AdS/CFT: Then and NowAdS/CFT: Then and Now

Igor KlebanovIgor Klebanov

Department of PhysicsDepartment of Physics

Talk at Miami 2007 Workshop Talk at Miami 2007 Workshop `Celebrating 10 Years of AdS/CFT’`Celebrating 10 Years of AdS/CFT’

Stacking Non-Dilatonic D-Stacking Non-Dilatonic D-BranesBranes

• A stack of N Dirichlet 3-branes realizes A stack of N Dirichlet 3-branes realizes NN=4 =4 supersymmetric SU(N) gauge theory in 4 supersymmetric SU(N) gauge theory in 4 dimensions. It also creates a smooth extremal dimensions. It also creates a smooth extremal background of 10-d theory of closed background of 10-d theory of closed superstrings with a constant dilaton superstrings with a constant dilaton (artwork by (artwork by E.Imeroni)E.Imeroni)

which for small r approaches which for small r approaches

• As Juan Maldacena pointed out 10 years ago, the As Juan Maldacena pointed out 10 years ago, the small r limit is the low-energy limit where the small r limit is the low-energy limit where the NN=4 SYM theory on D-branes decouples from the =4 SYM theory on D-branes decouples from the bulk closed strings. This is AdS/CFT `in a bulk closed strings. This is AdS/CFT `in a nutshell.’nutshell.’

Super-Conformal Super-Conformal InvarianceInvariance

• In the In the NN=4 SYM theory there are 6 scalar =4 SYM theory there are 6 scalar fields and 4 gluinos interacting with the fields and 4 gluinos interacting with the gluons. All the fields are in the adjoint gluons. All the fields are in the adjoint representation of the SU(N) gauge group.representation of the SU(N) gauge group.

• The Asymptotic Freedom is canceled by The Asymptotic Freedom is canceled by the extra fields; the beta function is the extra fields; the beta function is exactly zero for any complex coupling. The exactly zero for any complex coupling. The coupling may be very strong in the UV. coupling may be very strong in the UV.

• The theory is invariant under scale The theory is invariant under scale transformations xtransformations x -> a x -> a xIt is also It is also invariant under space-time inversions. The invariant under space-time inversions. The full super-conformal group is SU(2,2|4). full super-conformal group is SU(2,2|4).

• A way of probing relations between D-A way of probing relations between D-branes and geometry that was popular branes and geometry that was popular in the mid-90’s was to add a small in the mid-90’s was to add a small amount of energy density which amount of energy density which produces a near-extremal solution of produces a near-extremal solution of type IIB supergravity:type IIB supergravity:

• On the D3-branes we then find a gas of On the D3-branes we then find a gas of massless open strings, i.e. massless open strings, i.e. NN=4 SYM =4 SYM theory at finite temperature identified theory at finite temperature identified with the Hawking temperature of the with the Hawking temperature of the horizon located at rhorizon located at r00

• A brief calculation gives the entropy A brief calculation gives the entropy density density Gubser, IK, Peet; IK, Tseytlin Gubser, IK, Peet; IK, Tseytlin

• This gravitational entropy `knows’ aboutThis gravitational entropy `knows’ about the Stefan-Boltzman dependence on the Stefan-Boltzman dependence on

temperature in 3 spatial dimensions, and temperature in 3 spatial dimensions, and also that the massless gas has ~ Nalso that the massless gas has ~ N22 degrees of freedom. degrees of freedom.

• Yet, the precise coefficient is smaller by a Yet, the precise coefficient is smaller by a factor of ¾ than the corresponding result factor of ¾ than the corresponding result in free SYM theory, which initially caused in free SYM theory, which initially caused some confusion and consternation. some confusion and consternation.

• The correct intepretation of the BH entropy is as The correct intepretation of the BH entropy is as the strong coupling limit in the planar gauge the strong coupling limit in the planar gauge theory theory

• The weak `t Hooft coupling behavior of the The weak `t Hooft coupling behavior of the interpolating function is determined by Feynman interpolating function is determined by Feynman graph calculations in the graph calculations in the NN=4 SYM theory =4 SYM theory

• We deduce from AdS/CFT duality thatWe deduce from AdS/CFT duality that

• The entropy density is multiplied only by ¾ as The entropy density is multiplied only by ¾ as the coupling changes from zero to infinity. the coupling changes from zero to infinity. Gubser, Gubser, IK, TseytlinIK, Tseytlin

Stacking M2- and M5-Stacking M2- and M5-BranesBranes

• Similar calculations for the 2- and 5-brane Similar calculations for the 2- and 5-brane backgrounds in 11-d supergravity yield backgrounds in 11-d supergravity yield IK, IK, Tseytlin Tseytlin

• These are the dual gravity predictions for These are the dual gravity predictions for thermal entropy of the large thermal entropy of the large nn superconformal field theories on coincident superconformal field theories on coincident M2- and M5-branes, respectively. M2- and M5-branes, respectively.

• They are still not well-understood, even at a They are still not well-understood, even at a qualitative level. Particularly interesting is qualitative level. Particularly interesting is the nthe n33 growth of the number of degrees of growth of the number of degrees of freedom on M5-branes, which is faster than freedom on M5-branes, which is faster than nn22 found in U(n) gauge theory. found in U(n) gauge theory.

• For the For the NN=4 SYM theory we at =4 SYM theory we at least have some qualitative least have some qualitative understanding of the ¾.understanding of the ¾.

• Corrections to the interpolating Corrections to the interpolating function at strong coupling function at strong coupling come from the higher-come from the higher-derivative terms in the type IIB derivative terms in the type IIB effective action:effective action:

Gubser, IK, TseytlinGubser, IK, Tseytlin

• The interpolating function is The interpolating function is usually assumed to have a usually assumed to have a smooth monotonic form (see smooth monotonic form (see Juan’s 1998 sketch), but so far Juan’s 1998 sketch), but so far we do not know its form at the we do not know its form at the intermediate coupling.intermediate coupling.

• A similar reduction of A similar reduction of entropy by strong-entropy by strong-coupling effects is coupling effects is observed in lattice non-observed in lattice non-supersymmetric gauge supersymmetric gauge theories for N=3: the theories for N=3: the arrows show free field arrows show free field values.values.

Karsch (hep-lat/0106019).Karsch (hep-lat/0106019).

• N-dependence in the N-dependence in the pure glue theory enters pure glue theory enters largely through the largely through the overall normalization.overall normalization.

Bringoltz and Teper (hep-lat/0506034)Bringoltz and Teper (hep-lat/0506034)

The AdS/CFT dualityThe AdS/CFT dualityMaldacena; Gubser, IK, Polyakov; WittenMaldacena; Gubser, IK, Polyakov; Witten

• Relates conformal gauge theory in 4 dimensions Relates conformal gauge theory in 4 dimensions to string theory on 5-d Anti-de Sitter space times a to string theory on 5-d Anti-de Sitter space times a 5-d compact space. For the 5-d compact space. For the NN=4 SYM theory this =4 SYM theory this compact space is a 5-d sphere.compact space is a 5-d sphere.

• When a gauge theory is strongly coupled, the When a gauge theory is strongly coupled, the radius of curvature of the dual AdSradius of curvature of the dual AdS55 and of the 5-d and of the 5-d compact space becomes large:compact space becomes large:

• String theory in such a weakly curved background String theory in such a weakly curved background can be studied in the effective (super)-gravity can be studied in the effective (super)-gravity approximation, which allows for a host of explicit approximation, which allows for a host of explicit calculations. Corrections to it proceed in powers of calculations. Corrections to it proceed in powers of

• Feynman graphs instead develop a weak coupling Feynman graphs instead develop a weak coupling expansion in powers of expansion in powers of At weak coupling the At weak coupling the dual string theory becomes difficult.dual string theory becomes difficult.

• Gauge invariant operators in the CFTGauge invariant operators in the CFT44 are in are in one-to-one correspondence with fields (or one-to-one correspondence with fields (or extended objects) in AdSextended objects) in AdS55

• Operator dimension is determined by the Operator dimension is determined by the mass of the dual field; e.g. for scalar mass of the dual field; e.g. for scalar operators operators GKPWGKPW

• The BPS protected operators are dual to The BPS protected operators are dual to SUGRA fields of m~1/L. Their dimensions SUGRA fields of m~1/L. Their dimensions are independent of are independent of . .

• The unprotected operators (Konishi operator The unprotected operators (Konishi operator is the simplest) are dual to massive string is the simplest) are dual to massive string states. AdS/CFT predicts that at strong states. AdS/CFT predicts that at strong coupling their dimensions grow as coupling their dimensions grow as 1/41/4..

A Proof ?A Proof ?• These arguments provide a solid These arguments provide a solid

motivation for the AdS/CFT motivation for the AdS/CFT correspondence, but its proof correspondence, but its proof remains an open problem. New ideas remains an open problem. New ideas in this direction continue to emerge. in this direction continue to emerge. Berkovits, VafaBerkovits, Vafa

• A time-honored tradition is to simply A time-honored tradition is to simply assume that the correspondence assume that the correspondence holds. holds.

• A recent confidence boost has been A recent confidence boost has been provided by a smooth extrapolation provided by a smooth extrapolation of `the cusp anomaly’ between the of `the cusp anomaly’ between the perturbative data at weak coupling perturbative data at weak coupling and the AdS/CFT predictions at and the AdS/CFT predictions at strong coupling. This is based on the strong coupling. This is based on the plausible, but yet unproven plausible, but yet unproven assumption of Exact Integrability. assumption of Exact Integrability. Minahan, Zarembo; Beisert, Kristjanson, Staudacher; Minahan, Zarembo; Beisert, Kristjanson, Staudacher; Bena, Polchinski, Roiban; Beisert, Eden, Staudacher; Bena, Polchinski, Roiban; Beisert, Eden, Staudacher; Bern, Czakon, Dixon, Kosower, Smirnov; Benna, Bern, Czakon, Dixon, Kosower, Smirnov; Benna, Benvenuti, IK, Scardicchio; Basso, Korchemsky, Benvenuti, IK, Scardicchio; Basso, Korchemsky, Kotanski; …Kotanski; …

The quark anti-quark The quark anti-quark potentialpotential• The z-direction of AdS is The z-direction of AdS is

dual to the energy scale of dual to the energy scale of the gauge theory: small z the gauge theory: small z is the UV; large z is the IR. is the UV; large z is the IR.

• Because of the 5-th Because of the 5-th dimension z, the string dimension z, the string picture applies even to picture applies even to theories that are theories that are conformal. The quark and conformal. The quark and anti-quark are placed at anti-quark are placed at the boundary of Anti-de the boundary of Anti-de Sitter space (z=0), but the Sitter space (z=0), but the string connecting them string connecting them bends into the interior bends into the interior (z>0). Due to the scaling (z>0). Due to the scaling symmetry of the AdS symmetry of the AdS space, this gives Coulomb space, this gives Coulomb potential potential (Maldacena; Rey, Yee)(Maldacena; Rey, Yee)

String Theoretic Approaches toString Theoretic Approaches to ConfinementConfinement

• It is possible to generalize It is possible to generalize the AdS/CFT correspondence the AdS/CFT correspondence in such a way that the in such a way that the quark-antiquark potential is quark-antiquark potential is linear at large distance but linear at large distance but nearly Coulombic at small nearly Coulombic at small distance.distance.

• The 5-d metric should have The 5-d metric should have a warped form (Polyakov):a warped form (Polyakov):

• The space ends at a The space ends at a maximum value of z where maximum value of z where the warp factor is finite. the warp factor is finite. Then the confining string Then the confining string tension is tension is

Dynamical Confinement Dynamical Confinement • Introduction of minimal Introduction of minimal

supersymmetry (supersymmetry (NN=1) facilitates =1) facilitates construction of gauge/string construction of gauge/string dualities.dualities.

• A useful tool is to place D3-branes A useful tool is to place D3-branes and wrapped D5-branes at the tip and wrapped D5-branes at the tip of a 6-d cone, e.g. the conifold.of a 6-d cone, e.g. the conifold.

• The 10-d geometry dual to the The 10-d geometry dual to the gauge theory on these branes is gauge theory on these branes is the warped deformed conifold the warped deformed conifold (IK, (IK, Strassler)Strassler)

• is the metric of the deformed is the metric of the deformed conifold, a simple Calabi-Yau spaceconifold, a simple Calabi-Yau space

defined by the following constraint defined by the following constraint on 4 complex variables: on 4 complex variables:

• The warp factor is finite at the `tip of the The warp factor is finite at the `tip of the cigar’ t=0, as required for the confinement: cigar’ t=0, as required for the confinement: h(t)= 2h(t)= 2-8/3-8/3 I(t) I(t)

• The string tension, is proportional to h(t) The string tension, is proportional to h(t) -1/2 -1/2 and is minimized at t=0. It blows up at large and is minimized at t=0. It blows up at large t (near the boundary) where space is `near-t (near the boundary) where space is `near-AdS.’ AdS.’

• Dimensional transmutationDimensional transmutation in the IR. The in the IR. The dynamically generated confinement scale isdynamically generated confinement scale is

• The pattern of The pattern of R-symmetry breakingR-symmetry breaking is the is the same as in the SU(M) SYM theory: Zsame as in the SU(M) SYM theory: Z2M2M -> Z -> Z22

• Comparison of warp factors in the AdS, Comparison of warp factors in the AdS, warped conifold, and warped deformed warped conifold, and warped deformed conifold cases. The warped conifold solution conifold cases. The warped conifold solution with with =0 has an unacceptable naked =0 has an unacceptable naked singularity where h=0. singularity where h=0.

• This is how string theory tells us that the This is how string theory tells us that the chiral symmetry breaking and dynamical chiral symmetry breaking and dynamical scale generation must take place through scale generation must take place through turning on the deformation turning on the deformation . The finiteness of . The finiteness of the warp factor at r=0 translates into the warp factor at r=0 translates into confinement.confinement.

• The radius-squared of the SThe radius-squared of the S33 at t=0 is at t=0 is ggssM in string units.M in string units.

• When gWhen gssM is large, the curvatures are M is large, the curvatures are small everywhere, and the SUGRA small everywhere, and the SUGRA solution is reliable in `solving’ this solution is reliable in `solving’ this confining gauge theory.confining gauge theory.

• Even when gEven when gssM is small, the curvature M is small, the curvature gets small at large t (in the UV).gets small at large t (in the UV).

• In the dual gauge theory the coupling In the dual gauge theory the coupling stays strong in the UV. It is not stays strong in the UV. It is not asymptotically free, but rather asymptotically free, but rather undergoes a `cascading’ logarithmic RG undergoes a `cascading’ logarithmic RG flow that produces SU(M) gauge theory flow that produces SU(M) gauge theory in the IR.in the IR.

• What is the explanation in the dual What is the explanation in the dual SU(kM)xSU((k-1)M) SYM theory coupled SU(kM)xSU((k-1)M) SYM theory coupled to bifundamental chiral superfields Ato bifundamental chiral superfields A11, , AA22, B, B11, B, B22 ? A novel phenomenon, called ? A novel phenomenon, called a a duality cascadeduality cascade, takes place: k , takes place: k repeatedly changes by 1 as a result of repeatedly changes by 1 as a result of the Seiberg duality the Seiberg duality IK, Strassler IK, Strassler

(diagram of RG flows from a review by M. Strassler)(diagram of RG flows from a review by M. Strassler)

• The graph of quark anti-The graph of quark anti-quark potential is quark potential is qualitatively similar to qualitatively similar to that found in numerical that found in numerical simulations of QCD. The simulations of QCD. The upper graph, from the upper graph, from the recent Senior Thesis of V. recent Senior Thesis of V. Cvicek shows the string Cvicek shows the string theory result for the theory result for the warped deformed warped deformed conifold.conifold.

• The lower graph shows The lower graph shows lattice QCD results by G. lattice QCD results by G. Bali et al with rBali et al with r00 ~ 0.5 fm. ~ 0.5 fm.

Spectrum of GlueballsSpectrum of Glueballs• The confining string tension isThe confining string tension is

• The glueballs are the normalizable modes localized The glueballs are the normalizable modes localized near small t. In the supergravity limit (at large gnear small t. In the supergravity limit (at large gss M) M) their mass scale istheir mass scale is

• The mThe m22 of the n-th radial excitation scales as of the n-th radial excitation scales as nn22 T Tss/(g/(gss M) (see, for example the recent plots from M) (see, for example the recent plots from Benna, Dymarsky, IK, Soloviev). This is the behavior Benna, Dymarsky, IK, Soloviev). This is the behavior found in Kaluza-Klein theory, but not in QCD. found in Kaluza-Klein theory, but not in QCD.

• Glueballs with spin > 2 have much higher masses: Glueballs with spin > 2 have much higher masses: mm22 ~ T ~ Ts s

• This separation of scales is a new phenomenon This separation of scales is a new phenomenon found for theories with reliable gravity duals.found for theories with reliable gravity duals.

• All of this provides us with an All of this provides us with an exactexact solutionsolution of a class of 4-d large N confining of a class of 4-d large N confining supersymmetric gauge theories. supersymmetric gauge theories.

• This should be a good playground for This should be a good playground for studying strongly coupled gauge theory.studying strongly coupled gauge theory.

• Some results on glueball spectra are Some results on glueball spectra are already available, and further calculations already available, and further calculations are ongoing. are ongoing. Krasnitz; Caceres, Hernandez; Dymarsky, Melnikov; Berg, Krasnitz; Caceres, Hernandez; Dymarsky, Melnikov; Berg,

Haack, Muck; Benna, Dymarsky, IK, SolovievHaack, Muck; Benna, Dymarsky, IK, Soloviev

• Possible applications of these models to Possible applications of these models to new physics include RS warped extra new physics include RS warped extra dimension models, KKLT moduli dimension models, KKLT moduli stabilization in flux compactifications, as stabilization in flux compactifications, as well as warped throat D-brane cosmology well as warped throat D-brane cosmology (KKLMMT).(KKLMMT).

Applications to D-brane Applications to D-brane InflationInflation• The Slow-Roll Inflationary The Slow-Roll Inflationary

Universe Universe (Guth; Linde; Albrecht, (Guth; Linde; Albrecht, Steinhardt)Steinhardt) is a very promising idea is a very promising idea for generating the CMB for generating the CMB anisotropy spectrum observed anisotropy spectrum observed by the WMAP. by the WMAP.

• Finding models with very flat Finding models with very flat potentials has proven to be potentials has proven to be difficult. Recent string theory difficult. Recent string theory constructions use moving D-constructions use moving D-branes. branes. Dvali, Tye, …Dvali, Tye, …

• In the KKLT/KKLMMT model, the In the KKLT/KKLMMT model, the warped deformed conifold is warped deformed conifold is embedded into a string embedded into a string compactification. An anti-D3-compactification. An anti-D3-brane is added at the bottom to brane is added at the bottom to break SUSY and generate a break SUSY and generate a potential. A D3-brane rolls in the potential. A D3-brane rolls in the throat. Its radial coordinate throat. Its radial coordinate plays the role of an inflaton.plays the role of an inflaton.

Kachru, Kallosh, Linde, Maldacena, McAllister, Kachru, Kallosh, Linde, Maldacena, McAllister, TrivediTrivedi

Slow roll D-brane inflation?Slow roll D-brane inflation?

• Effects of D7-branes and of Effects of D7-branes and of compactification generically compactification generically spoil the flatness of the spoil the flatness of the potential. Non-perturbative potential. Non-perturbative effects introduce the KKLT-effects introduce the KKLT-type superpotentialtype superpotential

where X denotes the D3-where X denotes the D3-brane position. In any brane position. In any warped throat D-brane warped throat D-brane inflation model, it is inflation model, it is important to calculate important to calculate A(X).A(X).

• The effective potential for The effective potential for the inflaton generically has the inflaton generically has a local maximum and a local maximum and minium. It can be fine-minium. It can be fine-tuned to have an inflection tuned to have an inflection point.point.

• Motion near the inflection Motion near the inflection point can produce enough point can produce enough e-folds of inflation. e-folds of inflation. Baumann, Baumann, Dymarsky, IK, McAllister, SteinhardtDymarsky, IK, McAllister, Steinhardt

• Models of Inflection Point Models of Inflection Point Inflation were also Inflation were also considered in string theory considered in string theory by Itzhaki and Kovetz; …by Itzhaki and Kovetz; …

and in MSSM inflation by and in MSSM inflation by Allahverdi, Enqvist, Garcia-Allahverdi, Enqvist, Garcia-Bellido and Mazumdar; …Bellido and Mazumdar; …

Inflection Point InflationInflection Point Inflation• Assume a potentialAssume a potential

• The slow-roll parametersThe slow-roll parameters

• The number of e-folds until the end of The number of e-folds until the end of inflation isinflation is

• We find that NWe find that Ntottot is ~ is ~

The Scalar Spectral IndexThe Scalar Spectral Index• The usual slow-roll formula isThe usual slow-roll formula is

• For IPI, in terms ofFor IPI, in terms of

• For large NFor large Ntottot, , which is around 0.93.which is around 0.93.

• The running of the spectral The running of the spectral index is small for index is small for

Connection with Cosmic Connection with Cosmic StringsStrings

• A fundamental string at the bottom of the A fundamental string at the bottom of the

warped deformed conifold is dual to a confining warped deformed conifold is dual to a confining string. A D-string is dual to a certain solitonic string. A D-string is dual to a certain solitonic string. There is also a rich spectrum of (p,q) string. There is also a rich spectrum of (p,q) strings.strings.

• Upon embedding of the warped throat into a Upon embedding of the warped throat into a flux compactification, these objects can be flux compactification, these objects can be used to model cosmic strings. used to model cosmic strings. Copeland, Myers, Polchinski;Copeland, Myers, Polchinski; ……

• This throat is not the `standard model throat’ This throat is not the `standard model throat’ but another, `inflationary throat,’ dual to a but another, `inflationary throat,’ dual to a hidden sector gauge theory with confining hidden sector gauge theory with confining scale ~10scale ~101313 GeV. GeV.

• Some features of string web evolution are Some features of string web evolution are different from the typical models explored in different from the typical models explored in the 80’s. Now one finds low recombination the 80’s. Now one finds low recombination probability.probability.

Contact with the Observable Contact with the Observable WorldWorld

• Thermal gauge/string dualities are being used Thermal gauge/string dualities are being used to model QCD above deconfinement to model QCD above deconfinement temperature and to compare with Heavy-Ion temperature and to compare with Heavy-Ion collider experiments. Many successes have collider experiments. Many successes have already been achieved, e.g., the observed low already been achieved, e.g., the observed low shear viscosity appears naturally via AdS/CFT. shear viscosity appears naturally via AdS/CFT. Policastro, Son, StarinetsPolicastro, Son, Starinets

• Bottom-up model building in warped 5 Bottom-up model building in warped 5 dimensions (AdS/QCD) has provided new dimensions (AdS/QCD) has provided new fruitful ways to look at low-energy hadron fruitful ways to look at low-energy hadron physics. physics. Erlich et al, Da Rold, Pomarol, Brodsky et al, …Erlich et al, Da Rold, Pomarol, Brodsky et al, …

• Large N M2-brane theory at finite T and Large N M2-brane theory at finite T and chemical potential has shed light on chemical potential has shed light on properties of superconductors near Tproperties of superconductors near Tcc. . Herzog et al, Herzog et al, Hartnoll et al.Hartnoll et al.

• More generally, there exists a large More generally, there exists a large class of AdSclass of AdS44/CFT/CFT33 dualities that are not dualities that are not yet well-understood. Finding 3-d fixed yet well-understood. Finding 3-d fixed points in lattice/condensed matter points in lattice/condensed matter systems that have AdSsystems that have AdS44 duals is a good duals is a good goal. For large N Wilson-Fisher fixed goal. For large N Wilson-Fisher fixed points it was proposed that the dual is a points it was proposed that the dual is a higher-spin gauge theory in AdShigher-spin gauge theory in AdS44. . IK, IK, PolyakovPolyakov

• Another developing contact between Another developing contact between AdS/CFT and condensed matter physics AdS/CFT and condensed matter physics is the Entanglement Entropy. is the Entanglement Entropy. Ryu, TakayanagiRyu, Takayanagi

Lessons for the Future ?Lessons for the Future ?• I tried searching for them in the past I tried searching for them in the past

history…history…

• The original spirit of D-brane/black hole The original spirit of D-brane/black hole correspondence was: use D-branes to correspondence was: use D-branes to explain `black hole microstates.’ This led to explain `black hole microstates.’ This led to quick success for some specially constructed quick success for some specially constructed black holes. black holes. Strominger, Vafa; Callan, Maldacena; …Strominger, Vafa; Callan, Maldacena; …

• But soon various puzzles appeared, like the But soon various puzzles appeared, like the ¾ reduction in the 3-brane entropy, or the ¾ reduction in the 3-brane entropy, or the NN33 degrees of freedom for N coincident M5- degrees of freedom for N coincident M5-branes. branes.

Turning Things Around !Turning Things Around !• Instead of using D-branes to explain black Instead of using D-branes to explain black

holes, use black holes to make predictions holes, use black holes to make predictions about theories on D-branes, i.e. strongly about theories on D-branes, i.e. strongly coupled gauge theories. coupled gauge theories.

• This has opened a whole new window on This has opened a whole new window on aspects of gauge theory that were aspects of gauge theory that were previously inaccessible. In this picture the previously inaccessible. In this picture the extra dimensions of string or M-theory extra dimensions of string or M-theory neatly fit with the observable dimensions to neatly fit with the observable dimensions to produce new geometrical pictures of produce new geometrical pictures of various phenomena in strongly coupled various phenomena in strongly coupled gauge theory. gauge theory.

• So, AdS/CFT really was a paradigm shift.So, AdS/CFT really was a paradigm shift.

• Yet, we have not been able to solve QCD Yet, we have not been able to solve QCD or any other asymptotically free gauge or any other asymptotically free gauge theory.theory.

• For a simple reason: when the coupling in For a simple reason: when the coupling in such theories becomes weak in the UV, such theories becomes weak in the UV, curvature is expected to get large. The curvature is expected to get large. The machinery of Einstein’s GR is not enough. machinery of Einstein’s GR is not enough. We would need much more sophisticated We would need much more sophisticated string theoretic machinery to extract string theoretic machinery to extract something precise this way. something precise this way.

• This is an important project which may This is an important project which may take a long time…take a long time…

Turn Things Around Yet Turn Things Around Yet Again ?Again ?• Since QCD does not seem to want to be Since QCD does not seem to want to be

solved this way, why force it? We solved this way, why force it? We understand it pretty well anyway…understand it pretty well anyway…

• Rather, let us hope that Nature will Rather, let us hope that Nature will present us with a gift of new physics that present us with a gift of new physics that is `easy’ from the point of view of is `easy’ from the point of view of gauge/string dualities, i.e. is described by gauge/string dualities, i.e. is described by weakly curved string backgrounds. weakly curved string backgrounds.

• Further understanding of generic features Further understanding of generic features of such theories is a good goal. of such theories is a good goal.

• One such generic feature is the large One such generic feature is the large separation of scales between bound separation of scales between bound states with spin up to 2 and spin > 2. states with spin up to 2 and spin > 2. This is found in the RS model (cut-off This is found in the RS model (cut-off AdS) and extends to various other AdS) and extends to various other warped models. warped models.

• Similarly, we can search for Similarly, we can search for characteristic features of gauge/string characteristic features of gauge/string dualities in the sky: in models of D-dualities in the sky: in models of D-brane inflation, cosmic strings (which brane inflation, cosmic strings (which are sometimes dual to confining are sometimes dual to confining strings), etc. strings), etc.

• Let us follow the path where Let us follow the path where gauge/string dualities are taking us, gauge/string dualities are taking us, and surely the wonderful journey will and surely the wonderful journey will continue. continue.

• Happy Birthday, AdS/CFT, and Many Happy Birthday, AdS/CFT, and Many Happy Returns!Happy Returns!