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Remarks on the gauge/gravity duality Juan Maldacena

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Field Theory Gravity theory = Gauge Theories QCD Quantum Gravity String theory

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Large N and string theory SU(N) theory when N ∞, with g 2 N = fixed Planar diagrams dominate View them as the worldsheet of a string 1/N corrections non-planar diagrams = strings worldsheets with more complicated topologies + == + ‘t Hooft 74

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Look for a string theory in 4d not consistent At least 5 dimensions Polyakov

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D-branes in string theory Solitons in string theory. Excitations described by open strings Lowest energies U(N) gauge fields Can also be viewed as black branes ij A ij Polchinski 95 Horowitz Strominger 91

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N=4 SU(N) Yang Mills theory in 3+1 dimensions String theory on AdS 5 x S 5 ij A ij JM 97 g 2 N = ³ R l s ´ 4

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AdS/CFT Conformal symmetry g 2 N = ³ R l s ´ 4 Duality:

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Development of the dictionary Correlation functions Wilson loops Various deformations: masses, marginal deformations Many new examples, both conformal and non-conformal Continues today… Gubser, Klebanov, Polyakov, Witten

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Lessons for gravity Use the field theory as a definition of string theory or quantum gravity (finite N). Lessons for black holes. Emergent space time

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Black holes Entropy = area = statistical entropy in the field theory Unitary evolution: Quantum mechanics and gravity are compatible

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Counting supersymmetric black holes More detailed black hole counting, subleading corrections AdS 3 /CFT 2 examples Connections to the topological string Connections between matrix models, N=1 quantum field theories and geometry Gopakumar, Vafa, Dijkgraaf, Vafa

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Holography Physics in some region described by a theory on the boundary of the region with a number of q-bits that grows like the area of the region in Planck units. Not clear how to extend the idea to other cases - What are the degrees of freedom - How do we define a ``boundary’’

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Emergent space time Spacetime: like the fermi surface, only defined in the classical limit

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Initial singularity Asymptotic future De-Sitter Euclidean conformal field theory = ? No explicit example is known! How do we get “emergent” time ? Is there a dS/CFT ? Ψ[g] = Z[g] Wave function of the universe Partition function of a Euclidean field theory Witten Strominger JM Objections: - dS decays - dS is thermal

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Description of the string landscape, and eternal inflation? What are the future boundaries? Who measures that?

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What are the field theory duals of the AdS 4 vacua in the landscape (preserving N=1 susy) ?

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How do describe the interior of a black hole Crunching cosmology Intrinsically approximate description, up e -S There is no obvious “exact” sense in which the interior exists. Probably relevant for describing cosmology There is probably no other region behind the singularity. Can we have crunch-bang transitions ?

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Lessons for gauge theories We can view the geometry as the solution of the theory in the large ‘t Hooft coupling limit. View these as toy model field theories that can be explicitly solved. These theories capture many interesting physical phenomena. Explicit examples of confining theories Thermal aspects are calculable. Transport properties can be computed. e.g. RHIC applications or as toy model for condensed matter problems.

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Could be describing physics at higher energies (Randall-Sundrum models). Examples of constructions of metastable vacua in supersymmetric theories. Use to understand supersymmetry breaking, supersymmetry breaking mediation mechanisms. Many examples of conformal N=1 SUSY gauge theories with a geometric description. Used to test other dualities.

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All coupling Connect the weak and strong coupling regimes by computing exactly for all coupling. Integrability in N=4 SYM. Cusp anomalous dimension known exactly for all values of g 2 N.

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Future More on the dictionary Toy model for analyzing field theory dynamics. Further exact results in N=4.

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Big challenges Understand how to describe universes with cosmological singularities Clear description of the interior of black holes String theory describing the large N limit of ordinary Yang Mills theory, or other large N theories whose duals have stringy curvature.

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4 dimensions Extra dimension Graviton is localized in the extra dimension Massive spin 2 particle in 4 dimensions Gravitational potential in the extra dimension

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Jet Physics at strong coupling Diego Hofman and J. M., to appear

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Jet physics at strong coupling e+e+ e-e- γ q q J ¹ ( p )j 0 i

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Why? New physics at the LHC could involve a strong or moderately strongly coupled field theory We would like to understand the transition between the weak coupling picture of the event where one can qualitatively think in terms of underlying partons and the gravity picture where we do not see the partons in any obvious way. Weak coupling Strong couling ?

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How do we describe the produced state Not convenient to talk about partons Inclusive observable Energy correlation functions. θ1θ1 θ2θ2 h ² ( µ 1 ) ² ( µ 2 )i h ² ( µ 1 ) ¢¢¢ ² ( µ n )i h ² ( µ 1 )i Basham, Brown, Ellis, Love 78

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² ( µ ) = R d t T 0 i n i Integrated flux of energy at infinity Three point function Symmetries determine the three point function up to two constants h ² ( µ )i = h 0 j j y R T j j 0 i h ² ( µ )i = a + b ( cos 2 µ ¡ 1 3 )

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weak coupling in QCD: In any theory with a gravity dual: In N=4 SYM at weak and strong coupling: N=1 superconformal field theory, j = R-current: ² ( µ ) » 1 ² ( µ ) » 1 ² ( µ ) » 1 + cos 2 µ ² ( µ ) » 1 + 3 2 a ¡ c c ( cos 2 µ ¡ 1 3 ) (unpolarized e-beam) θ is angle to beam

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Can we get a non-zero b from AdS ? Bulk couplings between an on shell graviton and two on shell gluons - Only two possible couplings in 5 dimensions - a is fixed by the two point function of the currents - The second coupling is a higher derivative correction (which is present in bosonic string theory) - If a ~ b higher derivative correction as important as the leading term) S 5 ¡ d » R a F 2 + b R ¹º ± ¾ F ¹º F ± ¾

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Two point function Consider a state produced by a scalar operator. Two point function is a somewhat complicated function of the angle between the two detectors At very weak coupling is goes like µ 12 h ² ( µ 1 ) ² ( µ 2 )i » g 2 N 1 µ 2 12 ; µ 12 ¿ 1

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Strong coupling computation x x x x Since we integrate the stress tensor wavefunction localized on an H 3 subspace (SO(1,3) symmetry) h ² ( µ 1 ) ² ( µ 2 )i = R H 3 ª ¤ @ 2 ¡ ª f ( µ 1 ; x ) f ( µ 2 ; x ) g ( x, ¢ )

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H3H3 x ² ( µ ; x ) = f ( µ ; x ) In the gravity approximation the distribution of energy depends just on three random variables: a point on H 3 1/Δ General description

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Small angle singularity The small angle singularity is governed by the twist of the operators contributing to the light cone OPE of the stress tensor R d x ¡ T ¡¡ ( ~ y ) R d x ¡ T ¡¡ ( 0 ) » y ¡ 4 + ¿ n R d x ¡ O n ¡¡¡ Twist = Δ - S Non-local operator of spin 3 Balitsky Braun 88 h ² ( µ 1 ) ² ( µ 2 )i » g 2 N 1 µ 4 ¡ 2 ¢ 12

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Single trace and double trace operators. Double trace operators of the form O ¢ O ¢ At weak coupling single trace operators have dimension one but at strong coupling they get large dimension are the dominant contribution at strong coupling ¿ 3 = 2 + ¸ + ¢¢¢ f or ¸ ¿ 1 ( 1 ) ¿ 3 = 2 1 = 2 ¸ 1 = 4 ; f or ¸ À 1 ( 2 )

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Related to deep inelastic scattering It is related to a particular moment of the deep inelastic amplitude of gravitons. Polchinski Strassler 02 h ² ( ¡ q ) ² ( q )i » R 1 ¡ 1 d s A DIS ( s ; q 2 )

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Conclusions One can define inclusive, event shape variables for conformal theories They can be computed at weak and strong coupling Small angle features governed by the anomalous dimension of twist two operators Events are more spherically symmetric at strong coupling, but not completely uniform. In a non-conformal theory one would need to face the details of hadronization. Depending on these details there could be small or large changes.

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Strings and Strong Interactions Before 60s proton, neutron elementary During 60s many new strongly interacting particles Many had higher spins s = 2, 3, 4 …. All these particles different oscillation modes of a string. This model explained features of the spectrum of mesons. Rotating String model From E. Klempt hep-ex/0101031

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Strong Interactions from Quantum ChromoDynamics 3 colors (charges) They interact exchanging gluons Electrodynamics Chromodynamics (QCD) electron photon gluon g g g g Gauge group U(1) SU(3) 3 x 3 matrices Gluons carry color charge, so they interact among themselves Experiments at higher energies revealed quarks and gluons

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Coupling constant decreases at high energy g 0 at high energies QCD is easier to study at high energies Hard to study at low energies Indeed, at low energies we expect to see confinement q q Flux tubes of color field = glue At low energies we have something that looks like a string. There are approximate phenomenological models in terms of strings. V = T L How do strings emerge from QCD Can we have an effective low energy theory in terms of strings ? Gross, Politzer, Wilczek

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Large N and strings Gluon: color and anti-color Open strings mesons Closed strings glueballs Take N colors instead of 3, SU(N) Large N limit t’ Hooft ‘74 g 2 N = effective interaction strength when colors are correlated

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General Idea - Solve first the N=∞ theory. - Then do an expansion in 1/N. - Set 1/N =1/3 in that expansion.

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The N=∞ case - It is supposed to be a string theory - Try to guess the correct string theory - Two problems are encountered.

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1. Simplest action = Area Not consistent in D=4 ( D=26 ? ) At least one more dimension (thickness) Polyakov generate 2. Strings theories always contain a state with m=0, spin =2: a Graviton. But: - In QCD there are no massless particles. - This particle has the interactions of gravity For this reason strings are commonly used to study quantum gravity. Forget about QCD and use strings as a theory of quantum gravity. Superstring theory, unification, etc. But what kind of string theory should describe QCD ? Scherk-Schwarz Yoneya Lovelace

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We need to find the appropriate 5 dimensional geometry It should solve the equations of string theory They are a kind of extension of Einstein’s equations Very difficult so solve Consider a simpler case first. A case with more symmetry. We consider a version of QCD with more symmetries.

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Most supersymmetric QCD Supersymmetry Bosons Fermions Gluon Gluino Many supersymmetries B1 F1 B2 F2 Maximum 4 supersymmetries, N = 4 Super Yang Mills Susy might be present in the real world but spontaneously broken at low energies. So it is interesting in its own right to understand supersymmetric theories. We study this case because it is simpler. Ramond Wess, Zumino

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Similar in spirit to QCD Difference: most SUSY QCD is scale invariant Classical electromagnetism is scale invariant V = 1/r QCD is scale invariant classically but not quantum mechanically, g(E) Most susy QCD is scale invariant even quantum mechanically Symmetry group Lorentz + translations + scale transformations + other These symmetries constrain the shape of the five dimensional space. ds 2 = R 2 w 2 (z) ( dx 2 3+1 + dz 2 ) redshift factor = warp factor ~ gravitational potential Demanding that the metric is symmetric under scale transformations x x, we find that w(z) = 1/z

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This metric is called anti-de-sitter space. It has constant negative curvature, with a radius of curvature given by R. Boundary ds 2 = R 2 (dx 2 3+1 + dz 2 ) z 2 R4R4 AdS 5 z = 0 z z = infinity Gravitational potential w(z) z (The gravitational potential does not have a minimum can have massless excitations Scale invariant theory no scale to set the mass )

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Anti de Sitter space Solution of Einstein’s equations with negative cosmological constant. De Sitter solution with positive cosmological constant, accelerated expanding universe () Two dimensional negatively curved space

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Spatial section of AdS = Hyperbolic space

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R = radius of curvature Light raysMassive particles Time The Field theory is defined on the boundary of AdS. The space has a boundary. It is infinitely far in spatial distance A light ray can go to the boundary and back in finite time, as seen from an observer in the interior. The time it takes is proportional to R.

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Building up the Dictionary Graviton stress tensor z) y) x) Field theory = Probability amplitude that gravitons go between given points on the boundary Gubser, Klebanov, Polyakov - Witten Other operators Other fields (particles) propagating in AdS. Mass of the particle scaling dimension of the operator

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We expected to have string theory on AdS. Supersymmetry D=10 superstring theory on AdS x (something) 5 5 S 5 Type IIB superstrings on AdS x S 5 55 5-form field strength F = generalized magnetic field quantized (J. Schwarz) Most supersymmetric QCD

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String Theory Free strings String Tension = T = = string length Relativistic, so T = (mass)/(unit length) Excitations along a stretched string travel at the speed of light Closed strings Can oscillateNormal modes Quantized energy levels Mass of the object = total energy, M=0 states include a graviton (a spin 2 particle) First massive state has M ~ T 2 Veneziano Scherk Schwarz Green …..

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String Interactions Splitting and joining g String theory Feynman diagram Simplest case: Flat 10 dimensions and supersymmetric Precise rules, finite results, constrained mathematical structure At low energies, energies smaller than the mass of the first massive string state Gravity theory ( Incorporates gauge interactions Unification ) Radius of curvature >> string length gravity is a good approximation lsls R

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Most supersymmetry QCD theory String theory on AdS x S 5 5 = Radius of curvature (J.M.) Duality: g 2 N is small perturbation theory is easy – gravity is bad g 2 N is large gravity is good – perturbation theory is hard Strings made with gluons become fundamental strings. Particle theory = gravity theory N colors N = magnetic flux through S 5

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Where Do the Extra Dimensions Come From? 3+1 AdS 5 radial dimension z Boundary Interior Gluons live here Strings live here

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What about the S 5 ? Related to the 6 scalars S 5 other manifolds = Most susy QCD less susy QCD. Large number of examples Klebanov, Witten, Gauntlett, Martelli, Sparks, Hannany, Franco, Benvenutti, Tachikawa, Yau …..

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string Boundary q q Quark anti quark potential V = potential = proper length of the string in AdS Weak coupling result:

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Confining Theories Add masses to scalars and fermions pure Yang Mills at low energies confining theory. There are many concrete examples. At strong coupling gravity solution is a good description. w(z 0 ) > 0 boundary String at z 0 has finite tension from the point of view of the boundary theory. Graviton in the interior massive spin=2 particle in the boundary theory = glueball. Gravitational potential or warp factor w(z) zz0z0

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Checking the conjecture It is hard because either one side is strongly coupled or the other. Supersymmetry allows many checks. Quantities that do not depend on the coupling. More recently, ``integrability’’ allowed to check the conjecture for quantities that have a non-trivial dependence on the coupling, g 2 N. One can vividly see how the gluons that live in four dimensions link up to produce strings that move in ten dimensions. … Minahan, Zarembo, Beisert, Staudacher, Arutyunov, Frolov, Hernandez, Lopez, Eden

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What can we learn about gravity from the field theory ? Useful for understanding quantum aspects of black holes The relation connects a quantum field theory to gravity.

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Black holes Gravitational collapse leads to black holes Classically nothing can escape once it crosses the event horizon Quantum mechanics implies that black holes emit thermal radiation. (Hawking) Black holes evaporate Evaporation time Temperature is related to entropy dM = T dS S = Area of the horizon 4 L Planck 2 What is the statistical interpretation of this entropy? (Hawking-Bekenstein)

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Black holes in AdS Thermal configurations in AdS. Entropy: S GRAVITY = Area of the horizon = S FIELD THEORY = Log[ Number of states] Evolution: Unitary Solve the information paradox raised by S. Hawking

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Confining Theories and Black Holes Confinement Deconfinement= black hole (black brane) Low temperatures High temperatures Gravitational potential Extra dimension Horizon

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z=z 0 z=0 Black hole Very low shear viscosity similar to what is observed at RHIC: “ the most perfect fluid” Kovtun, Son, Starinets, Policastro High energy collision produces a black hole = droplet of deconfined phase ~ quark gluon plasma. z=z 0, z=0 Black holes in the Laboratory QCD 5d string theory Very rough model, we do not yet know the precise string theory

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Emergent space time Spacetime: like the fermi surface, only defined in the classical limit Lin, Lunin, J.M.

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A theory of some universe Suppose that we lived in anti-de-sitter space Then the ultimate description of the universe would be in terms of a 2+1 dimensional field theory living on the sphere at infinity. (With around 10 120 fields to give a universe of the size of ours) Out universe is close to de-Sitter. Could we have a similar description in that case ?

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Conclusions - Gravity and particle physics are “unified” Usual: Quantum gravity particle physics. New: Particle physics quantum gravity. -Black holes and confinement are related -Emergent space-time. Started from a theory without gravity got a theory in higher dimensions with gravity. -Tool to do computations in gauge theories. -Tool to do computations in gravity.

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Future Field theory: Gravity : Theories closer to the theory of strong interactions Solve large N QCD Quantum gravity in other spacetimes Understand cosmological singularities

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