Testing AdS/CFT

Igor Klebanov

Department of Physics

Talk at Conference `10 Years of AdS/CFT’ Buenos Aires, Dec. 19, 2007 Stacking NonNon--DilatonicDilatonic D-D-BranesBranes

• A stack of N Dirichlet 33--branesbranes realizes N=4 supersymmetric SU(N) gauge theory in 4 dimensions. It also creates a smooth extremal background of 10-d theory of closed superstrings with a constant dilaton (artwork by E.Imeroni)

which for small r approaches

• As Juan Maldacena ppyg,ointed out 10 years ago, the small r limit is the lowlow--energyenergy limit where the N=4 SYM theory on D-D-branesbranes decouples from the bulk closed strings. This is AdS/CFT `in a nutshell.’ Super-Conformal Invariance

• In the N=4 SYM theory there are 6 scalar fields (it is useful to combine them into 3 complex scalars: Z , W , V) and 4 gluinos interacting with the gluons. All the fields are in the adjoint representation of the SU(N) gauge group. • The Asymptotic Freedom is canceled by the extfildthbtftiitra fields; the beta function is exactl y zero f or any complex coupling. The theory is invariant undltftider scale transformations xμ -> a xμ . It is a lso invariant under spacespace--timetime inversions. The full supersuper--conconflformal group i iSU(22|4)s SU(2,2|4). • A way of probing relations between D - branes and geometry that was popular in the mid-90’s was to add a small amount of energy density which produces a nearnear-- extremal solution of type IIB supergravity:

• On the D3 -branes we then find a gas of massless open strings, i.e. N=4 SYM theory at finite temperature identified with the Hawking temperature of the horizon located at r0 • A brief calculation gives the entropy density Gubser, IK, Peet; IK, Tseytlin

• This gravitational entropy `knows’ about the Stefan-Boltzmann dependence on temperature in 3 spatial dimensions, and also that the massless gas has ~ N2 degrees of freedom. • Yet, the ppyrecise coefficient is smaller by a factor of ¾ than the corresponding result in free SYM theory, which initially caused some confusion andttid consternation. • The correct intepretation of the BH entropy is as the stronggpg coupling limit in the p lanar g ggauge theory

• The weak `t Hooft coupling behavior of the itinterpol ltiating f uncti on i s d dtetermi ned dbF by Feynman graph calculations in the N=4 SYM theory

• We deduce from AdS/CFT duality that

• The entropy density is multiplied only by ¾ as the coupling changes from zero to infinity. Gubser, IK, Tseytlin Stac king M2--anandM5d M5-Branes

• Similar calculations for the 2-2-andand 55--branebrane backgrounds in 1111--dd supergravity yield IK, Tseytlin

• These are the dual gravity predictions for thermal entropy of the large n superconformal field theories on coincident M2M2--andand M5M5--branes,branes, respectively. • They are still not well-well-understood,understood, even at a qualitative level . Particularly interesting is the n 3 growth of the number of degrees of freedom on M5M5--branes,branes, which is faster than n2 found in U(n) gauge theory. • For the N=4 SYM theory we at least have some qualitative understanding of the ¾. • Corrections to the interpolating function at strong coupling come from the higherhigher--derivativederivative terms in the type IIB effective action:

Gubser, IK, Tseytlin • The interpolating function is usually assumed to have a smoothif(h monotonic form (see Juan’s 1998 sketch), but so far we do not know its form at the intermediate coupling. • A similar reduction of entropy by strong --coucouppgling effects is observed in lattice nonnon--supersymmetricsupersymmetric gauge theories for N=3: the arrows show free field values. Karsch (hep(hep--lat/0106019).lat/0106019).

• NN--dependencedependence in the pure glue theory enters largely through the overall normalization. Bringoltz and Teper (hep(hep--lat/0506034)lat/0506034) Conebrane Dualities • To reduce the number of supersymmetries in AdS//,CFT, we ma ypy place the stack of N D3D3--branesbranes at the tip of a 66--dd RicciRicci--flatflat cone X whose base is a 55--dd Einstein space Y:

• Taking the near -horizon limit of the background created by the N D3D3--branes,branes, we find the space AdS5 x Y, with N units ofRR5f RR 5-fflhform flux, whose radi diiius is given by • This type IIB background is conjectured to be dual to the IR limit of the gauge theory on N D3D3--branesbranes at the tip of the cone X. • The N=1 SCFT on N D3D3--branesbranes at the apex offf the conifold has gauge group SU( ()N)xSU( ()N) coup led to chiral

superfields A1, A2, in , and B1, B2 in . IK, Witten • The R-R-chargecharge of each fields is ½. This

iU(1)insures U(1)R anomallltily cancellation.

• The unique SU(2) AxSU(2)B invariant, exactly marginal quartic superpotential is added: • This defines the gggauge theory whose 1,1 dual is AdS5 x T The AdS//yCFT duality Maldacena; Gubser, IK, Polyakov; Witten • Relates conformal gauge theory in 4 dimensions to string theory on 5 -ddAnti Anti-de Sitter space times a 5-dcompactd compact space. For the N=4 SYM theory this compact space is a 55--dd sphere. • When a gauge theory is strongly coupled, the radius of curvature of the dual AdS5 and of the 55--dd compact space becomes large:

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

• Feynman graphs instead develop a weak coupling expansion in powers of λ. At weak coupling the dual string theory becomes difficult. • Gauge invariant operators in the CFT4 are in one-to-one corr espondence w ith fields (or

extended objects) in AdS5 • Operator dimension is determined by the mass of the dual field; e.g. for scalar operators GKPW

• The BPS protected operators are dual to SUGRA fields of m~1/L. Their dimensions are independent of l. Matching them provided some of the earliest tests of AdS/CFT. • The unprotected operators are dual to massive string states. AdS/CFT predicts that at strong coupling their dimensions grow as l1/4. APA Proo f?f ? • These arguments provide a solid motivation for the AdS/CFT correspondence, but its rigorous proof remains an open problem. New ideas in this direction continue to emerge. Berkovits, Vafa • It h as b ecome a ti meme--hdtditithonored tradition to simply assume that the correspondence holds. OdthihtbdOver and over, this was shown to be a good idea. • To illustrate this, let me entertain you with the story of the `cusp anomaly’ in N=4 SYM theory. Spinning Strings vs . Highly Charged Operators • Vibratinggggg closed strings with large angular momentum on the 55--spheresphere are dual to SYM oppgerators with large R--charchargg(e (the number of fields Z) Berenstein, Maldacena, Nastase • Generally, semi -classical spinning strings are dual to highly charged operators, e.g. the dual of a high -spin operator

iflddtiiiis a folded string spinning around dth the

center of AdS5. Gubser, IK, Polyakov • The structure of dimensions of highhigh--spinspin operatitors is • The function f(g) is indep endent of the twist; it is universal in the planar limit. • It also enters the cusp anomaly of Wilson loops in Minkowski space.

Polyakov; Korchemsky, Radyushkin, … This can be calculated using

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

• At weak couppgling the exp ansion of the universal function f(g) up to 3 loops is Kotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, Smirnov

• Clear ly, a good set of ma themati cal t ool s are needed to interpolate between these two limi ts. Exact Integrability • Perturbative calculations of anomalous dimensions are mapped to integ rable sp in chains, sugg esting exact integrability of the N=4 SYM theory. Minahan, Zarembo; Beisert, Kristjansen, Staudacher • FlfthFor example, for the `SU(2) sect or’ operat ors Tr (ZZZWZW…ZW) , where Z and W are two complex scalars, the Heisenberg spin chain emerges at 1 loop. Higher loops correct the Hamiltonian but seem to preserve its integrability. • This meshes nicely with earlier findings of integrability in certain subsectors of QCD. Lipatov; Faddeev, Korchemsky; Braun, Deeaco,aasorkachov, Manashov • The dual string theory approach indicates that in the SYM theory the exact integrability is present at very stlitrong coupling (Bena, Polchinski, Roiban). Hence it is lik el y t o exist for all values of the coupling. • The coefficients in f(g) appear to be related to the corresponding coefficients in QQgCD through selecting at each order the term with the highest transcendentality. Kotikov, Lipatov, Onishchenko, Velizhanin • Recently, great progress has been achieved in finding f(g) at 4 loops and beyond. • Usi ng th e th e spi n ch ai n symmet ri es, th e Bethe ansatz equations were restricted to the form Staudacher, Beisert • The integrability of the planar N=4 SYM is a powerful conjecture, but it does not seem sufficient by itself. The magnon SS--matrixmatrix contains an undetermined phase factor which affects the observables. • A simple assumption, initially advocated by some physicists, is that the phase is trivial. The only problem is thhihat this contradi dihAdS/CFTcts the AdS/CFT correspond ence whi hihch implies that it is non-non-trivialtrivial at strong coupling. Arutyunov, Frolov, Staudacher; Hofman, Maldacena • Using the trivial phase, Eden and Staudacher proposed an equation which gives the cusp anomaly f(g) and showed that the first 3 perturbative coefficients agree with gauge theory calculations. • Bern, Czakon, Dixon, Kosower and Smirnov embarked on aa4 4-loop calculation to check whether agreement with the ES equation continues to hold. The fate of the AdS/CFT correspondence seemed to be hanging in the balance! • The monumental BCDKS 4 -loop calculation took many months to complete. In the meantime, Beisert, Hernandez and Lopez decided to assume the strong coupling behavior of the phase factor predicted by AdS/CFT and to use Janik's crossing symmetry assumption for developing the strong coupling expansion of the phase factor. • Finally, the different approaches converged late in 2006. BCDKS found the 44--looploop coefficient in f(g) and ruled out the ``trivial phase'' conjecture. They guessed a simple prescription for how to modify the ES expansion of f(g) to all orders. • Indeppy,,endently, Beisert, Eden and Staudacher g uessed the small g expansion of the phase factor consistent with the strong coupling expansion found by BHL. They derived the corrected form of the eqqpuation that determines the cusp anomaly and found the same series as the one conjectured by BCDKS. The BES Equation • f(g) is determined throug h solving an integral equation

• The BES kernel is • The first term is the ES kernel while the second one is due to the dressing phase in the magnon SS--matrixmatrix • This eqqyuation yields analy yptic predictions for all planar perturbative coefficients

• The gauge th eory 4 -lilkloop answer is only known numerically and agrees with this analytical prediction to around 0 .001% .

Bern, Czakon, Dixon, Kosower and Smirnov;Cachazo, Spradlin, Volovich • The alternation of the series and the geometric beh avi or of th e coeffi ci ents remove all singularities from the real axis, allowing smooth extrapolation to infinite coupling . • The radius of convergence is ¼. The closest singgqularities are squareuare--rootroot branch points at

• To comppgg(g)yare the large g behavior of f(g) directly with the AdS/CFT predictions, one needs to resum the perturbative expansion. One approach to this is to look for the solution of the BES equation for all g. This is hard, but a simple and veryypp accurate numerical approach was found. Benna, Benvenuti, Klebanov, Scardicchio • To solve the equation at finite coupling, we use a basis of linearly independent functions • Determination of is tantamount to inverting an infinite matrix.

• Truncation to finite matrices converges very rapidly. Benna, Benvenuti, IK, Scardicchio • The blue line refers to the BES equation, red line to the ES , green line to the equation where the dressing kernel is divided by 2 .

• The first two terms of the numerical large g asymptotics are in very precise agreement with the AdS/CFT spinning string predictions. The third is an approximate prediction. • Expanding at strong coupling , The leading solution is

Alday, Arutyunov, Benna, IK • The difficult problem of strong coupling expansion around this solution was recently solved by Basso, Korchemsky and Kotanski who found that the coefficient of1/f 1/g is -K/(4 p2), i n agreement with th e numeri cal result. • The expression containing the Catalan constant K is in exact agreement with the string sigma model 2-2-looploop correction to f(g). Roiban, Tirziu, Tseytlin

• Thus, f(g) is tested to the first 4 orders at small g, and the first 3 orders at large g. Conebranes and Trace Anomaly • In a 4-4-dd CFT there are two trace anomaly coeffi ci ents, a and c:

• Calculations on AdS5 x Y give their leading large N values Henningson, Skenderis; Gubser • In super -conformal theories the anomalies are related to the spectfRtrum of R--cchfthharges of the chiral fermions: Anselmi, Freedman, Grisaru, Johansen • CY cones over S asaki -Einst ei n spaces Yp,q of topology S2 x S3 have been consttd(dtructed (p and q are coco--prpriiime integers). Gauntlett, Martelli, Sparks, Waldram • Their volumes are

• The SU(N)2p SCFT’s on N D3D3--branesbranes at the tip of the cones have been found. Benvenuti, Franco, Hanany, Martelli, Sparks • Here is the quiver diagram for the 4,3 SCFT dual to AdS5 x Y R-charges from a-maximization • The conformal invariance conditions do not fully determine the RR--charges.charges. Let RZ=x, R Y=y, R U=1-(x+ y)/2, R V=1+ (x-y)/2 • The technique of aa--maximizationmaximization Intriligator, Wecht gives

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

Benvenuti, Franco, Hanany, Martelli Sparks; Bertolini, Bigazzi, Cotrone Conclusions • The AdS/CFT correspondence makes a multitude of dynamical predictions about strongly coupled conformal gauge theories. They always appear to make sense, but are often difficult to check quantiil(itatively (e.g., th h¾ihe ¾ in the entropy) . • For non-non-BPSBPS quantities in N=4 SYM, non-non-trivialtrivial interpolating functions appear . Recently, the conjectured integrability and constraints from AdS//pCFT and perturbative ggygauge theory led to a compelling proposal for the `cusp anomaly’ function. It provides new evidence for the validity of the AdS/CFT duality. • This approaches makes further predictions for perturbative gauge theory that are perfectly testable. • Thus, AdS/CFT has enabled intricate tests of string theory: not yet in the lab, but with pppaper, ,p pencil, comp uter and millions of Feynman diagrams. • There are many other impressive tests of AdS/CFT: the matching of BPS protected operator dimensions, of the trace anomalies, etc. See also V. Pestun’s paper rigorously establishing matrix integral representation for circular Wilson loops. • Happy Bi rthda y, AdS/CFT , and Man y Happy Returns!