From Weak to Strong From Weak to Strong

Coupling in Coupling in NN=4 SYM Theory =4 SYM Theory

Igor KlebanovIgor Klebanov

Department of PhysicsDepartment of Physics

Talk at Rutgers University Talk at Rutgers University March 6, 2007March 6, 2007

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.

Breaking the IceBreaking the Ice• 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 dimensions. It also creates a curved background of 10-d theory of closed superstringsof 10-d theory of closed superstrings (artwork by (artwork by E.Imeroni)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-branes, related to 2-point function of stress-energy tensor in the SYM theory, with a gravity energy tensor in the SYM theory, with a gravity calculation in the 3-brane metric (IK; Gubser, IK, calculation in the 3-brane metric (IK; Gubser, IK, Tseytlin) was a precursor of the AdS/CFT Tseytlin) was a precursor of the AdS/CFT correspondence.correspondence.

Super-Conformal InvarianceSuper-Conformal Invariance

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

• It is interesting to study observables of the It is interesting to study observables of the planar theory as functions of the `t Hooft planar theory as functions of the `t Hooft coupling coupling ..

Entropy of thermal Entropy of thermal supersymmetric SU(N) theorysupersymmetric SU(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 The CFT temperature is identified with the Hawking T of the horizon located at zHawking 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 This is interpreted as the strong coupling limit of of

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

• Until recently, the only (probably) exact Until recently, the only (probably) exact non-BPS interpolating function was the non-BPS interpolating function was the expectation value of a circular Wilson loop expectation value of a circular Wilson loop obtained from rainbow graph summation obtained from rainbow graph summation Erickson, Semenoff, ZaremboErickson, Semenoff, Zarembo

• Here the radius of convergence of ln W is Here the radius of convergence of ln W is related to the zero of the Bessel function Jrelated to the zero of the Bessel function J11..

• There is an infinite number of branch points There is an infinite number of branch points at negative at negative ..

• The strong coupling limitThe strong coupling limit

is in agreement with the AdS calculation. is in agreement with the AdS calculation.

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

are are mapped to integrable spin chains, suggesting mapped to integrable spin chains, suggesting

exact exact integrability of the integrability of the NN=4 SYM theory. =4 SYM theory. Minahan, Zarembo; Minahan, Zarembo;

Beisert, StaudacherBeisert, 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 Hence it is likely to exist for all values of the coupling.is likely to exist for all values of the coupling.

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

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

is a folded string spinning around is a folded string spinning around the center of AdSthe center 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 universal function f(g) is The universal function f(g) is independent of the twist; it is universal independent of the twist; it is universal in the planar limit.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.

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

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

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

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

• 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)

The Dressing PhaseThe Dressing Phase• The `dressing phase’ in the magnon S-matrix appears The `dressing phase’ in the magnon S-matrix appears

first at 4-loop order in perturbation theory: first at 4-loop order in perturbation theory:

where qwhere qrr(u) are eigenvalues of the magnon charges. (u) are eigenvalues of the magnon charges. It was determined first in the large g expansion (It was determined first in the large g expansion (Beisert, Beisert,

Hernandez, LopezHernandez, Lopez), and then via appropriate resummation ), and then via appropriate resummation in the small g expansion (in the small g expansion (Beisert, Eden, StaudacherBeisert, Eden, Staudacher). ).

• f(g) is determined through solving an integral f(g) is determined through solving an integral equation, which corrects an earlier similar equation equation, which corrects an earlier similar equation derived by Eden and Staudacher (who assumed that derived by Eden and Staudacher (who assumed that the phase vanishes)the phase vanishes)

• 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

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

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

• So far the 4-loop answer is only known So far the 4-loop answer is only known numerically. Recently, a new method yielded numerically. Recently, a new method yielded improved numerical precision and agrees with improved numerical precision and agrees with the analytical prediction to around 0.001%.the analytical prediction to around 0.001%.

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

• The analytic structure of f(g) is not completely The analytic structure of f(g) is not completely clear, but there appear to be an infinite clear, but there appear to be an infinite number of branch cuts along the imaginary number of branch cuts along the imaginary axis. There is an essential singularity at axis. There is an essential singularity at infinity. These qualitative features also appear infinity. These qualitative features also appear for the circular Wilson loop.for the circular Wilson loop.

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

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

• The first two terms The first two terms of the BES large g of the BES large g asymptotics are in asymptotics are in very precise very precise agreement with the agreement with the AdS/CFT spinning AdS/CFT spinning string predictions. string predictions.

• Recently, the exact analytic solution Recently, the exact analytic solution was obtained for the leading order was obtained for the leading order BES equation. BES equation. Alday, Arutyunov, Benna, Eden, IK; Kostov, Alday, Arutyunov, Benna, Eden, IK; Kostov, Serban, VolinSerban, Volin

• Expanding at strong coupling,Expanding at strong coupling,

The solution isThe solution is

• Since sSince s11=1/2, this proves that =1/2, this proves that

f(g)= 4 g + O(1)f(g)= 4 g + O(1)

Quark Anti-Quark PotentialQuark Anti-Quark Potential

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

• In a pleasant surprise, because In a pleasant surprise, because of the 5-th dimension z, the of the 5-th dimension z, the string picture applies even to string picture applies even to theories that are conformal theories that are conformal (not confining!). The quark and (not confining!). The quark and anti-quark are placed at the anti-quark are placed at the boundary of Anti-de Sitter boundary of Anti-de Sitter space (z=0), but the string space (z=0), but the string connecting them bends into connecting them bends into the interior (z>0). Due to the the interior (z>0). Due to the scaling symmetry of the AdS scaling symmetry of the AdS space, this gives Coulomb space, this gives Coulomb potential potential (Maldacena; Rey, Yee)(Maldacena; Rey, Yee)

Excitations of the Gluonic Excitations of the Gluonic StringString Callan, Guijosa; Brower, Tan, Thorn; IK, Maldacena, ThornCallan, Guijosa; Brower, Tan, Thorn; IK, Maldacena, Thorn• The spectrum of small oscillations of The spectrum of small oscillations of

the string with Dirichlet boundary the string with Dirichlet boundary conditions at z=0 is known:conditions at z=0 is known:

• By conformal invariance, all energies By conformal invariance, all energies scale inversely with quark anti-quark scale inversely with quark anti-quark distance L. Their spectrum is known, distance L. Their spectrum is known, and they correspond to gluonic and they correspond to gluonic excitations at strong `t Hooft coupling.excitations at strong `t Hooft coupling.

• But at weak coupling there are no But at weak coupling there are no such excitations: only the ground state such excitations: only the ground state of energy where of energy where

• A simplified model where only A simplified model where only ladder graphs are summed ladder graphs are summed indicates that the coupling indicates that the coupling where an infinite number of where an infinite number of excitations appear is excitations appear is

• Their appearance is related to Their appearance is related to the fall-to-the center instability the fall-to-the center instability in the bound state equationin the bound state equation

• The near-threshold bound The near-threshold bound states are in exact agreement states are in exact agreement with the spectrum of a highly with the spectrum of a highly excited string containing a excited string containing a single very long fold:single very long fold:

Is such a `fall to the center’ Is such a `fall to the center’ transition possible in QCD? transition possible in QCD?

• Apparently not, since the Apparently not, since the asymptotic freedom makes asymptotic freedom makes the coupling weak when the coupling weak when the ends of the string the ends of the string approach each other. approach each other.

• A recent lattice calculation A recent lattice calculation of the string excitation of the string excitation spectrum shows this spectrum shows this explicitly. Only one level explicitly. Only one level exists with energy much exists with energy much lower than others at short lower than others at short separation. separation. Juge, Kuti, MorningstarJuge, Kuti, Morningstar

Why are the flux tube Why are the flux tube excitations stable in a CFT ?excitations stable in a CFT ?

• For a combination of energetic and For a combination of energetic and large N reasons!large N reasons!

• After a gluon is emitted the resulting After a gluon is emitted the resulting state would be in a color adjoint, and state would be in a color adjoint, and would have positive energy. Hence, would have positive energy. Hence, energy conservation forbids one-gluon energy conservation forbids one-gluon emission from an excited bound state. emission from an excited bound state.

• The energy conservation does not The energy conservation does not forbid glueball (closed string) emission. forbid glueball (closed string) emission. But it is suppressed at large N.But it is suppressed at large N.

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 difficult to check quantitatively (e.g., the ¾ in difficult to check quantitatively (e.g., the ¾ in the entropy).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 has led to conjectured integrability has led to determination of the cusp anomaly function. determination of the cusp anomaly function. This provides strong new evidence for the This provides strong new evidence for the validity of the AdS/CFT duality. validity of the AdS/CFT duality.

• The recent developments raise hope of further The recent developments raise hope of further exact results, and perhaps an eventual proof of exact results, and perhaps an eventual proof of AdS/CFT duality.AdS/CFT duality.

• The planar The planar NN=4 SYM is a rich and =4 SYM is a rich and complicated theory, and some complicated theory, and some properties of its spectrum can properties of its spectrum can depend dramatically on the value of depend dramatically on the value of the coupling.the coupling.

• The z-direction of AdS is dual The z-direction of AdS is dual to the energy scale of the to the energy scale of the gauge theory: small z is the gauge theory: small z is the UV; large z is the IR. UV; large z is the IR.

• In a pleasant surprise, In a pleasant surprise, 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 (not confining!). The quark (not confining!). The quark and anti-quark are placed at and 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 (z>0). bends into the interior (z>0). Due to the scaling symmetry Due to the scaling symmetry of the AdS space, this gives of the AdS space, this gives Coulomb potential Coulomb potential (Maldacena; (Maldacena; Rey, Yee)Rey, Yee)

String Theoretic Approaches to String Theoretic Approaches to ConfinementConfinement

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

Semi-classical Nambu String Semi-classical Nambu String • Semi-classical quantization around Semi-classical quantization around

long straight Nambu string predicts long straight Nambu string predicts the quark-antiquark potentialthe quark-antiquark potential

• The coefficient of the universal The coefficient of the universal Luescher term depends only on the Luescher term depends only on the space-time dimension d and is space-time dimension d and is proportional to the Regge intercept:proportional to the Regge intercept:

Comparison with the Lattice Comparison with the Lattice DataData

LLueuescherscher, Weisz (2002), Weisz (2002)• Lattice Lattice

calculations calculations of the force of the force vs. distance vs. distance produce good produce good agreement agreement with the with the semi-classical semi-classical Nambu string Nambu string for r>0.7 fm:for r>0.7 fm:

Embedding inEmbedding in Flux Flux CompactificationsCompactifications• A long warped throat embedded into A long warped throat embedded into

a compactification with NS-NS and a compactification with NS-NS and R-R fluxes leads to a small ratio R-R fluxes leads to a small ratio between the IR scale at the bottom between the IR scale at the bottom of the throat and the string scale.of the throat and the string scale.

Randall, Sundrum; Verlinde; IK, Strassler; Giddings, Randall, Sundrum; Verlinde; IK, Strassler; Giddings, Kachru, Polchinski; KKLT; KKLMMTKachru, Polchinski; KKLT; KKLMMT

• In the dual cascading gauge theory In the dual cascading gauge theory the IR scale is the confinement the IR scale is the confinement scale: confinement stabilizes the scale: confinement stabilizes the hierarchy between the Planck scale hierarchy between the Planck scale and the SM or the inflationary scale.and the SM or the inflationary scale.

• Cascading gauge theories dual to Cascading gauge theories dual to “standard model throats” may offer “standard model throats” may offer interesting possibilities for new interesting possibilities for new physics beyond the standard model. physics beyond the standard model. Cascales, Franco, Hanany, Saad, Uranga, …Cascales, Franco, Hanany, Saad, Uranga, …

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

are are mapped to integrable spin chains, suggesting mapped to integrable spin chains, suggesting

exact exact integrability of the integrability of the NN=4 SYM theory. =4 SYM theory. Minahan, Zarembo; Minahan, Zarembo;

Beisert, StaudacherBeisert, 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 Hence it is likely to exist for all values of the coupling.is likely to exist for all values of the coupling.

Connection with Cosmic stringsConnection with Cosmic strings Copeland, Myers, PolchinskiCopeland, Myers, Polchinski

• A fundamental string at the bottom of the A fundamental string at the bottom of the warped deformed conifold is dual to a warped deformed conifold is dual to a confining string. A D-string is dual to a confining string. A D-string is dual to a solitonic string which couples to the solitonic string which couples to the Goldstone mode.Goldstone mode.

• 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.

• This throat is not the “standard model This throat is not the “standard model throat’’ but another throat, the “inflationary throat’’ but another throat, the “inflationary throat,” dual to some hidden sector throat,” dual to some hidden sector cascading gauge theory.cascading gauge theory.

Speed of Sound Speed of Sound

• In a CFT, the pressure is In a CFT, the pressure is related to the energy related to the energy density, p=3density, p=3. Hence, . Hence,

CCss22 = dp/d = dp/d = 1/3 . = 1/3 .

• The plot of CThe plot of Css22 in pure glue in pure glue

SU(3) lattice gauge theorySU(3) lattice gauge theory Gavai, Gupta, MukherjeeGavai, Gupta, Mukherjee

• For T/TFor T/Tcc = 3, the coupling = 3, the coupling is still quite strong, is still quite strong, ~ 7. ~ 7. This suggests that AdS/CFT This suggests that AdS/CFT methods may be useful.methods may be useful.

Folded Strings on SFolded Strings on S55 and and MagnonsMagnons• Folded string spinning around the north Folded string spinning around the north

pole has, for large R-charge J: pole has, for large R-charge J: Gubser, IK, PolyakovGubser, IK, Polyakov

• Recently this string was identified by Recently this string was identified by Hofman and Maldacena with a 2-Hofman and Maldacena with a 2-magnon operator where each magnon operator where each magnon carries p=magnon carries p=The The exact energy of a free magnon is exact energy of a free magnon is Santambrogio, ZanonSantambrogio, Zanon

BeisertBeisert

• Its dual is an open string spinning on SIts dual is an open string spinning on S55 : :• Hence the 2-magnon operator Hence the 2-magnon operator

dimension isdimension is in in exactexact agreement with the folded agreement with the folded

string result for large string result for large

• The gauge theory on D7-branes The gauge theory on D7-branes wrapping a 4-cycle wrapping a 4-cycle 44 has coupling has coupling

• The non-perturbative superpotentialThe non-perturbative superpotential

depends on the D3-brane location depends on the D3-brane location through the warped volume through the warped volume

• In the throat approximation, the warp In the throat approximation, the warp factor can be calculated and integrated factor can be calculated and integrated over a 4-cycle explicitly. over a 4-cycle explicitly. Baumann, Dymarsky, Baumann, Dymarsky,

Klebanov, Maldacena, McAllister, Murugan.Klebanov, Maldacena, McAllister, Murugan.

• For a class of conifold embeddings For a class of conifold embeddings Arean, Crooks, RamalloArean, Crooks, Ramallo ((ww11=z=z11+iz+iz2 2 , etc.), etc.)

the result isthe result is

• This formula applies both to This formula applies both to nn wrapped wrapped D7-branes, and to a wrapped Euclidean D3 D7-branes, and to a wrapped Euclidean D3 ((n=1n=1).).

• For the latter case, Ganor showed that For the latter case, Ganor showed that AA has a simple zero when the D3-brane has a simple zero when the D3-brane approaches the 4-cycle. Our result agrees approaches the 4-cycle. Our result agrees with this.with this.

• We have also carried out such calculations We have also carried out such calculations for 4-cycles within the Calabi-Yau cones for 4-cycles within the Calabi-Yau cones over Yover Yp,qp,q with analogous results: with analogous results: A(X)A(X) is is proportional to the embedding equation proportional to the embedding equation raised to the power raised to the power 1/n1/n. This appears to be . This appears to be a general rule for 4-cycles in the throat.a general rule for 4-cycles in the throat.

• For operators with large R-charge J, the new For operators with large R-charge J, the new results imply that the so-called `perturbative results imply that the so-called `perturbative BMN scaling’ (the assumption that all BMN scaling’ (the assumption that all quantitites have expansion in powers of quantitites have expansion in powers of /J/J22) is ) is violated. This resolved certain discrepancies violated. This resolved certain discrepancies that followed from this unjustified that followed from this unjustified assumption.assumption.

• In fact, in our 2002 paper, Spradlin, Volovich In fact, in our 2002 paper, Spradlin, Volovich and I showed that this scaling must be and I showed that this scaling must be violated for related observables. Now this violated for related observables. Now this violation is believed to be a general violation is believed to be a general phenomenon in the AdS/CFT duality, which phenomenon in the AdS/CFT duality, which allows for smooth extrapolation from weak to allows for smooth extrapolation from weak to strong coupling. strong coupling.

• The anomalous dimension of such a high spin The anomalous dimension of such a high spin twist-2 operator istwist-2 operator is

• AdS/CFT predicts that at strong couplingAdS/CFT predicts that at strong coupling

• A 3-loop perturbative A 3-loop perturbative NN=4 SYM calculation =4 SYM calculation gives gives Kotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, SmirnovKotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, Smirnov

• An An approximateapproximate extrapolation formula works extrapolation formula works with about 10% accuracy:with about 10% accuracy:

• Constrains on f(Constrains on f() imposed by the ) imposed by the integrability were studied by Eden, integrability were studied by Eden, Staudacher; Belitsky et al Staudacher; Belitsky et al

News FlashNews Flash• Remarkably, Bern, Czakon, Dixon, Kosower and Remarkably, Bern, Czakon, Dixon, Kosower and

Smirnov have calculated the four-loop correction Smirnov have calculated the four-loop correction to the universal function f(to the universal function f():):

-16-16-3-3 44 (73 (73 66/630 + 4 /630 + 4 (3)(3)22))

• This allows to determine the scattering phase of This allows to determine the scattering phase of magnon S-matrix in the integrability approach, magnon S-matrix in the integrability approach, and appears to completely determine the and appears to completely determine the expansion of f(expansion of f() ! ) ! Beisert, Eden, Staudacher; Beisert, Hernandez, Beisert, Eden, Staudacher; Beisert, Hernandez, LopezLopez

• Extrapolation to large Extrapolation to large gives good numerical gives good numerical agreement with the string theory prediction. There agreement with the string theory prediction. There are indications the agreement is exact.are indications the agreement is exact.

• I expect many more results following this I expect many more results following this breakthrough in the near future. breakthrough in the near future.

• Correlation functions are calculated Correlation functions are calculated from the dependence of string theory from the dependence of string theory path integral on boundary conditions path integral on boundary conditions 00 in AdS in AdS55, imposed near z=0: , imposed near z=0: Gubser, IK, Gubser, IK,

Polyakov; WittenPolyakov; Witten

• In the large N limit the path integral In the large N limit the path integral is found from the classical string is found from the classical string action:action: