From Weak to Strong Coupling in N =4 SYM Theory

1. From Weak to Strong Coupling in N=4 SYM Theory Igor Klebanov Department of Physics Talk at Rutgers University March 6, 2007

2. Large N Gauge Theories • Connection of gauge theory with string theory is most apparent in `t Hooft’s generalization from 3 colors (SU(3) gauge group) to N colors (SU(N) gauge group). • Make N large, while keeping the `t Hooft coupling fixed: • The probability of snapping a flux tube by quark-antiquark creation (meson decay) is 1/N. The string coupling is 1/N. • In the large N limit only planar diagrams contribute, but 4-d gauge theory is still very difficult.

3. Breaking the Ice • Dirichlet branes (Polchinski) led string theory back to gauge theory in the mid-90’s. • A stack of N Dirichlet 3-branes realizes N=4 supersymmetric SU(N) gauge theory in 4 dimensions. It also creates a curved background of 10-d theory of closed superstrings (artwork by E.Imeroni) which for small r approaches • Successful matching of graviton absorption by D3-branes, related to 2-point function of stress-energy tensor in the SYM theory, with a gravity calculation in the 3-brane metric (IK; Gubser, IK, Tseytlin) was a precursor of the AdS/CFT correspondence.

4. 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 extra fields; the beta function is exactly zero for any complex coupling. The theory is invariant under scale transformations xm -> a xm . It is also invariant under space-time inversions. The full super-conformal group is SU(2,2|4). • It is interesting to study observables of the planar theory as functions of the `t Hooft coupling l.

5. Entropy of thermal supersymmetric SU(N) theory • Thermal CFT is described by a near-extremal 3-brane background whose near-horizon form is a black hole in AdS5 • The CFT temperature is identified with the Hawking T of the horizon located at zh • Any event horizon contains Bekenstein-Hawking entropy • A brief calculation gives the entropy density Gubser, IK, Peet

6. This is interpreted as the strong coupling limit of • The weak `t Hooft coupling behavior of the interpolating function is determined 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

7. Corrections to the interpolating function at strong coupling come from the higher-derivative terms in the type IIB effective action: Gubser, IK, Tseytlin • The interpolating function is usually assumed to have a smooth monotonic form, but so far we do not know its form at the intermediate coupling.

8. A similar reduction of entropy by strong-coupling effects is observed in lattice non-supersymmetric gauge theories for N=3: the arrows show free field values. Karsch (hep-lat/0106019). • N-dependence in the pure glue theory enters largely through the overall normalization. Bringoltz and Teper (hep-lat/0506034)

9. The AdS/CFT dualityMaldacena; Gubser, IK, Polyakov; Witten • Relates conformal gauge theory in 4 dimensions to string theory on 5-d Anti-de Sitter space times a 5-d compact space. For the N=4 SYM theory this compact space is a 5-d sphere. • When a gauge theory is strongly coupled, the radius of curvature of the dual AdS5 and of the 5-d 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 l. At weak coupling the dual string theory becomes difficult.

10. Gauge invariant operators in the CFT4 are in one-to-one correspondence with 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. • The unprotected operators (Konishi operator is the simplest) are dual to massive string states. AdS/CFT predicts that at strong coupling their dimensions grow as l1/4.

11. Until recently, the only (probably) exact non-BPS interpolating function was the expectation value of a circular Wilson loop obtained from rainbow graph summation Erickson, Semenoff, Zarembo • Here the radius of convergence of ln W is related to the zero of the Bessel function J1. • There is an infinite number of branch points at negative l. • The strong coupling limit is in agreement with the AdS calculation.

12. Exact Integrability • Perturbative calculations of anomalous dimensions are mapped to integrable spin chains, suggesting exact integrability of the N=4 SYM theory. Minahan, Zarembo; Beisert, Staudacher • For example, for the `SU(2) sector’ operators 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, Derkachov, Manashov • The dual string theory approach indicates that in the SYM theory the exact integrability is present at very strong coupling (Bena, Polchinski, Roiban). Hence it is likely to exist for all values of the coupling.

13. Spinning Strings vs. Highly Charged Operators • Vibrating closed strings with large angular momentum on the 5-sphere are dual to SYM operators with large R-charge (the number of fields Z) Berenstein, Maldacena, Nastase • Generally, semi-classical spinning strings are dual to long operators, e.g. the dual of a high-spin operator is a folded string spinning around the center of AdS5. Gubser, IK, Polyakov

14. The structure of dimensions of high-spin operators is • The universal function f(g) is independent of the twist; it is universal in the planar limit. • It also enters the cusp anomaly of Wilson loops in Minkowski space. This can be calculated using AdS/CFT. Kruczenski

15. At strong coupling, the AdS/CFT corresponds predicts via the spinning string energy calculations Gubser, IK, Polyakov; Frolov, Tseytlin • At weak coupling the expansion of the universal function f(g) up to 3 loops is

16. The coefficients in f(g) are related to the corresponding coefficients in QCD through selecting at each order the term with the highest transcendentality. Kotikov, Lipatov, Onishchenko, Velizhanin • Recently, great progress has been achieved on finding f(g) at 4 loops and beyond! • Using the of the spin chain symmetries, the Bethe ansatz equations were restricted to the form Staudacher, Beisert

17. The magnon dispersion relation is • The complex x-variables encode the momentum p and energy C: • Of particular importance is the crossing symmetry (Janik)

18. The Dressing Phase • The `dressing phase’ in the magnon S-matrix appears first at 4-loop order in perturbation theory: where qr(u) are eigenvalues of the magnon charges. It was determined first in the large g expansion (Beisert, Hernandez, Lopez), and then via appropriate resummation in the small g expansion (Beisert, Eden, Staudacher). • f(g) is determined through solving an integral equation, which corrects an earlier similar equation derived by Eden and Staudacher (who assumed that the phase vanishes)

19. The BES kernel is • The first term is the ES kernel while the second one is due to the dressing phase in the magnon S-matrix

20. Perturbative order-by-order solution of the BES equation gives the 4-loop term in f(g) (it differs by relative sign from the ES prediction which did not include the `dressing phase’) • Remarkably, an independent 4-loop calculation by Bern, Dixon, Czakon, Kosower and Smirnov yielded a numerical value that prompted them to conjecture exactly the same analytical result. • This has led the two groups to the same conjecture for the complete structure of the perturbative expansion of f(g): it is the one yielded by the BES integral equation.

21. This approach yields analytic predictions for all planar perturbative coefficients • So far the 4-loop answer is only known numerically. Recently, a new method yielded improved numerical precision and agrees with the analytical prediction to around 0.001%. Cachazo, Spradlin, Volovich

22. The alternation of the series and the geometric behavior of the coefficients remove all singularities from the real axis, allowing smooth extrapolation to infinite coupling. • The radius of convergence is ¼. The closest singularities are square-root branch points at • The analytic structure of f(g) is not completely clear, but there appear to be an infinite number of branch cuts along the imaginary axis. There is an essential singularity at infinity. These qualitative features also appear for the circular Wilson loop.

23. 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

24. 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 BES large g asymptotics are in very precise agreement with the AdS/CFT spinning string predictions.

25. Recently, the exact analytic solution was obtained for the leading order BES equation. Alday, Arutyunov, Benna, Eden, IK; Kostov, Serban, Volin • Expanding at strong coupling, The solution is • Since s1=1/2, this proves that f(g)= 4 g + O(1)

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

27. Excitations of the Gluonic StringCallan, Guijosa; Brower, Tan, Thorn; IK, Maldacena, Thorn • The spectrum of small oscillations of the string with Dirichlet boundary conditions at z=0 is known: • By conformal invariance, all energies scale inversely with quark anti-quark distance L. Their spectrum is known, and they correspond to gluonic excitations at strong `t Hooft coupling. • But at weak coupling there are no such excitations: only the ground state of energy where

28. A simplified model where only ladder graphs are summed indicates that the coupling where an infinite number of excitations appear is • Their appearance is related to the fall-to-the center instability in the bound state equation • The near-threshold bound states are in exact agreement with the spectrum of a highly excited string containing a single very long fold:

29. Is such a `fall to the center’ transition possible in QCD? • Apparently not, since the asymptotic freedom makes the coupling weak when the ends of the string approach each other. • A recent lattice calculation of the string excitation spectrum shows this explicitly. Only one level exists with energy much lower than others at short separation. Juge, Kuti, Morningstar

30. Why are the flux tube excitations stable in a CFT ? • For a combination of energetic and large N reasons! • After a gluon is emitted the resulting state would be in a color adjoint, and would have positive energy. Hence, energy conservation forbids one-gluon emission from an excited bound state. • The energy conservation does not forbid glueball (closed string) emission. But it is suppressed at large N.

31. 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 difficult to check quantitatively (e.g., the ¾ in the entropy). • For non-BPS quantities in N=4 SYM, non-trivial interpolating functions appear. Recently, the conjectured integrability has led to determination of the cusp anomaly function. This provides strong new evidence for the validity of the AdS/CFT duality. • The recent developments raise hope of further exact results, and perhaps an eventual proof of AdS/CFT duality.

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

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

34. String Theoretic Approaches to Confinement • It is possible to generalize the AdS/CFT correspondence in such a way that the quark-antiquark potential is linear at large distance. • The 5-d metric should have a warped form (Polyakov): • The space ends at a maximum value of z where the warp factor is finite. Then the confining string tension is

35. Semi-classical Nambu String • Semi-classical quantization around long straight Nambu string predicts the quark-antiquark potential • The coefficient of the universal Luescher term depends only on the space-time dimension d and is proportional to the Regge intercept:

36. Comparison with the Lattice DataLuescher, Weisz (2002) • Lattice calculations of the force vs. distance produce good agreement with the semi-classical Nambu string for r>0.7 fm:

37. Embedding in Flux Compactifications • A long warped throat embedded into a compactification with NS-NS and R-R fluxes leads to a small ratio between the IR scale at the bottom of the throat and the string scale. Randall, Sundrum; Verlinde; IK, Strassler; Giddings, Kachru, Polchinski; KKLT; KKLMMT • In the dual cascading gauge theory the IR scale is the confinement scale: confinement stabilizes the hierarchy between the Planck scale and the SM or the inflationary scale. • Cascading gauge theories dual to “standard model throats” may offer interesting possibilities for new physics beyond the standard model. Cascales, Franco, Hanany, Saad, Uranga, …

38. Exact Integrability • Perturbative calculations of anomalous dimensions are mapped to integrable spin chains, suggesting exact integrability of the N=4 SYM theory. Minahan, Zarembo; Beisert, Staudacher • For example, for the SU(2) sector operators 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, Derkachov, Manashov • The dual string theory approach indicates that in the SYM theory the exact integrability is present at very strong coupling (Bena, Polchinski, Roiban). Hence it is likely to exist for all values of the coupling.

39. Connection with Cosmic stringsCopeland, Myers, Polchinski • A fundamental string at the bottom of the warped deformed conifold is dual to a confining string. A D-string is dual to a solitonic string which couples to the Goldstone mode. • Upon embedding of the warped throat into a flux compactification, these objects can be used to model cosmic strings. • This throat is not the “standard model throat’’ but another throat, the “inflationary throat,” dual to some hidden sector cascading gauge theory.

40. Speed of Sound • In a CFT, the pressure is related to the energy density, p=3e. Hence, Cs2 = dp/de = 1/3 . • The plot of Cs2 in pure glue SU(3) lattice gauge theory Gavai, Gupta, Mukherjee • For T/Tc = 3, the coupling is still quite strong, l ~ 7. This suggests that AdS/CFT methods may be useful.

41. Folded Strings on S5 and Magnons • Folded string spinning around the north pole has, for large R-charge J: Gubser, IK, Polyakov • Recently this string was identified by Hofman and Maldacena with a 2-magnon operator where each magnon carries p=p . The exact energy of a free magnon is Santambrogio, Zanon Beisert • Its dual is an open string spinning on S5 : • Hence the 2-magnon operator dimension is in exact agreement with the folded string result for large l.

42. The gauge theory on D7-branes wrapping a 4-cycle S4 has coupling • The non-perturbative superpotential depends on the D3-brane location through the warped volume • In the throat approximation, the warp factor can be calculated and integrated over a 4-cycle explicitly. Baumann, Dymarsky, Klebanov, Maldacena, McAllister, Murugan. • For a class of conifold embeddings Arean, Crooks, Ramallo(w1=z1+iz2 , etc.) the result is

43. This formula applies both to n wrapped D7-branes, and to a wrapped Euclidean D3 (n=1). • For the latter case, Ganor showed that A has a simple zero when the D3-brane approaches the 4-cycle. Our result agrees with this. • We have also carried out such calculations for 4-cycles within the Calabi-Yau cones over Yp,q with analogous results: A(X) is proportional to the embedding equation raised to the power 1/n. This appears to be a general rule for 4-cycles in the throat.

44. For operators with large R-charge J, the new results imply that the so-called `perturbative BMN scaling’ (the assumption that all quantitites have expansion in powers of l/J2) is violated. This resolved certain discrepancies that followed from this unjustified assumption. • In fact, in our 2002 paper, Spradlin, Volovich and I showed that this scaling must be violated for related observables. Now this violation is believed to be a general phenomenon in the AdS/CFT duality, which allows for smooth extrapolation from weak to strong coupling.

45. The anomalous dimension of such a high spin twist-2 operator is • AdS/CFT predicts that at strong coupling • A 3-loop perturbative N=4 SYM calculation gives Kotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, Smirnov • An approximate extrapolation formula works with about 10% accuracy: • Constrains on f(l) imposed by the integrability were studied by Eden, Staudacher; Belitsky et al

46. News Flash • Remarkably, Bern, Czakon, Dixon, Kosower and Smirnov have calculated the four-loop correction to the universal function f(l): -16-3p-8l4 (73 p6/630 + 4 z(3)2) • This allows to determine the scattering phase of magnon S-matrix in the integrability approach, and appears to completely determine the expansion of f(l) ! Beisert, Eden, Staudacher; Beisert, Hernandez, Lopez • Extrapolation to large l gives good numerical agreement with the string theory prediction. There are indications the agreement is exact. • I expect many more results following this breakthrough in the near future.

48. Correlation functions are calculated from the dependence of string theory path integral on boundary conditions f0 in AdS5, imposed near z=0: Gubser, IK, Polyakov; Witten • In the large N limit the path integral is found from the classical string action: