Gauge Theories, D-Branes Gauge Theories, D-Branes

and Stringsand Strings

Igor KlebanovIgor Klebanov

Department of PhysicsDepartment of Physics

Talk at Jefferson Lab Talk at Jefferson Lab November 16, 2007November 16, 2007

QCD and String TheoryQCD and String Theory• At short distances, At short distances,

must smaller than 1 must smaller than 1 fermi, the quark-fermi, the quark-antiquark potential is antiquark potential is approximately approximately Coulombic, due to the Coulombic, due to the Asymptotic Freedom.Asymptotic Freedom.

• At large distances the At large distances the potential should be potential should be linear (Wilson) due to linear (Wilson) due to formation of confining formation of confining flux tubes.flux tubes.

Flux Tubes in QCDFlux Tubes in QCD• When these objects When these objects

are a lot longer than are a lot longer than their diameter their diameter (which is around a (which is around a fermi), their fermi), their dynamics is dynamics is approximately approximately described by the described by the Nambu-Goto area Nambu-Goto area action. So, strings action. So, strings have been have been observed, at least in observed, at least in numerical numerical simulations of simulations of gauge theory gauge theory (animation from lattice work by (animation from lattice work by D. Leinweber et al, Univ. of D. Leinweber et al, Univ. of Adelaide)Adelaide)

Large N Gauge TheoriesLarge N Gauge Theories• Connection of gauge theory with string Connection of gauge theory with string

theory is most apparent in `t Hooft’s theory is most apparent in `t Hooft’s generalization from 3 colors (SU(3) generalization from 3 colors (SU(3) gauge group) to N colors (SU(N) gauge gauge group) to N colors (SU(N) gauge group).group).

• Make N large, while keeping the `t Make N large, while keeping the `t Hooft coupling fixed:Hooft coupling fixed:

• The probability of snapping a flux tube The probability of snapping a flux tube by quark-antiquark creation (meson by quark-antiquark creation (meson decay) is 1/N. The string coupling is decay) is 1/N. The string coupling is 1/N.1/N.

• In the large N limit only planar In the large N limit only planar diagrams contribute, but 4-d gauge diagrams contribute, but 4-d gauge theory is still very difficult.theory is still very difficult.

Stacking D-BranesStacking D-Branes• Dirichlet branesDirichlet branes (Polchinski) led string theory back (Polchinski) led string theory back

to gauge theory in the mid-90’s.to gauge theory in the mid-90’s.• 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 curved background of dimensions. It also creates a curved background of 10-d theory of closed superstrings10-d theory of closed superstrings (artwork by E.Imeroni) (artwork by E.Imeroni)

which for small r approaches which for small r approaches

• Successful matching of graviton absorption by D3-Successful matching of graviton absorption by D3-branes, related to 2-point function of stress-energy branes, related to 2-point function of stress-energy tensor in the SYM theory, with a gravity calculation tensor in the SYM theory, with a gravity calculation in the 3-brane metric (IK; Gubser, IK, Tseytlin) was in the 3-brane metric (IK; Gubser, IK, Tseytlin) was a precursor of the AdS/CFT correspondence.a precursor of the AdS/CFT correspondence.

Super-Conformal Super-Conformal InvarianceInvariance

• 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 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 group.representation of the SU(N) gauge group.

• The Asymptotic Freedom is canceled by the The Asymptotic Freedom is canceled by the extra fields; the beta function is exactly zero extra fields; the beta function is exactly zero for any complex coupling. The theory is for any complex coupling. The theory is invariant under scale transformations xinvariant under scale transformations x -> a -> a xxIt is also invariant under space-time It is also invariant under space-time inversions. The full super-conformal group is inversions. The full super-conformal group is SU(2,2|4). SU(2,2|4).

Entropy of thermal Entropy of thermal NN=4 SUSY =4 SUSY SU(N) theorySU(N) theory

• Thermal CFT is described by a near-Thermal CFT is described by a near-extremal 3-brane background whose extremal 3-brane background whose near-horizon form is a black hole in AdSnear-horizon form is a black hole in AdS55

• The CFT temperature is identified with The CFT temperature is identified with the Hawking T of the horizon located at zthe Hawking T of the horizon located at zhh

• Any event horizon contains Bekenstein-Any event horizon contains Bekenstein-Hawking entropyHawking entropy

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

• This is interpreted as the strong coupling limit of This is interpreted as the strong coupling limit of

• 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

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

Gubser, IK, TseytlinGubser, IK, Tseytlin

• The interpolating function is usually The interpolating function is usually assumed to have a smooth assumed to have a smooth monotonic form, but so far we do not monotonic form, but so far we do not know its form at the intermediate know its form at the intermediate coupling.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 to Relates conformal gauge theory in 4 dimensions to string theory on 5-d Anti-de Sitter space times a 5-string theory on 5-d Anti-de Sitter space times a 5-d compact space. For the 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 are SUGRA fields of m~1/L. Their dimensions are independent of 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..

• While the above arguments provide a While the above arguments provide a solid motivation for the AdS/CFT solid motivation for the AdS/CFT correspondence, its proof has not yet correspondence, its proof has not yet been found. been found.

• It has become a time-honored It has become a time-honored tradition to simply assume that the tradition to simply assume that the correspondence holds. Over and over, correspondence holds. Over and over, this was shown to be a good idea. this was shown to be a good idea.

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

Spinning Strings vs. Highly Charged Spinning Strings vs. Highly Charged OperatorsOperators

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

• Generally, semi-classical spinning Generally, semi-classical spinning strings are dual to highly charged strings are dual to highly charged operators, e.g. the dual of a high-operators, e.g. the dual of a high-spin operator spin operator

is 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 high-spin The structure of dimensions of high-spin operators isoperators is

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

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

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

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

• At weak coupling the expansion of the At weak coupling the expansion of the universal function f(g) up to 3 loops isuniversal function f(g) up to 3 loops is

Kotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, SmirnovKotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, Smirnov

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

are are mapped to integrable spin chains, suggesting exact mapped to integrable spin chains, suggesting exact integrability of the integrability of the NN=4 SYM theory. =4 SYM theory. Minahan, Zarembo; Minahan, Zarembo;

Beisert, Kristjansen, StaudacherBeisert, Kristjansen, Staudacher• 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 , where Z and W are two

complex scalars, the Heisenberg spin chain complex scalars, the Heisenberg spin chain emerges at 1 loop. Higher loops correct the emerges at 1 loop. Higher loops correct the Hamiltonian but seem to preserve its integrability.Hamiltonian but seem to preserve its integrability.

• This meshes nicely with earlier findings of This meshes nicely with earlier findings of integrability in certain subsectors of QCD. integrability in certain subsectors of QCD. Lipatov; Lipatov; Faddeev, Korchemsky; Braun, Derkachov, ManashovFaddeev, Korchemsky; Braun, Derkachov, Manashov

• The dual string theory approach indicates that in The dual string theory approach indicates that in the SYM theory the exact integrability is present at the SYM theory the exact integrability is present at very strong coupling very strong coupling (Bena, Polchinski, Roiban).(Bena, Polchinski, Roiban). Hence it is Hence it is likely to exist for all values of the coupling.likely to exist for all values of the coupling.

• The coefficients in f(g) appear to be The coefficients in f(g) appear to be related to the corresponding related to the corresponding coefficients in QCD through selecting coefficients in QCD through selecting at each order the term with the at each order the term with the highest transcendentality. highest transcendentality. Kotikov, Lipatov, Kotikov, Lipatov, Onishchenko, Velizhanin Onishchenko, Velizhanin

• Recently, great progress has been Recently, great progress has been achieved on finding f(g) at 4 loops achieved on finding f(g) at 4 loops and beyond. and beyond.

• Using the the spin chain symmetries, Using the the spin chain symmetries, the Bethe ansatz equations were the Bethe ansatz equations were restricted to the form restricted to the form Staudacher, BeisertStaudacher, Beisert

• 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. conjecture, but it does not seem sufficient by itself. The magnon S-matrix contains an undetermined The magnon S-matrix contains an undetermined phase factor which affects the observables. phase factor 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 physicists, is that the phase is trivial. The only problem is that this contradicts the AdS/CFT problem is that this contradicts the AdS/CFT correspondence which implies that it is non-trivial at correspondence which implies that it is non-trivial at strong coupling. strong coupling. Arutyunov, Frolov, StaudacherArutyunov, Frolov, Staudacher

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

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

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

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

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

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

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 the while the second one is due to the

dressing phase in the magnon S-dressing phase in the magnon S-matrixmatrix

• This equation yields analytic predictions for This equation yields analytic predictions for all planar perturbative coefficients all planar perturbative coefficients

• The gauge theory 4-loop answer is only The gauge theory 4-loop answer is only known numerically and agrees with this known numerically and agrees with this analytical prediction to around 0.001%.analytical 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 The alternation of the series and the geometric behavior of the coefficients remove geometric behavior of the coefficients remove all singularities from the real axis, allowing all singularities from the real axis, allowing smooth extrapolation to infinite coupling. smooth extrapolation to infinite coupling.

• The radius of convergence is ¼. The closest The radius of convergence is ¼. The closest singularities are square-root branch points at singularities are square-root branch points at

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

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

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

• 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, green line to the ES, green line to the equation line to the equation where the dressing where the dressing kernel is divided by 2. kernel is divided by 2.

• The 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 precise agreement with the AdS/CFT with the AdS/CFT spinning string spinning string predictions. The third predictions. The third is an approximate is an approximate prediction.prediction.

• 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 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 Korchemsky and Kotanski who found that the coefficient of 1/g is -K/(4 coefficient of 1/g is -K/(4 22), in agreement with the ), in agreement with the numerical result. numerical result.

• The expression containing the Catalan constant K The expression containing the Catalan constant K is in exact agreement with the string sigma model is in exact agreement with the string sigma model 2-loop correction to f(g). 2-loop correction to f(g). Roiban, Tirziu, TseytlinRoiban, Tirziu, Tseytlin

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

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 is the gauge theory: small z is the UV; large z is the IR. 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 conformal. theories that are conformal. The quark and anti-quark The quark and anti-quark are placed at the boundary are placed at the boundary of Anti-de Sitter space of Anti-de Sitter space (z=0), but the string (z=0), but the string connecting them bends into connecting them bends into the interior (z>0). Due to the interior (z>0). Due to the scaling symmetry of the scaling symmetry of the AdS space, this gives the AdS space, this gives Coulomb potential Coulomb potential (Maldacena; (Maldacena; Rey, Yee)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

Confinement in SYM Confinement in SYM theoriestheories• 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 of and wrapped D5-branes at the tip of a 6-d cone, e.g. the conifold.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 the gauge theory on these branes is the warped deformed conifold warped deformed conifold (IK, Strassler)(IK, 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:

• 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 conifold cases. The warped conifold solution has a naked singularity which is solution has a naked singularity which is resolved via deformation. This is how string resolved via deformation. This is how string theory tells us that the chiral symmetry theory tells us that the chiral symmetry breaking and dynamical scale generation breaking and dynamical scale generation must take place! The finiteness of the warp must take place! The finiteness of the warp factor at r=0 translates into confinement.factor at r=0 translates into confinement.

• 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.

• 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 testing This should be a good playground for testing various ideas about strongly coupled gauge various ideas about strongly coupled gauge theory.theory.

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

• High energy scattering of bound states in High energy scattering of bound states in confining gauge/gravity models has also confining gauge/gravity models has also been studied successfully, e.g. the recent been studied successfully, e.g. the recent work on BFKL pomeron. work on BFKL pomeron. Brower, Polchinski, Strassler, Tan Brower, Polchinski, Strassler, Tan

• Could there be applications of these models Could there be applications of these models to new physics?to new physics?

ConclusionsConclusions• The AdS/CFT correspondence makes a The AdS/CFT correspondence makes a

multitude of dynamical predictions about multitude of dynamical predictions about strongly coupled conformal gauge theories. strongly coupled conformal gauge theories. They always appear to make sense, but are They always appear to make sense, but are often difficult to check quantitatively (e.g., the often difficult to check quantitatively (e.g., the ¾ in the entropy).¾ in the entropy).

• For non-BPS quantities in For non-BPS quantities in NN=4 SYM, non-trivial =4 SYM, non-trivial interpolating functions appear. Recently, the interpolating functions appear. Recently, the conjectured integrability and other constraints conjectured integrability and other constraints led to determination of the cusp anomaly led to determination of the cusp anomaly function. This provides strong new evidence for function. This provides strong new evidence for the validity of the AdS/CFT duality. the validity of the AdS/CFT duality.

• Gauge/string duality gives a new geometrical Gauge/string duality gives a new geometrical view of such important phenomena as view of such important phenomena as confinement, dimensional transmutation and confinement, dimensional transmutation and chiral symmetry breaking. chiral symmetry breaking.

Breaking the IceBreaking the Ice• Dirichlet branesDirichlet branes (Polchinski) led (Polchinski) led

string theory back to gauge theory in string theory back to gauge theory in the mid-90’s the mid-90’s (artwork by E.Imeroni)(artwork by E.Imeroni)

• A stack of N Dirichlet 3-branes A stack of N Dirichlet 3-branes realizes realizes NN=4 supersymmetric SU(N) =4 supersymmetric SU(N) gauge theory in 4 dimensions. It also gauge theory in 4 dimensions. It also creates a curved background of 10-d creates a curved background of 10-d theory of closed superstrings theory of closed superstrings Horowitz, Horowitz, Strominger; Duff, LuStrominger; Duff, Lu

which for small r approaches which for small r approaches

• The magnon dispersion relation is The magnon dispersion relation is

• The complex x-variables encode the The complex x-variables encode the momentum p and energy C:momentum p and energy C:

• Of particular importance is the Of particular importance is the crossing symmetry (Janik)crossing symmetry (Janik)

• Perturbative order-by-order solution of the Perturbative order-by-order solution of the BES equation gives the 4-loop term in f(g)BES equation gives the 4-loop term in f(g)

(it differs by relative sign from the ES (it differs by relative sign from the ES

prediction which did not include the `dressing prediction which did not include the `dressing phase’)phase’)

• Remarkably, an independent 4-loop Remarkably, an independent 4-loop calculation by Bern, Dixon, Czakon, Kosower calculation by Bern, Dixon, Czakon, Kosower and Smirnov yielded a numerical value that and Smirnov yielded a numerical value that prompted them to conjecture exactly the prompted them to conjecture exactly the same analytical result. same analytical result.

• This has led the two groups to the same This has led the two groups to the same conjecture for the complete structure of the conjecture for the complete structure of the perturbative expansion of f(g): it is the one perturbative expansion of f(g): it is the one yielded by the BES integral equation.yielded by the BES integral equation.