Testing AdS/CFTTesting AdS/CFTIgor KlebanovIgor Klebanov

Department of PhysicsDepartment of Physics

Talk at Conference `10 Years of AdS/CFT’Talk at Conference `10 Years of AdS/CFT’Buenos Aires, Dec. 19, 2007Buenos Aires, Dec. 19, 2007

Stacking NonStacking Non--Dilatonic DDilatonic D--BranesBranesgg•• A stack of N Dirichlet 3A stack of N Dirichlet 3--branes realizes branes realizes NN=4 =4

supersymmetric SU(N) gauge theory in 4 dimensions. supersymmetric SU(N) gauge theory in 4 dimensions. It also creates a smooth extremal background of It also creates a smooth extremal background of 1010--d theory of closed superstrings with a constantd theory of closed superstrings with a constant1010 d theory of closed superstrings with a constant d theory of closed superstrings with a constant dilaton dilaton (artwork by E.Imeroni)(artwork by E.Imeroni)

which for small r approacheswhich for small r approacheswhich 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 p y g ,p y g ,small r limit is the lowsmall r limit is the low--energy limit where the energy limit where the NN=4 =4 SYM theory on DSYM theory on D--branes decouples from the bulk branes decouples from the bulk closed strings This is AdS/CFT `in a nutshell ’closed strings This is AdS/CFT `in a nutshell ’closed strings. This is AdS/CFT in a nutshell.closed strings. This is AdS/CFT in a nutshell.

SuperSuper--Conformal InvarianceConformal InvarianceSuperSuper Conformal InvarianceConformal Invariance

•• In theIn the NN=4 SYM theory there are 6 scalar fields=4 SYM theory there are 6 scalar fields•• In the In the NN=4 SYM theory there are 6 scalar fields =4 SYM theory there are 6 scalar fields (it is useful to combine them into 3 complex (it is useful to combine them into 3 complex scalars: Z W V) and 4 gluinos interacting withscalars: Z W V) and 4 gluinos interacting withscalars: Z, W, V) and 4 gluinos interacting with scalars: Z, W, V) and 4 gluinos interacting with the gluons. All the fields are in the adjoint the gluons. All the fields are in the adjoint representation of the SU(N) gauge grouprepresentation of the SU(N) gauge grouprepresentation of the SU(N) gauge group.representation of the SU(N) gauge group.

•• The Asymptotic Freedom is canceled by the The Asymptotic Freedom is canceled by the t fi ld th b t f ti i tl ft fi ld th b t f ti i tl fextra fields; the beta function is exactly zero for extra fields; the beta function is exactly zero for

any complex coupling. The theory is invariant any complex coupling. The theory is invariant d l t f tid l t f ti μμ >> μ μ It i lIt i lunder scale transformations xunder scale transformations xμμ --> a x> a xμ μ . . It is also It is also

invariant under spaceinvariant under space--time inversions. The full time inversions. The full f l i SU(2 2|4)f l i SU(2 2|4)supersuper--conformal group is SU(2,2|4). conformal group is SU(2,2|4).

•• A way of probing relations between DA way of probing relations between D--•• A way of probing relations between DA way of probing relations between Dbranes and geometry that was popular in branes and geometry that was popular in the midthe mid--90’s was to add a small amount of90’s was to add a small amount ofthe midthe mid 90 s was to add a small amount of 90 s was to add a small amount of energy density which produces a nearenergy density which produces a near--extremal solution of type IIB supergravity:extremal solution of type IIB supergravity:extremal solution of type IIB supergravity:extremal solution of type IIB supergravity:

•• On the D3On the D3--branes we then find a gas ofbranes we then find a gas ofOn the D3On the D3 branes we then find a gas of 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 withtheory at finite temperature identified withtheory at finite temperature identified with theory at finite temperature identified with the Hawking temperature of the horizon the Hawking temperature of the horizon located at rlocated at r00located at rlocated at r00

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

•• This gravitational entropy `knows’ aboutThis gravitational entropy `knows’ aboutthe Stefanthe Stefan--Boltzmann dependence onBoltzmann dependence onthe Stefanthe Stefan Boltzmann dependence on Boltzmann dependence on temperature in 3 spatial dimensions, and also temperature in 3 spatial dimensions, and also that the massless gas has ~ Nthat the massless gas has ~ N22 degrees of degrees of gg ggfreedom. freedom.

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

d t tid t tiand consternation. 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 g p g p g gg p g p g gtheory theory

•• The weak `t Hooft coupling behavior of the The weak `t Hooft coupling behavior of the i t l ti f ti i d t i d b Fi t l ti f ti i d t i d b Finterpolating 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•• We deduce from AdS/CFT duality thatWe deduce from AdS/CFT duality that

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

St ki M2St ki M2 d M5d M5 BBStacking M2Stacking M2-- and M5and M5--BranesBranes•• Similar calculations for the 2Similar calculations for the 2-- and 5and 5--brane brane

backgrounds in 11backgrounds in 11--d supergravity yield d supergravity yield IK, Tseytlin IK, Tseytlin

•• These are the dual gravity predictions for thermalThese are the dual gravity predictions for thermalThese are the dual gravity predictions for thermal These are the dual gravity predictions for thermal entropy of the large entropy of the large nn superconformal field superconformal field theories on coincident M2theories on coincident M2-- and M5and M5--branes, branes, respectivelyrespectivelyrespectively. respectively.

•• They are still not wellThey are still not well--understood, even at a understood, even at a qualitative level Particularly interesting is the nqualitative level Particularly interesting is the n33qualitative level. Particularly interesting is the nqualitative level. Particularly interesting is the ngrowth of the number of degrees of freedom on growth of the number of degrees of freedom on M5M5--branes, which is faster than nbranes, which is faster than n22 found in U(n) found in U(n)

ththgauge theory.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 qqunderstanding of the ¾.understanding of the ¾.

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

Gubser IK TseytlinGubser IK TseytlinGubser, IK, TseytlinGubser, IK, Tseytlin

•• The interpolating function is The interpolating function is usually assumed to have a usually assumed to have a

h i f (h i f (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 thewe do not know its form at thewe 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 strongentropy by strong--coupling coupling py y gpy y g p gp geffects is observed in lattice effects is observed in lattice nonnon--supersymmetric gauge supersymmetric gauge theories for N=3: the arrowstheories for N=3: the arrowstheories for N=3: the arrows theories for N=3: the arrows show free field values.show free field values.Karsch (hepKarsch (hep--lat/0106019).lat/0106019).

•• NN--dependence in the pure dependence in the pure glue theory enters largelyglue theory enters largelyglue theory enters largely glue theory enters largely through the overall through the overall normalization.normalization.normalization.normalization.Bringoltz and Teper (hepBringoltz and Teper (hep--lat/0506034)lat/0506034)

Conebrane DualitiesConebrane Dualities•• To reduce the number of supersymmetries To reduce the number of supersymmetries

in AdS/CFT, we may place the stack of N in AdS/CFT, we may place the stack of N / , y p/ , y pD3D3--branes at the tip of a 6branes at the tip of a 6--d Riccid Ricci--flat flat cone X whose base is a 5cone X whose base is a 5--d Einstein space d Einstein space Y:Y:Y: Y:

•• Taking the nearTaking the near--horizon limit of thehorizon limit of the•• Taking the nearTaking the near horizon limit of the horizon limit of the background created by the N D3background created by the N D3--branes, branes, we find the space AdSwe find the space AdS55 x Y, with N units x Y, with N units f RR 5f RR 5 f fl h di i if fl h di i iof RR 5of RR 5--form flux, whose radius is given form flux, whose radius is given

byby•• This type IIB background is conjectured toThis type IIB background is conjectured to•• This type IIB background is conjectured to This type IIB background is conjectured to

be dual to the IR limit of the gauge theory be dual to the IR limit of the gauge theory on N D3on N D3--branes at the tip of the cone X. branes at the tip of the cone X.

•• The The NN=1 SCFT on N D3=1 SCFT on N D3--branes at the branes at the f ff fapex of the conifold has gauge apex of the conifold has gauge

group SU(N)xSU(N) coupled to chiral group SU(N)xSU(N) coupled to chiral g p ( ) ( ) pg p ( ) ( ) psuperfields Asuperfields A11, A, A22, in , and B, in , and B11, B, B22inin IK WittenIK Wittenin . in . IK, WittenIK, Witten

•• The RThe R--charge of each fields is ½. This charge of each fields is ½. This i U(1)i U(1) l ll til ll tiinsures U(1)insures U(1)RR anomaly cancellation. anomaly cancellation.

•• The unique SU(2)The unique SU(2)AAxSU(2)xSU(2)BB invariant,invariant,The unique SU(2)The unique SU(2)AAxSU(2)xSU(2)BB invariant, invariant, exactly marginal quartic superpotential is exactly marginal quartic superpotential is added:added:added:added:

•• This defines the gauge theory whose This defines the gauge theory whose g g yg g ydual is AdSdual is AdS55 x x TT1,11,1

The AdS/CFT dualityThe AdS/CFT duality/ y/ yMaldacena; Gubser, IK, Polyakov; WittenMaldacena; Gubser, IK, Polyakov; Witten

•• Relates conformal gauge theory in 4 dimensions to string Relates conformal gauge theory in 4 dimensions to string theory on 5theory on 5 d Antid Anti de Sitter space times a 5de Sitter space times a 5 d compactd compacttheory on 5theory on 5--d Antid Anti--de Sitter space times a 5de Sitter space times a 5--d compact d compact space. For the space. For the NN=4 SYM theory this compact space is a =4 SYM theory this compact space is a 55--d sphere.d sphere.

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

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

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

•• Gauge invariant operators in the CFTGauge invariant operators in the CFT44 are in are in oneone toto one co espondence ith fields (oone co espondence ith fields (ooneone--toto--one correspondence with fields (or one correspondence with fields (or extended objects) in AdSextended objects) in AdS55

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

•• The BPS protected operators are dual to SUGRAThe BPS protected operators are dual to SUGRAThe BPS protected operators are dual to SUGRA The BPS protected operators are dual to SUGRA fields of m~1/L. Their dimensions are fields of m~1/L. Their dimensions are independent ofindependent of ll. Matching them provided some. Matching them provided someindependent of independent of ll. Matching them provided some . Matching them provided some of the earliest tests of AdS/CFT.of the earliest tests of AdS/CFT.

•• The unprotected operators are dual to massiveThe unprotected operators are dual to massive•• The unprotected operators are dual to massive The unprotected operators are dual to massive string states. AdS/CFT predicts that at strong string states. AdS/CFT predicts that at strong coupling their dimensions grow ascoupling their dimensions grow as ll1/41/4coupling their dimensions grow as coupling their dimensions grow as ll // ..

A P f ?A P f ?A Proof ?A Proof ?•• These arguments provide a solid motivation forThese arguments provide a solid motivation for•• These arguments provide a solid motivation for These arguments provide a solid motivation for

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

It h b tiIt h b ti h d t diti th d t diti t•• It has become a timeIt has become a time--honored tradition to honored tradition to simply assume that the correspondence holds. simply assume that the correspondence holds. O d thi h t b dO d thi h t b dOver and over, this was shown to be a good Over and over, this was shown to be a good idea. idea.

•• To illustrate this, let me entertain you with the To illustrate this, let me entertain you with the story of the `cusp anomaly’ in story of the `cusp anomaly’ in NN=4 SYM theory.=4 SYM theory.

Spinning Strings vs Highly Charged OperatorsSpinning Strings vs Highly Charged OperatorsSpinning Strings vs. Highly Charged OperatorsSpinning Strings vs. Highly Charged Operators

•• Vibrating closed strings with large angular Vibrating closed strings with large angular g g g gg g g gmomentum on the 5momentum on the 5--sphere are dual to sphere are dual to SYM operators with large RSYM operators with large R--charge (the charge (the p gp g g (g (number of fields Z) number of fields Z) Berenstein, Maldacena, NastaseBerenstein, Maldacena, Nastase

•• Generally semiGenerally semi--classical spinning stringsclassical spinning stringsGenerally, semiGenerally, semi classical spinning strings classical spinning strings are dual to highly charged operators, e.g. are dual to highly charged operators, e.g. the dual of a highthe dual of a high--spin operatorspin operatorthe dual of a highthe dual of a high spin operator spin operator

i f ld d t i i i d thi f ld d t i i i d this a folded string spinning around the is a folded string spinning around the center of AdScenter of AdS55. . Gubser, IK, PolyakovGubser, IK, Polyakov

•• The structure of dimensions of highThe structure of dimensions of high--spin spin t it ioperators isoperators is

•• The function f(g) is independent of the twist; The function f(g) is independent of the twist; (g) p ;(g) p ;it is universal in the planar limit.it is universal in the planar limit.

•• It also enters the cusp anomaly ofIt also enters the cusp anomaly of•• It also enters the cusp anomaly ofIt also enters the cusp anomaly ofWilson loops in Minkowski space. Wilson loops in Minkowski space. Polyakov; Korchemsky, Radyushkin, …Polyakov; Korchemsky, Radyushkin, …

This can be calculated usingThis can be calculated usingThis can be calculated usingThis can be calculated usingAdS/CFT. AdS/CFT. KruczenskiKruczenski//

•• At strong coupling, the AdS/CFT duality At strong coupling, the AdS/CFT duality predicts via the spinning string energypredicts via the spinning string energypredicts via the spinning string energy predicts via the spinning string energy calculations calculations Gubser, IK, Polyakov; Frolov, TseytlinGubser, IK, Polyakov; Frolov, Tseytlin

•• At weak coupling the expansion of the At weak coupling the expansion of the p g pp g puniversal function f(g) up to 3 loops isuniversal function f(g) up to 3 loops isKotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, Smirnov Kotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, Smirnov

Cl l d t f th ti l t lCl l d t f th ti l t l•• Clearly, a good set of mathematical tools Clearly, a good set of mathematical tools are needed to interpolate between these are needed to interpolate between these

li ili itwo limits.two limits.

Exact IntegrabilityExact Integrability•• Perturbative calculations of anomalous dimensions are Perturbative calculations of anomalous dimensions are

mapped to integrable spin chains, suggesting exact mapped to integrable spin chains, suggesting exact pp g p , gg gpp g p , gg gintegrability of the integrability of the NN=4 SYM theory. =4 SYM theory. Minahan, Zarembo; Minahan, Zarembo; Beisert, Kristjansen, StaudacherBeisert, Kristjansen, Staudacher

F l f th `SU(2) t ’ tF l f th `SU(2) t ’ t•• For example, for the SU(2) sector’ operators For example, for the SU(2) sector’ operators Tr (ZZZWZW…ZW)Tr (ZZZWZW…ZW) , where Z and W are two complex , where Z and W are two complex scalars the Heisenberg spin chain emerges at 1 loopscalars the Heisenberg spin chain emerges at 1 loopscalars, the Heisenberg spin chain emerges at 1 loop. scalars, the Heisenberg spin chain emerges at 1 loop. Higher loops correct the Hamiltonian but seem to Higher loops correct the Hamiltonian but seem to preserve its integrability.preserve its integrability.

•• This meshes nicely with earlier findings of integrability in This meshes nicely with earlier findings of integrability in certain subsectors of QCD. certain subsectors of QCD. Lipatov; Faddeev, Korchemsky; Braun, Lipatov; Faddeev, Korchemsky; Braun, Derkachov, ManashovDerkachov, Manashove ac o , a as oe ac o , a as o

•• The dual string theory approach indicates that in the The dual string theory approach indicates that in the SYM theory the exact integrability is present at very SYM theory the exact integrability is present at very t lit li H it i lik l tH it i lik l tstrong coupling strong coupling (Bena, Polchinski, Roiban).(Bena, Polchinski, Roiban). Hence it is likely to Hence it is likely to

exist for all values of the coupling.exist for all values of the coupling.

•• The coefficients in f(g) appear to be The coefficients in f(g) appear to be ffffrelated to the corresponding coefficients in related to the corresponding coefficients in

QCD through selecting at each order the QCD through selecting at each order the Q g gQ g gterm with the highest transcendentality. term with the highest transcendentality. Kotikov, Lipatov, Onishchenko, Velizhanin Kotikov, Lipatov, Onishchenko, Velizhanin

•• Recently, great progress has been Recently, great progress has been achieved in finding f(g) at 4 loops andachieved in finding f(g) at 4 loops andachieved in finding f(g) at 4 loops and achieved in finding f(g) at 4 loops and beyond. beyond. U i th th i h i t i thU i th th i h i t i th•• Using the the spin chain symmetries, the Using the the spin chain symmetries, the Bethe ansatz equations were restricted to Bethe ansatz equations were restricted to the form the form Staudacher, BeisertStaudacher, Beisert

•• The integrability of the planarThe integrability of the planar NN=4 SYM is a powerful=4 SYM is a powerful•• The integrability of the planar The integrability of the planar NN=4 SYM is a powerful =4 SYM is a powerful conjecture, but it does not seem sufficient by itself. The conjecture, but it does not seem sufficient by itself. The magnon Smagnon S--matrix contains an undetermined phase factor matrix contains an undetermined phase factor which affects the observableswhich affects the observableswhich affects the observables. which affects the observables.

•• A simple assumption, initially advocated by some A simple assumption, initially advocated by some physicists, is that the phase is trivial. The only problem is physicists, is that the phase is trivial. The only problem is h hi di h AdS/CFT d hi hh hi di h AdS/CFT d hi hthat this contradicts the AdS/CFT correspondence which that this contradicts the AdS/CFT correspondence which

implies that it is nonimplies that it is non--trivial at strong coupling. trivial at strong coupling. Arutyunov, Arutyunov, Frolov, Staudacher; Hofman, MaldacenaFrolov, Staudacher; Hofman, Maldacena

•• Using the trivial phase, Eden and Staudacher proposed Using the trivial phase, Eden and Staudacher proposed an equation which gives the cusp anomaly f(g) and an equation which gives the cusp anomaly f(g) and showed that the first 3 perturbative coefficients agreeshowed that the first 3 perturbative coefficients agreeshowed that the first 3 perturbative coefficients agree showed that the first 3 perturbative coefficients agree with gauge theory calculations. with gauge theory calculations.

•• Bern, Czakon, Dixon, Kosower and Smirnov embarked on Bern, Czakon, Dixon, Kosower and Smirnov embarked on a 4a 4 loop calculation to check whether agreement withloop calculation to check whether agreement witha 4a 4--loop calculation to check whether agreement with loop calculation to check whether agreement with the ES equation continues to hold. The fate of the the ES equation continues to hold. The fate of the AdS/CFT correspondence seemed to be hanging in the AdS/CFT correspondence seemed to be hanging in the balance!balance!balance! balance!

•• The monumental BCDKS 4The monumental BCDKS 4--loop calculation took manyloop calculation took manyThe monumental BCDKS 4The monumental BCDKS 4 loop calculation took many loop calculation took many months to complete. In the meantime, Beisert, Hernandez months to complete. In the meantime, Beisert, Hernandez and Lopez decided to assume the strong coupling behavior and Lopez decided to assume the strong coupling behavior of the phase factor predicted by AdS/CFT and to useof the phase factor predicted by AdS/CFT and to useof the phase factor predicted by AdS/CFT and to use of the phase factor predicted by AdS/CFT and to use Janik's crossing symmetry assumption for developing the Janik's crossing symmetry assumption for developing the strong coupling expansion of the phase factor. strong coupling expansion of the phase factor.

•• Finally the different approaches converged late in 2006Finally the different approaches converged late in 2006•• Finally, the different approaches converged late in 2006. Finally, the different approaches converged late in 2006. BCDKS found the 4BCDKS found the 4--loop coefficient in f(g) and ruled out loop coefficient in f(g) and ruled out the ``trivial phase'' conjecture. They guessed a simple the ``trivial phase'' conjecture. They guessed a simple prescription for how to modify the ES expansion of f(g) toprescription for how to modify the ES expansion of f(g) toprescription for how to modify the ES expansion of f(g) to prescription for how to modify the ES expansion of f(g) to all orders. all orders.

•• Independently, Beisert, Eden and Staudacher guessed the Independently, Beisert, Eden and Staudacher guessed the p y, , gp y, , gsmall g expansion of the phase factor consistent with the small g expansion of the phase factor consistent with the strong coupling expansion found by BHL. They derived the strong coupling expansion found by BHL. They derived the corrected form of the equation that determines the cusp corrected form of the equation that determines the cusp q pq panomaly and found the same series as the one conjectured anomaly and found the same series as the one conjectured by BCDKS. by BCDKS.

The BES EquationThe BES EquationThe BES EquationThe BES Equation•• f(g) is determined through solving an f(g) is determined through solving an (g) g g(g) g g

integral equationintegral equation

•• The BES kernel isThe BES kernel is•• The first term is the ES kernelThe first term is the ES kernel

while the second one is due to thewhile the second one is due to thewhile the second one is due to the while the second one is due to the dressing phase in the magnon Sdressing phase in the magnon S--matrixmatrix

•• This equation yields analytic predictions for all This equation yields analytic predictions for all q y y pq y y pplanar perturbative coefficients planar perturbative coefficients

•• Th th 4Th th 4 l i l kl i l k•• The gauge theory 4The gauge theory 4--loop answer is only known loop answer is only known numerically and agrees with this analytical numerically and agrees with this analytical prediction to around 0 001%prediction to around 0 001%prediction to around 0.001%.prediction to around 0.001%.Bern, Czakon, Dixon, Kosower and Smirnov;Bern, Czakon, Dixon, Kosower and Smirnov; Cachazo, Spradlin, VolovichCachazo, Spradlin, Volovich

•• The alternation of the series and the geometric The alternation of the series and the geometric b h i f h ffi i llb h i f h ffi i llbehavior of the coefficients remove all behavior of the coefficients remove all singularities from the real axis, allowing smooth singularities from the real axis, allowing smooth extrapolation to infinite couplingextrapolation to infinite couplingextrapolation to infinite coupling. extrapolation to infinite coupling.

•• The radius of convergence is ¼. The closest The radius of convergence is ¼. The closest singularities are squaresingularities are square--root branch points at root branch points at g qg q pp

•• To compare the large g behavior of f(g) directly To compare the large g behavior of f(g) directly p g g (g) yp g g (g) ywith the AdS/CFT predictions, one needs to with the AdS/CFT predictions, one needs to resum the perturbative expansion. One resum the perturbative expansion. One approach to this is to look for the solution of theapproach to this is to look for the solution of theapproach to this is to look for the solution of the approach to this is to look for the solution of the BES equation for all g. This is hard, but a simple BES equation for all g. This is hard, but a simple and very accurate numerical approach was and very accurate numerical approach was y ppy ppfound. found. Benna, Benvenuti, Klebanov, ScardicchioBenna, Benvenuti, Klebanov, Scardicchio

•• To solve the equation at finite coupling,To solve the equation at finite coupling,To solve the equation at finite coupling, To solve the equation at finite coupling, we use a basis of linearly independent we use a basis of linearly independent functionsfunctionsfunctionsfunctions

•• Determination of is tantamount to Determination of is tantamount to inverting an infinite matrix. inverting an infinite matrix.

•• Truncation to finite matrices convergesTruncation to finite matrices converges•• Truncation to finite matrices converges Truncation to finite matrices converges very rapidly. very rapidly. Benna, Benvenuti, IK, Scardicchio Benna, Benvenuti, IK, Scardicchio

•• The blue line refers to The blue line refers to the BES equation, red the BES equation, red line to the ES greenline to the ES greenline to the ES, green line to the ES, green line to the equation line to the equation where the dressing where the dressing kernel is divided by 2kernel is divided by 2kernel is divided by 2. kernel is divided by 2.

•• The first two terms ofThe first two terms ofThe first two terms of The first two terms of the numerical large g the numerical large g asymptotics are in very asymptotics are in very precise agreement withprecise agreement withprecise agreement with precise agreement with the AdS/CFT spinning the AdS/CFT spinning string predictions. The string predictions. The third is an approximatethird is an approximatethird is an approximate third is an approximate prediction.prediction.

•• Expanding at strong couplingExpanding at strong coupling•• Expanding at strong coupling,Expanding at strong coupling,The leading solution isThe leading solution is

Alday, Arutyunov, Benna, IKAlday, Arutyunov, Benna, IK

•• The difficult problem of strong coupling expansionThe difficult problem of strong coupling expansion•• The difficult problem of strong coupling expansion The difficult problem of strong coupling expansion around this solution was recently solved by Basso, around this solution was recently solved by Basso, Korchemsky and Kotanski who found that the coefficient Korchemsky and Kotanski who found that the coefficient f 1/ if 1/ i K/(4K/(4 22) i t ith th i l) i t ith th i lof 1/g is of 1/g is --K/(4 K/(4 pp22), in agreement with the numerical ), in agreement with the numerical

result. result. •• The expression containing the Catalan constant K is inThe expression containing the Catalan constant K is in•• The expression containing the Catalan constant K is in The expression containing the Catalan constant K is in

exact agreement with the string sigma model 2exact agreement with the string sigma model 2--loop loop correction to f(g). correction to f(g). Roiban, Tirziu, TseytlinRoiban, Tirziu, Tseytlin

•• Thus, f(g) is tested to the first 4 orders at small g, and Thus, f(g) is tested to the first 4 orders at small g, and the first 3 orders at large gthe first 3 orders at large gthe first 3 orders at large g. the first 3 orders at large g.

Conebranes and Trace AnomalyConebranes and Trace AnomalyConebranes and Trace AnomalyConebranes and Trace Anomaly•• In a 4In a 4--d CFT there are two d CFT there are two

l ffi il ffi itrace anomaly coefficients, a trace anomaly coefficients, a and c:and c:

•• Calculations on AdSCalculations on AdS5 5 x Y give x Y give their leading large N valuestheir leading large N valuestheir leading large N values their leading large N values Henningson, Skenderis; GubserHenningson, Skenderis; Gubser

•• In superIn super--conformal theories theconformal theories the•• In superIn super--conformal theories the conformal theories the anomalies are related to the anomalies are related to the

t f Rt f R h f thh f thspectrum of Rspectrum of R--charges of the charges of the chiral fermions: chiral fermions: Anselmi, Freedman, Anselmi, Freedman, Grisaru, JohansenGrisaru, Johansen

CY S kiCY S ki Ei t iEi t i•• CY cones over SasakiCY cones over Sasaki--Einstein spaces Einstein spaces YYp,qp,q of topology Sof topology S22 x Sx S33 have been have been

t t d ( dt t d ( d iiconstructed (p and q are coconstructed (p and q are co--prime prime integers). integers). Gauntlett, Martelli, Sparks, Waldram Gauntlett, Martelli, Sparks, Waldram

•• Their volumes are Their volumes are

•• The SU(N)The SU(N)2p2p SCFT’s on N D3SCFT’s on N D3--branes branes at the tip of the cones have beenat the tip of the cones have beenat the tip of the cones have been at the tip of the cones have been found. found. Benvenuti, Franco, Hanany, Martelli, SparksBenvenuti, Franco, Hanany, Martelli, Sparks

•• Here is the quiver diagram for theHere is the quiver diagram for the•• Here is the quiver diagram for the Here is the quiver diagram for the SCFT dual to AdSSCFT dual to AdS55 x Yx Y4,34,3

RR--charges from acharges from a--maximizationmaximizationRR charges from acharges from a maximizationmaximization•• The conformal invariance conditions do not The conformal invariance conditions do not

fully determine the Rfully determine the R--charges. Let charges. Let RRZZ=x, R=x, RYY=y, R=y, RUU=1=1--(x+y)/2, R(x+y)/2, RVV=1+ (x=1+ (x--y)/2y)/2RRZZ x, Rx, RYY y, Ry, RUU 11 (x y)/2, R(x y)/2, RVV 1 (x1 (x y)/2 y)/2

•• The technique of aThe technique of a--maximization maximization Intriligator, WechtIntriligator, Wecht

givesgivesgivesgives

•• Remarkably, this gives the trace anomalyRemarkably, this gives the trace anomalyRemarkably, this gives the trace anomaly Remarkably, this gives the trace anomaly agreeing with the AdS/CFTagreeing with the AdS/CFT

Benvenuti, Franco, Hanany, Martelli Sparks; Bertolini, Bigazzi, CotroneBenvenuti, Franco, Hanany, Martelli Sparks; Bertolini, Bigazzi, Cotrone

ConclusionsConclusionsConclusionsConclusions•• The AdS/CFT correspondence makes a multitude of The AdS/CFT correspondence makes a multitude of

dynamical predictions about strongly coupleddynamical predictions about strongly coupleddynamical predictions about strongly coupled dynamical predictions about strongly coupled conformal gauge theories. They always appear to conformal gauge theories. They always appear to make sense, but are often difficult to check make sense, but are often difficult to check

i i l ( h ¾ i h )i i l ( h ¾ i h )quantitatively (e.g., the ¾ in the entropy).quantitatively (e.g., the ¾ in the entropy).•• For nonFor non--BPS quantities in BPS quantities in NN=4 SYM, non=4 SYM, non--trivial trivial

interpolating functions appear Recently theinterpolating functions appear Recently theinterpolating functions appear. Recently, the interpolating functions appear. Recently, the conjectured integrability and constraints from conjectured integrability and constraints from AdS/CFT and perturbative gauge theory led to a AdS/CFT and perturbative gauge theory led to a / p g g y/ p g g ycompelling proposal for the `cusp anomaly’ function. compelling proposal for the `cusp anomaly’ function. It provides new evidence for the validity of the It provides new evidence for the validity of the AdS/CFT dualityAdS/CFT dualityAdS/CFT duality. AdS/CFT duality.

•• This approaches makes further predictions for This approaches makes further predictions for perturbative gauge theory that are perfectlyperturbative gauge theory that are perfectlyperturbative gauge theory that are perfectly perturbative gauge theory that are perfectly testable. testable.

•• Thus, AdS/CFT has enabled intricate tests Thus, AdS/CFT has enabled intricate tests of string theory: not yet in the lab, but of string theory: not yet in the lab, but with paper, pencil, computer and millions with paper, pencil, computer and millions p p , p , pp p , p , pof Feynman diagrams. of Feynman diagrams.

•• There are many other impressive tests ofThere are many other impressive tests ofThere are many other impressive tests of There are many other impressive tests of AdS/CFT: the matching of BPS protected AdS/CFT: the matching of BPS protected operator dimensions of the traceoperator dimensions of the traceoperator dimensions, of the trace operator dimensions, of the trace anomalies, etc. See also V. Pestun’s paper anomalies, etc. See also V. Pestun’s paper rigorously establishing matrix integralrigorously establishing matrix integralrigorously establishing matrix integral rigorously establishing matrix integral representation for circular Wilson loops.representation for circular Wilson loops.

•• Happ Bi thda AdS/CFT and ManHapp Bi thda AdS/CFT and Man•• Happy Birthday, AdS/CFT, and Many Happy Birthday, AdS/CFT, and Many Happy Returns!Happy Returns!