Presentation on theme: "Remarks on the gauge/gravity duality"— Presentation transcript:
1 Remarks on the gauge/gravity duality Juan Maldacena
2 = Field Theory Gravity theory Gauge Theories Quantum Gravity QCD String theory
3 Large N and string theory SU(N) theory when N ∞ , with g2 N = fixedPlanar diagrams dominateView them as the worldsheet of a string1/N corrections non-planar diagrams = strings worldsheets with more complicated topologies‘t Hooft 74==++
4 Look for a string theory in 4d not consistent At least 5 dimensionsPolyakov
5 D-branes in string theory Solitons in string theory. Excitations described by open stringsLowest energies U(N) gauge fieldsCan also be viewed as black branesPolchinski 95HorowitzStrominger 91ijAij
6 A N=4 SU(N) Yang Mills String theory theory in 3+1 dimensions jAijN=4 SU(N) Yang Millstheory in 3+1 dimensionsString theoryon AdS5 x S5g2N=Rls4JM 97
8 Development of the dictionary Gubser, Klebanov,Polyakov, WittenCorrelation functionsWilson loopsVarious deformations: masses, marginal deformationsMany new examples, both conformal and non-conformalContinues today…
9 Lessons for gravityUse the field theory as a definition of string theory or quantum gravity (finite N).Lessons for black holes.Emergent space time
10 Black holes Entropy = area = statistical entropy in the field theory Unitary evolution:Quantum mechanicsand gravity arecompatible
11 Counting supersymmetric black holes More detailed black hole counting, subleading correctionsAdS3/CFT2 examplesConnections to the topological stringConnections between matrix models, N=1 quantum field theories and geometryGopakumar, Vafa,Dijkgraaf, Vafa
12 HolographyPhysics 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’’
13 Emergent space time Spacetime: like the fermi surface, only defined in the classical limit
14 = ? Ψ[g] = Z[g] Is there a dS/CFT ? De-Sitter WittenStromingerJMAsymptotic futureEuclidean conformalfield theory= ?De-SitterInitial singularityWave function ofthe universePartition function of aEuclidean field theoryΨ[g] = Z[g]No explicit example is known!How do we get “emergent” time ?Objections: - dS decays- dS is thermal
15 Description of the string landscape, and eternal inflation? What are the futureboundaries?Who measures that?
16 What are the field theory duals of the AdS4 vacua in the landscape (preserving N=1 susy) ?
17 How do describe the interior of a black hole Crunching cosmologyIntrinsically approximate description, up e-SThere is no obvious “exact” sense in which the interior exists.Probably relevant for describing cosmologyThere is probably no other region behind the singularity.Can we have crunch-bang transitions ?
18 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 theoriesThermal aspects are calculable. Transport properties can be computed. e.g. RHIC applications or as toy model for condensed matter problems.
19 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.
20 All couplingConnect 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 g2 N.
21 Future More on the dictionary Toy model for analyzing field theory dynamics.Further exact results in N=4.
22 Big challengesUnderstand how to describe universes with cosmological singularitiesClear description of the interior of black holesString theory describing the large N limit of ordinary Yang Mills theory, or other large N theories whose duals have stringy curvature.
27 Why?New physics at the LHC could involve a strong or moderately strongly coupled field theoryWe 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 couplingStrong couling?
28 How do we describe the produced state Not convenient to talk about partonsInclusive observableEnergy correlation functions.Basham, Brown, Ellis, Love 78h(1)iθ1θ2h(1)2ih(1)ni
29 h ² ( µ ) i = j R T ² ( µ ) = R d t T n Integrated flux of energy at infinity()=RdtTinh()i=jyRTThree point functionSymmetries determine the three point functionup to two constantsh()i=a+bcos213
30 In any theory with a gravity dual: weak coupling in QCD:In any theory with a gravity dual:In N=4 SYM at weak and strong coupling:(unpolarized e-beam)θ is angle to beam()1+cos2()1()1N=1 superconformal field theory, j = R-current:()1+32acos
31 S » R a F + b - a is fixed by the two point function of the currents Can we get a non-zero b from AdS ?Bulk couplings between an on shell gravitonand two on shell gluonsS5dRaF2+b- Only two possible couplings in 5 dimensions- a is fixed by the two point function of the currents- The second coupling is a higher derivativecorrection (which is present in bosonic string theory)- If a ~ b higher derivative correction as importantas the leading term)
32 Two point function Consider a state produced by a scalar operator. Two point function is a somewhat complicatedfunction of the angle between the two detectorsAt very weak coupling is goes likeh(1)2igN;12
33 Strong coupling computation Since we integrate the stresstensor wavefunction localized onan H3 subspace (SO(1,3) symmetry)xxxxg(x,)h(1)2i=RH3@f;x
34 In the gravity approximation the distribution of General description1/ΔxH3(;x)=fIn the gravity approximation the distribution ofenergy depends just on three random variables:a point on H3
35 Small angle singularity h(1)2igN4The small angle singularity is governed by thetwist of the operators contributing to the lightcone OPE of the stress tensorTwist = Δ - SRdxT(~y)4+nONon-local operator of spin 3Balitsky Braun 88
36 O Single trace and double trace operators. Double trace operators of the formOare the dominant contribution at strong couplingAt weak coupling single trace operatorshave dimension one but at strong couplingthey get large dimension3=2+for1()4;À
37 Related to deep inelastic scattering Polchinski Strassler 02Related to deep inelastic scatteringIt is related to a particular moment of the deepinelastic amplitude of gravitons.h(q)iR1dsADIS;2
38 ConclusionsOne can define inclusive, event shape variables for conformal theoriesThey can be computed at weak and strong couplingSmall angle features governed by the anomalous dimension of twist two operatorsEvents 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.
42 Strings and Strong Interactions Before 60s proton, neutron elementaryDuring 60s many new strongly interacting particlesMany had higher spins s = 2, 3, 4 ….All these particles different oscillation modes of a string.This model explained features of the spectrumof mesons.Rotating String modelFrom E. Klempt hep-ex/
43 Strong Interactions from Quantum ChromoDynamics 3 colors (charges)They interact exchanging gluonsExperiments at higher energiesrevealed quarks and gluonsChromodynamics (QCD)ElectrodynamicselectronphotongluongGauge group3 x 3 matricesGluons carrycolor charge, sothey interact amongthemselvesU(1)SU(3)
44 g at high energies V = T L q Coupling constant decreases at high energyGross, Politzer, Wilczekgat high energiesQCD is easier to study at high energiesHard to study at low energiesIndeed, at low energies we expect to see confinementqFlux tubes of color field = glueV = T LAt low energies we have something that looks like a string. There areapproximate phenomenological models in terms of strings.How do strings emerge from QCDCan we have an effective low energy theory in terms of strings ?
45 Large N and strings Gluon: color and anti-color Open strings mesons Take N colors instead of 3, SU(N)t’ Hooft ‘74Large N limitg2N = effective interaction strengthwhen colors are correlatedOpen strings mesonsClosed strings glueballs
46 General Idea - Solve first the N=∞ theory. - Then do an expansion in 1/N.- Set 1/N =1/3 in that expansion.
47 The N=∞ case - It is supposed to be a string theory - Try to guess the correct string theory- Two problems are encountered.
48 Not consistent in D=4 ( D=26 ? ) 1. Simplest action = AreaNot consistent in D= ( D=26 ? )LovelacegeneratePolyakovAt least one more dimension (thickness)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 gravityFor this reason strings are commonly used to studyquantum gravity. Forget about QCD and use strings as a theory ofquantum gravity. Superstring theory, unification, etc.But what kind of string theory should describe QCD ?Scherk-SchwarzYoneya
49 We need to find the appropriate 5 dimensional geometry It should solve the equations of string theoryThey are a kind of extension of Einstein’s equationsVery difficult so solveConsider a simpler case first. A case with more symmetry.We consider a version of QCD with more symmetries.
50 Most supersymmetric QCD SupersymmetryRamondWess, ZuminoBosons FermionsGluon GluinoMany supersymmetriesB F1B F2Maximum supersymmetries, N = 4 Super Yang MillsSusy 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.
51 Similar in spirit to QCD Difference: most SUSY QCD is scale invariantClassical electromagnetism is scale invariantV = 1/rQCD is scale invariant classically but not quantum mechanically, g(E)Most susy QCD is scale invariant even quantum mechanicallySymmetry groupLorentz + translations + scale transformations + otherThese symmetries constrain the shape of the five dimensional space.ds2 = R2 w2 (z) ( dx dz2 )redshift factor = warp factor ~ gravitational potentialDemanding that the metric is symmetric under scale transformationsx x , we find that w(z) = 1/zl
52 This metric is called anti-de-sitter space. It has constant negative ds2 = R2 (dx dz2)z2R4AdS5Boundaryzz = 0z = infinityThis metric is called anti-de-sitter space. It has constant negativecurvature, with a radius of curvature given by R.Gravitational potentialw(z)z(The gravitational potential does not have a minimum can have massless excitationsScale invariant theory no scale to set the mass )
53 Anti de Sitter spaceSolution of Einstein’s equations with negative cosmological constant.()De Sitter solution with positive cosmological constant, accelerated expandinguniverseTwo dimensionalnegatively curved space
55 The Field theory is defined on the boundary of AdS. R = radius of curvatureTimeLight raysMassive particlesThe space has a boundary.It is infinitely far in spatial distanceA light ray can go to the boundary and back in finite time, as seen froman observer in the interior. The time it takes is proportional to R.The Field theory is defined on the boundary of AdS.
56 < > Building up the Dictionary Graviton stress tensormnTGubser, Klebanov,Polyakov - WittenmnTmnT<>mnT (x)mnT (y)mnT (z)= Probability amplitude that gravitonsgo between given points on the boundaryField theoryOther operatorsOther fields (particles) propagating in AdS.Mass of the particle scaling dimension of the operator
57 Most supersymmetric QCD We expected to have string theory on AdS.Supersymmetry D=10 superstring theory on AdS x (something)555S555Type IIB superstrings on AdS x S(J. Schwarz)5-form field strength F = generalized magnetic field quantized
58 String Theory , First massive state has M ~ T Free strings VenezianoScherkSchwarzGreen…..Free stringsTension = T =,String= string lengthRelativistic, so T = (mass)/(unit length)Excitations along a stretched string travel at the speed of lightClosed stringsCan oscillateNormal modesQuantized energy levelsMass of the object = total energyM=0 states include a graviton (a spin 2 particle)First massive state has M ~ T2
59 ( Incorporates gauge interactions Unification ) String InteractionsSplitting and joiningString theory Feynman diagramgSimplest case: Flat 10 dimensions and supersymmetricPrecise rules, finite results, constrained mathematical structureAt low energies, energies smaller than the mass of the first massive string statelsGravity theoryRRadius of curvature >> string length gravity is a good approximation( Incorporates gauge interactions Unification )
60 = Particle theory = gravity theory Most supersymmetry QCD theory String theory on(J.M.)AdS x S5N = magnetic flux through S5N colorsRadius of curvatureDuality:g2 N is small perturbation theory is easy – gravity is badg2 N is large gravity is good – perturbation theory is hardStrings made with gluons become fundamental strings.
61 z Where Do the Extra Dimensions Come From? 3+1 AdS5 radial dimensionStrings live hereInteriorGluons live herezBoundary
62 What about the S5 ? Related to the 6 scalars S5 other manifolds = Most susy QCD less susy QCD.Large number of examplesKlebanov, Witten,Gauntlett, Martelli, Sparks,Hannany, Franco, Benvenutti,Tachikawa, Yau …..
63 Quark anti quark potential stringBoundaryqV = potential = proper length of the stringin AdSWeak coupling result:
64 Confining TheoriesAdd 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.Gravitational potentialor warp factorw(z)zz0boundaryw(z0) > 0String at z0 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.
65 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, g2N.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
66 What can we learn about gravity from the field theory ? The relation connects a quantum field theory to gravity.What can we learn about gravity from the field theory ?Useful for understanding quantum aspects of black holes
67 Black holes Black holes evaporate S = Gravitational collapse leads to black holesClassically nothing can escape once it crosses the event horizonQuantum mechanics implies that black holes emit thermalradiation.(Hawking)Black holes evaporateEvaporation timeTemperature is related to entropyArea of the horizon4 LdM = T dSS =(Hawking-Bekenstein)2PlanckWhat is the statistical interpretation of this entropy?
68 Black holes in AdS Thermal configurations in AdS. Entropy:SGRAVITY = Area of the horizon =SFIELD THEORY =Log[ Number of states]Evolution: UnitarySolve the information paradox raised by S. Hawking
69 Confining Theories and Black Holes Low temperaturesHigh temperaturesDeconfinement=black hole (black brane)ConfinementGravitationalpotentialHorizonExtra dimensionExtra dimension
70 Black holes in the Laboratory z=z0 ,z=0QCD 5d string theoryHigh energy collision produces a black hole =droplet of deconfined phase ~quark gluon plasma .z=z0z=0Black hole Very low shear viscositysimilar to what is observed at RHIC:“ the most perfect fluid”Kovtun, Son, Starinets, PolicastroVery rough model, we do not yet know the precise string theory
71 Emergent space time Spacetime: like the fermi surface, only defined in the classical limitLin, Lunin, J.M.
72 A theory of some universe Suppose that we lived in anti-de-sitter spaceThen 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 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 ?
73 Conclusions - Gravity and particle physics are “unified” Usual: Quantum gravity particle physics.New: Particle physics quantum gravity.Black holes and confinement are relatedEmergent 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.
74 FutureField theory:Theories closer to the theory of strong interactionsSolve large N QCDGravity:Quantum gravity in other spacetimesUnderstand cosmological singularities