH metrics is characterizedby a singularity at z = 0 which implies a rapid growth in the z directiontowards the D branes, then stopped near the horizon at z . Thus, to agood approximation, and for a large enough impact parameter (comparedto the horizon distance), the main contribution to the minimal area is fromthe metrics in the bulk near z which is nearly flat. Hence, near z , therelevant minimal area can be drawn on a classical helicoid , whose analyticexpression is known. This expression contains a logarithmic singularity interms of kinematical variables, which turns out to be essential to generatean imaginary part in the action after analytic continuation to Minkowskispace. roceedings printed on October 25, 2018 After analytic continuation, one obtains A P ( s, q ) = 2 is Z d~l e i~q · ~l − ( √ g Y M Nχ L R ) ∝ s − q R √ g Y M N . (33)which represents a Reggeized elastic amplitude, with a linear Regge trajec-tory α P ( q ) = α P (0) − q α ′ P ≡ − q R q g Y M N (34)characterized by a Pomeron intercept α P (0) = 1 and a Regge slope, definedin terms of the gravity dual parameters by α ′ P = R √ g Y M N , where g Y M N ≡ g s is the string or ‘t Hooft coupling.Let us now consider the dipole-dipole inelastic amplitude(11 ′ ) + (22 ′ ) −→ (33 ′ ) + (44 ′ ) , (35)represented in configuration space on Fig. 7, right. The helicoidal geometryremains valid due to the eikonal approximation for the “spectator quarks”,namely the 1 → → A R ( s, q ) = 2 i Z d~l e i~q · ~l − ( √ g Y M Nχ L R ) ∝ s − q R √ g Y M N , (36)corresponding to a linear Regge trajectory α R ( q ) = α R (0) − α ′ R q ≡ − q R q g Y M N (37)characterized by a “Reggeon” intercept α R = 0 and a Regge slope α ′ R = R √ g Y M N . Note that the slope α ′ R is related to the quark potential calculatedwithin the same AdS/CFT framework and, quite interestingly we find α ′ R =4 α ′ P , to be compared with the string result at weak coupling α ′ R = 2 α ′ P . Up to now, we restricted ourselves to a classical approximation based onthe evaluation of minimal surfaces solutions for the various Wilson loops in-volved in the preceeding calculations. It is interesting to note that a furtherstep can be done by evaluating the contribution of quadratic fluctuations of proceedings printed on October 25, 2018 the string worldsheet around the minimal surfaces in the case where thesesurfaces are embedded in helicoids, as discussed for the confining back-grounds. The semi classical correction comes from the fluctuations near theminimal surface. The main outcome is that this semi classical correctioncan be computed and is intimately related to the well-known “universal”L¨uscher term contribution to the interquark potential.After some non-trivial steps, the formulae (33,36) get corrected as follows A P ( s, q ) ∝ s α P ( − q ) = s n ⊥ − q α ′R A R ( s, q ) ∝ s α R ( − q ) = s n ⊥ − q α ′R (38)where n ⊥ is the number of zero modes of the gravity dual theory in thetransverse-to-the-branes directions. The result is just equivalent to theknown L¨uscher term in 4 d found in the large-time expansion of the rect-angular Wilson loop, except that the number of zero modes n ⊥ = d − d space.It is interesting to note that this theoretical feature is in qualitativeagreement with the phenomenology of soft scattering. Indeed once we fixthe α ′ from the phenomenological value of the static q ¯ q potential ( α ′ ∼ . GeV − ) we get for the slopes α R = α ′ ∼ . GeV − and α P = α ′ / ∼ . GeV − in good agreement with the phenomenological slopes.A second feature is the relation between the Pomeron and Reggeon in-tercepts. At the classical level of our approach these are respectively 1 and0 . Note that this classical contribution matches what is obtained from sim-ple exchanges of two gluons and quark-antiquark pair, respectively, in the t ≡ − q channel. The fluctuation (quantum) contributions to the Reggeonand Pomeron are also related by the factor 4.Adding both classical and fluctuation contributions gives an estimatewhich is in qualitative agreement with the observed intercepts. Indeed,when calculating the fluctuations around a minimal surface near the horizonin the BH backgrounds there could be n ⊥ = 7 , n ⊥ = 7 , . − . 083 for the Pomeron and 0 . − . 33 forthe Reggeon. This result is in agreement with the observed intercept forthe “Pomeron” and somewhat below the intercepts ∼ . roceedings printed on October 25, 2018 Lecture III: Quark-Gluon Plasma/Black Hole Duality10. QGP formation and Hydrodynamics pre-equilibrium stageQGPmixed phasehadronic gas describedby hydrodynamics Fig. 8. Description of QGP formation in heavy ion collisions . The kinematic land-scape is defined by τ = p x − x ; η = log x + x x − x ; x T = { x , x } , where the co-ordinates along the light-cone are x ± x , the transverse ones are { x , x } and τ is the proper time, η the “space-time rapidity”. The formation of a QGP (Quark Gluon Plasma) is expected to be real-ized in high-energy heavy-ion collisons, e.g. at RHIC and soon at the LHC.One of the main tools for the description of such a formation is the rele-vance of relativistic hydrodynamic equations in some intermediate stage ofthe collisions, see Fig. 8. The problem of the hydrodynamic description isthe somewhat indirect relation with the underlying fundamental theory. In-deed, the experimental observations seem to indicate an almost perfect-fluidbehaviour with small shear viscosity, which naturally leads to consider a the-ory at strong coupling and thus within the yet unknown non-perturbativeregime of QCD. Moreover the QGP formation appears to be fast, whichmay also point towards strong coupling properties. Another key point ofthe standard description is the approximate boost-invariance of the processin the central rapidity region, that is the well-known Bjorken flow . The goalof the string theoretic approach is to make use of the Gauge/Gravity corre-spondence as a way to tackle the problem of the hydrodynamic behaviourfrom the fundamental theory point of view. It allows to draw quantita-tive relations between a strongly coupled gauge field theory and a weaklycoupled string theoryMore specifically, the AdS/CFT correspondence between the N = 4 su-persymmetric SU ( N ) gauge theory and superstrings in 10 dimensions canbe used as a calculational laboratory for this kind of approach, at least as proceedings printed on October 25, 2018 a first stage before a more realistic application to QCD. The unconfinedcharacter of the QGP gives some hope that the explicit AdS/CFT examplecould be useful despite the lack of asymptotic freedom and other aspectsspecific of QCD. AdS/CFT and Holographic Hydrodynamics One typical and fascinating aspect of the Gauge/Gravity duality is theproperty of holography as we have seen in section 8. It states that theamount of information contained in the boundary gauge theory (on thebrane) is the same as the one contained in the bulk string theory. In ourproblem, we shall make use in a quantitative way of this property by tak-ing advantage of one of the remarkable relations due to the “holographicrenormalization”. Using the Fefferman-Graham coordinate system for themetric ds = g µν ( z ) dx µ dx ν + dz z . One can write g µν = g (0) µν (= η µν ) + z g (2) µν (= 0) + z h T µν i + z g (3) µν . . . + , (39)where g µν is the bulk metric in 5 dimensions, η µν , the boundary metricin physical (3+1) Minkowski space and h T µν i , the v.e.v. of the physicalenergy-momentum tensor. The higher coefficients of the expansion over thefifth dimension z can be obtained by the Einstein equations in the bulkprovided the boundary energy-momentum tensor fulfils the zero-trace andcontinuity equations. It is important to note that the relation (39) to bevalid requires for the boundary energy-momentum tensor, by consistency T µµ = 0 ; D ν T µν = 0 , which are nothing else than the properties of a physical 4-d T µν with thezero trace condition of a conformal theory, verified e.g. by the perfect fluid.The interesting observation on which we shall elaborate, namely thatthere is a non-trivial dual relation between a perfect fluid at rest in (3+1)dimensions and a static 5d black hole in the bulk can be proven usingholographic renormalization. Indeed, let us consider the perfect fluid witha stress-energy tensor equipped with diagonal elements h T µν i ∝ g (4) µν = /z = ǫ /z = p /z = p 00 0 0 1 /z = p , roceedings printed on October 25, 2018 where ǫ is the energy density and p = p = p = p is the pressure density.One can resum the whole holographic expansion (39) and get the followingbulk metric in Fefferman-Graham coordinates ds = − (1 − z /z ) (1 + z /z ) z dt + (1 + z /z ) dx z + dz z . (40) [Exercise 12.1: Recover the Energy-Momentum tensor corre-sponding to the metric (40), by using the expansion (39)] A change of variable z → ˜ z ≡ z/ r z z gives ds = − − ˜ z / ˜ z ˜ z dt + dx ˜ z + 11 − ˜ z / ˜ z d ˜ z ˜ z , (41)where one recognizes the Black Hole, in fact a black brane, with a statichorizon at ˜ z in the 5th dimension. [Exercise 12.2: prove the equivalence of the metric (40) and(41) by the change of variable z → ˜ z ] In fact there exists a one-to-one correspondence between the thermo-dynamic properties of the Black Hole ( BH ) and those of the perfect fluid( P F ), namely its temperature ( T BH = ǫ = T P F ) and entropy ( S BH ∼ Area = ǫ = S P F ).It is in this context of a static Black Hole configuration that one cango further than the perfect fluid approximation and derive the viscosityusing the Kubo formula. Indeed, the duality properties extend to a relationbetween the correlators of the energy-momentum tensor in two space-timepoints at zero frequency ω = 0 and the absorption cross section σ abs of agraviton by the static BH in the bulk. One writes σ abs ( ω ) ∝ Z d x e iωt ω h [ T x x ( x ) , T x x (0)] i ⇒ ηS ≡ σ abs (0) / πGA/ G = 14 π , (42)where S = S BH ≡ A/ G is the famous entropy-area relation of a BlackHole. From this relation, and putting numbers, it appears that the viscosityis weak, much weaker than the one computed in the weak coupling regimeand eventually realizing an absolute viscosity lower bound. QGP and Black Holes: From Statics to Dynamics The previous results were obtained for static configurations, i.e. for athermalized QGP at rest. In order to take into account, as much as possible,the actual kinematics of a heavy-ion collision, it is required to introduce proceedings printed on October 25, 2018 Fig. 9. The curvature scalar R . The singular structure of the Riemann scalar atthe horizon apart from the perfect fluid case is exemplified for s = ± . . hence,Nonsingular Geometry (absence of naked singularity , i.e. not hidden within the BHhorizon) implies the Perfect Fluid condition in the considered family of behavioursat large proper time. the proper time expansion of the plasma. On the gravity side, it calls forstudying non-equilibrium geometries, eventually of 5d BH configurations,which represent in itself a non-trivial and interesting issue. Dual geometriesto the standard “Bjorken flow” where recently constructed. The Bjorkenflow is the description of a boost-invariant expansion of the QGP, whichis expected to correspond to the physical situation in the central rapidityregion of the collision. In this context the questions why the QGP fluidappears to be nearly perfect (small viscosity) and why its thermalizationtime can be short have been adressed.Let us consider the equations obeyed by a physical energy-momentumtensor expressed in the { τ, η, x = x = x } coordinate system: T µµ ≡ − T ττ + τ T ηη + 2 T xx = 0 D ν T µν ≡ τ ddτ T ττ + T ττ + τ T ηη = 0 (43)In a boost-invariant framework, one may consider a general family of solu- roceedings printed on October 25, 2018 tions of proper time dependent, boundary energy-momentum tensors T µν = f ( τ ) 0 0 00 − τ ddτ f ( τ ) − τ f ( τ ) 0 00 0 f ( τ )+ τ ddτ f ( τ ) 00 0 0 ... (44)where the function f ( τ ) ∝ τ − s , satisfying the positivity condition T µν t µ t ν ≥ ⇒ < s < , corresponds to an interpolation between different relevant regimes, namely f ( τ ) ∝ τ − : Perfect fluid ǫ = p = p = p f ( τ ) ∝ τ − : Free streaming ǫ = p = p ; p = 0 f ( τ ) ∝ τ − : “Full anisotropy” ǫ = p ⊥ = − p L Using the holographic renormalization to compute the coefficients of thecorresponding metrics in the expansion on the fifth dimension and afterresummation, it was possible to solve the dual geometry for given s atasymptotic proper time τ. It reveals the existence of a scaling property ofthe solutions in terms of the proper time dependent variable v = zτ / . Analyzing the family of solutions as a function of s, it appears that theonly non-singular solution for invariant scalar quantities (here the squareof the Ricci tensor R = R µναβ R µναβ , see Fig. 9), is obtained for s = 4 / . Indeed, we find in Fefferman-Graham coordinates: ds = 1 z − (cid:16) − e z τ / (cid:17) e z τ / dτ + (cid:16) e z τ / (cid:17) ( τ dη + dx ) + dz z which is similar to the metrics of the static Black Hole (40), but substituting z → z /τ / . This solution is the only one of the family corresponding to aBlack Hole moving away in the fifth dimension. Hence the perfect-fluid caseis singled out and the moving BH in the bulk corresponds through duality tothe expansion of the QGP taking place in the boundary. Consequently, theBH horizon moves as z h ( τ ) ∝ τ , the temperature as T ( τ ) ∼ /z h ∼ τ − , and the entropy stays constant since S ( τ ) ∼ Area ∼ τ · /z h ∼ const. Notethat again the physical thermodynamical variables of the QGP are the sameas those one may attribute to the BH in the bulk (with the reservationthat thermodynamics of a moving BH may rise non-trivial interpretation proceedings printed on October 25, 2018 problems). Hence one finds a concrete realization of the idea of a dualitybetween the QGP formation and a moving Black Hole. Thermalization and Isotropization There has been a lot of activity along the lines of the AdS/CFT cor-respondence and its extensions to various geometric configurations. Dualstudies of jet quenching, quark dragging , etc... have been and are still beingperformed. Sticking to the configurations corresponding to an expandingplasma and going beyond the first order terms in proper time, one has ob-tained results on the viscosity, confirming the universal value (42), on therelaxation time of the plasma and very recently on the inclusion of flavordegrees of freedom.Let us finally focus on the thermalization problem, which can be usefullytaken up using the Gauge/Gravity duality in the strong coupling hypothesis.the problem is to give an explanation to the strikingly small thermalizationtime required for the formation of a QGP as can be abstracted from theexperimental observations. Analyzing the stability of the expanding plasmaconfiguration, it has been found that performing a small deviation from theBH metric by coupling with a scalar field and analyzing the corresponding quasi-normal modes defining the way how the system relaxes towards itsinitial state, one finds a numerically small value of the relaxation time inunits of the local temperature. Even if a definite value of this relaxationtime cannot be inferred at this stage due to scale-invariance, this resultwas suggestive of a stability of the QGP in the strong coupling regime withrespect to perturbations out of equilibrium.In order to go further, one has to deal with the problem of the QGP evo-lution at small proper times. The holographic renormalization program hasbeen pursued for the small proper time expansion. Relaxing the selectionof the appropriate metric by requiring only the metric tensor to be a realand single valued function of the coordinates everywhere in the bulk, onefinds an unique solution corresponding to the “fully anisotropic case” s = 0 . In the same paper, an evaluation of the range of the isotropization timehas been proposed, by extrapolation of realistic estimates abstracted fromexperiments to the supersymmetric case. The idea is to match the large andsmall proper time regimes at some value of the proper time τ iso . This propertime is mathematically defined as the crossing value for the branch-pointsingularities of both regimes. Physically, it is expected to give an estimateof the proper time range during which which the medium evolves from thefull anisotropic regime (small τ ) to the perfect fluid one (large τ ).In order to give an idea of the possible physical implications of this strongcoupling scheme, let us shortly reproduce the argument. Implementingthe estimated physical value of the energy density at some proper time roceedings printed on October 25, 2018 ε τε ~~ const τ iso ε ∼ τ −4/3 Fig. 10. Evaluation of the isotropization/thermalization time . The behaviour atlarge proper times and the one found at small ones are matched with the conditionthat the branching points where the solutions become multi-valued are avoided. thematching happens for a value τ iso whose range gives an evaluation of the isotropiza-tion time. ( e.g. ǫ ( τ ) = e τ / | τ = . ∼ GeV fermi − ) one finds τ iso = (cid:18) N c π e (cid:19) / ∼ . fermi . (45)This short isotropization time thus seems a characteristic feature of thestrong coupling scenarios. It is clear that more realistic estimates shouldtake into account less idealized dual models, corresponding to QCD, such asthe lack of supersymetry and the finite numbers of colors. However, the nonconfined character of the QGP and the robustness of some predictions (suchas the η/S ratio) may give some confidence that this short isotropizationtime could be a reasonable estimate at strong coupling. Outlook From the present rapid (and partial) survey of some of the results ob-tained in the AdS/CFT approach to the formation and expansion of theQuark-Gluon plasma in heavy-ion collisions, it appears that the GaugeGravity correspondence is a promising way to explore some features of QCDat strong coupling. Indeed some general features of this correspondence, re-lating at long distances the closed and open string geometries (see Fig. 3)are expected to be valid in principle for various dual schemes and thus,hopefully, QCD. proceedings printed on October 25, 2018 In practice, the quantitative dual schemes have been more precisely elab-orated for the specific AdS/CFT case, i.e. the gauge theory with N = 4supersymmetries. Among the results, it gives a calculable link betweenthe hydrodynamic quasi-perfect fluid behaviour on the “gauge theory side”with a BH geometry in the higher dimensional “gravity side” in an AdSbackground. This relation can be extended from the static case to a dy-namical regime reflecting (within the AdS/CFT framework) the relativisticexpansion of the corresponding quark-gluon plasma. This, and many otherapplications, some of them using more complex geometries, less supersym-metric backgrounds and examining other observables, gives hope for thefruitful possibilities of the Gauge/Gravity approach to the QGP formation.As an outlook, it is worth mentionning some of the possible new di-rections of study one is led to consider. Starting with the more technicalones, it is known that the Bjorken flow is not exactly verified in heavy-ioncollisions, since the observed distribution of particles is nearly gaussian inrapidity and thus not reflecting exactly the boost-invariance of the Bjorkenflow. It would be interesting to investigate dual properties for non-boostinvariant flows, such as the Landau flow . On a more general ground, thewhole approach still concerns only the hydrodynamical stage of the QGPexpansion. It would be important to attack both the initial (partonic) andfinal (hadronic) stages of the reaction in the same framework and thus theproblem of phase transitions during the collision. Finally, one would liketo have more realistic dual frameworks including a finite number of colors,flavor degrees of freedom and no (or broken) supersymmetry. Acknowledgements I would like to thank Romuald Janik (Jagelonian University, Cracow)for a fruitful and inspiring long-term collaboration between a string theoristand a particle phenomenologist. Emmanouil Saridakis, Alessandro Papaand Christophe Royon are thanked for helping to correct the manuscript. roceedings printed on October 25, 2018 Note on Lecture I The lecture I is based on an introduction to string theory via the stronginteraction peoblems, and thus follows more or less the historical develop-ment. Starting with Veneziano’s initial paper, we were largely inspired bythe two next references of the list. The first one is a book which has themerit to give the whole derivation (with detailed calculations) of the intro-duction to string theory via the strong interaction paradigm. The secondone is very helpful in understanding the story of the discovery of string the-ory and containing an abundant and documented list of references placed intheir historical perspective (as captivating as a detective novel). Moreovera large part of the derivation that I used is here given with useful details.Nowadays, string theory has developped as an autonomous theory, andmany deep properties and its modern ways of introduction are far from theinitial problematics. For primary instance the intimate connection of stringtheory to 2-dimensional conformal field theory do (and did) not appearimmediately from the construction of the Hilbert space through scatteringamplitudes as sketched in lecture I. Hence it is strongly advised for thestudents who would like to learn string theory to refer to modern textbooks.My colleagues string theorists would, among others, recommend the two lastreferences of the list, and find there the appropriate references. I make myapologizes to all contributors to string theory for not quoting them directly in the lecture for sake of simplicity (and possible misunderstanding from mypart). Note on Lecture II The lecture II is based on an elementary introduction to the AdS/CFTcorrespondence. In a considerably shortened and subjective selection on theenormous literature on the subject the three first references are the original proceedings printed on October 25, 2018 ones. The fourth one is a now rather old but still precious and readablereview on the subject. In the general argumentation about gauge/gravityduality, we were inspired by the nice introduction of Ref.[10].In introducing some concepts useful in the approach to the formulationof hadronic scattering amplitudes at strong coupling, we used mainly theduality vocabulary related to Wilson loops. We quote some relevant originalreferences, and for the rest refer to the review [7].The specific approach to dipole scattering amplitudes described in thetext comes from a collaboration with Romuald Janik, as quoted in thereferences. Note on Lecture III The lecture III is based on a series of papers using the AdS/CFT cor-respondence as a laboratory for the flow of a strongly coupled quark-gluonplasma. It relies both on the phenomenological validity of the relativistichydrodynamic approach to the real experiments and its surprisingly goodrelevance for the Gauge/Gravity correspondence.The first set of references recall some basic facts about 1+1 relativistichydrodynamic equations applied to particle collisions concerning the canon-ical Bjorken and Landau flows and including a recent interpolating familyof exact solutions.The second set of references begins with holographic renormalization,which is a basic tool used in holographic reconstruction followed by originalreferences on the emergence and calculations of the dual Black Hole geom-etry of a static plasma. In a third set are given some the original referencescorresponding to the expanding plasma geometry and its evolving black holedual.We display at the end a very limited number of references concerningother interesting aspects of the quark gluon plasma problems treated fromthe point of view of the AdS/CFT correspondence and which were notaddressed in this lecture. However the general set-up described here mayhelp for the understanding of the various approaches. roceedings printed on October 25, 2018 REFERENCES [1] G. Veneziano, “Construction of a crossing - symmetric, Regge behaved ampli-tude for linearly rising trajectories,” Nuovo Cim. A (1968) 190.[2] P. H. Frampton, “Dual Resonance Models And Superstrings,” Singapore, Sin-gapore: World Scientific ( 1986) 539p .[3] P. Di Vecchia, “The birth of string theory,” arXiv:0704.0101 [hep-th];P. Di Vecchia and A. Schwimmer, “The beginning of string theory: a historicalsketch,” arXiv:0708.3940 [physics.hist-ph]. Contribution to the volume ‘Stringtheory and fundamental interactions’, dedicated to Gabriele Veneziano on his65th birthday.[4] D. Lust and S. Theisen, “Lectures on string theory,” Lect. Notes Phys. ,Springer-Verlag (1989) 1.[5] J. Polchinski, “String theory. Vol. 1: An introduction to the bosonic string,” Cambridge, UK: Univ. Pr. (1998) 402 p ;“String theory. Vol. 2: Superstring theory and beyond,” Cambridge, UK: Univ.Pr. (1998) 531 p .———AdS/CFT————–[6] J. M. Maldacena, “The large N limit of superconformal field theories andsupergravity,” Adv. Theor. Math. Phys. , 231 (1998) [Int. J. Theor. Phys. , 1113 (1999)].[7] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlatorsfrom non-critical string theory,” Phys. Lett. B , 105 (1998).[8] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. , 253 (1998);“Anti-de Sitter space, thermal phase transition, and confinement in gaugetheories,” Adv. Theor. Math. Phys. , 505 (1998).[9] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, “Large Nfield theories, string theory and gravity,” Phys. Rept. , 183 (2000).[10] V. Schomerus, “Strings for Quantumchromodynamics,” Int. J. Mod. Phys. A , 5561 (2007).———Wilson Loops and AdS/CFT————–[11] J. M. Maldacena, “Wilson loops in large N field theories,” Phys. Rev. Lett. , 4859 (1998).[12] S. J. Rey and J. T. Yee, “Macroscopic strings as heavy quarks in large N gaugetheory and anti-de Sitter supergravity,” Eur. Phys. J. C , 379 (2001).[13] J. Sonnenschein and A. Loewy, “On the supergravity evaluation of Wilsonloop correlators in confining theories,” JHEP , 042 (2000).6 proceedings printed on October 25, 2018 ———Dual model for dipole amplitudes————–[14] R. A. Janik and R. Peschanski, “High energy scattering and the AdS/CFTcorrespondence,” Nucl. Phys. B , 193 (2000);“Minimal surfaces and Reggeization in the AdS/CFT correspondence ,” Nucl.Phys. B , 163 (2000);Reggeon exchange from AdS/CFT,” “Nucl. Phys. B , 279 (2002).———-Relativistic Hydrodynamics—————[15] J. D. Bjorken, “Highly Relativistic Nucleus-Nucleus Collisions: The CentralRapidity Region,” Phys. Rev. D , 140 (1983).[16] L. D. Landau, “On the multiparticle production in high-energy collisions,”Izv. Akad. Nauk Ser. Fiz. , 51 (1953).[17] A. Bialas, R. A. Janik and R. Peschanski, “ Unified description of Bjorkenand Landau 1+1 hydrodynamics,” Phys. Rev. C , 054901 (2007).———-AdS/CFT and Hydrodynamics—————[18] K. Skenderis, “Lecture notes on holographic renormalization,” Class. Quant.Grav. , 5849 (2002).[19] R. C. Myers, “ Stress tensors and Casimir energies in the AdS/CFT corre-spondence,” Phys. Rev. D , 046002 (1999);V. Balasubramanian, J. de Boer and D. Minic, “ Mass, entropy and hologra-phy in asymptotically de Sitter spaces,” Phys. Rev. D , 123508 (2002).[20] G. Policastro, D. T. Son and A. O. Starinets, “The shear viscosity of stronglycoupled N = 4 supersymmetric Yang-Mills plasma,” Phys. Rev. Lett. ,081601 (2001).[21] D. T. Son and A. O. Starinets, “ Viscosity, Black Holes, and Quantum FieldTheory,” Ann. Rev. Nucl. Part. Sci. , 95 (2007).———-AdS/CFT for an Expanding Plasma—————[22] H. Nastase, “ The RHIC fireball as a dual black hole,”E. Shuryak, S. J. Sin and I. Zahed, “A gravity dual of RHIC collisions,” J.Korean Phys. Soc. , 384 (2007).[23] R. A. Janik and R. B. Peschanski, “Asymptotic perfect fluid dynamics as aconsequence of AdS/CFT,” Phys. Rev. D , 045013 (2006);“Gauge / gravity duality and thermalization of a boost-invariant perfect fluid,”Phys. Rev. D , 046007 (2006).[24] S. Nakamura and S. J. Sin, “ A holographic dual of hydrodynamics JHEP , 020 (2006) ;R. A. Janik, “Viscous plasma evolution from gravity using AdS/CFT,” Phys. roceedings printed on October 25, 2018 , 022302 (2007) ;M. P. Heller and R. A. Janik, “Viscous hydrodynamics relaxation time fromAdS/CFT,” Phys. Rev. D , 025027 (2007);J. Große, R. A. Janik and P. Surowka, “Flavors in an expanding plasma,”Phys. Rev. D , 066010 (2008).[25] Y. V. Kovchegov and A. Taliotis, “ Early time dynamics in heavy ion collisionsfrom AdS/CFT correspondence,” Phys. Rev. C , 014905 (2007).———-Other AdS/CFT studies—————[26] H. Liu, K. Rajagopal and U. A. Wiedemann, “Calculating the jet quenchingparameter from AdS/CFT,” Phys. Rev. Lett. , 182301 (2006).[27] S. S. Gubser, “Drag force in AdS/CFT,” Phys. Rev. D , 126005 (2006).[28] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz and L. G. Yaffe, “ Energy lossof a heavy quark moving through N = 4 supersymmetric Yang-Mills plasma,”JHEP0607