Gauge Theories DBranes and Strings Igor Klebanov Department

Gauge Theories, D-Branes and Strings Igor Klebanov Department of Physics Talk at Jefferson Lab November 16, 2007

QCD and String Theory • At short distances, • must smaller than 1 fermi, the quarkantiquark potential is approximately Coulombic, due to the Asymptotic Freedom. At large distances the potential should be linear (Wilson) due to formation of confining flux tubes.

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

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.

Stacking D-Branes • 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 D 3 - 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 Ad. S/CFT correspondence.

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).

Entropy of thermal N=4 SUSY SU(N) theory • Thermal CFT is described by a near- extremal 3 -brane background whose nearhorizon form is a black hole in Ad. S 5 • 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

• 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 Ad. S/CFT duality that • The entropy density is multiplied only by ¾ as the coupling changes from zero to infinity. Gubser, IK, Tseytlin

• Corrections to the interpolating function at strong coupling come from the higherderivative 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.

• 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)

The Ad. S/CFT duality Maldacena; 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 Ad. S 5 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.

• Gauge invariant operators in the CFT 4 are in one • -to-one correspondence with fields (or extended objects) in Ad. S 5 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. Ad. S/CFT predicts that at strong coupling their dimensions grow as l 1/4.

• While the above arguments provide a solid motivation for the Ad. S/CFT correspondence, its proof has not yet been found. • It has become a time-honored tradition to simply assume that the correspondence holds. Over and over, this was shown to be a good idea. • To illustrate this, let me entertain you with ``the legend of the cusp anomaly in N=4 SYM theory. ''

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 highly charged operators, e. g. the dual of a high-spin operator is a folded string spinning around the center of Ad. S 5. Gubser, IK, Polyakov

• The structure of dimensions of high-spin operators is • The 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. Polyakov; Korchemsky, Radyushkin, … This can be calculated using Ad. S/CFT. Kruczenski

• At strong coupling, the Ad. S/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 Kotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, Smirnov

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, Kristjansen, 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.

• The coefficients in f(g) appear to be 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 spin chain symmetries, the 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 S-matrix 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 that this contradicts the Ad. S/CFT correspondence which implies that it is non-trivial at strong coupling. Arutyunov, Frolov, Staudacher • 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 a 4 -loop calculation to check whether agreement with the ES equation continues to hold. The fate of the Ad. S/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 Ad. S/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 4 -loop 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. Independently, Beisert, Eden and Staudacher guessed the small g expansion of the phase factor consistent with the strong coupling expansion found by BHL. They derived the corrected form of the equation that determines the cusp anomaly and found the same series as the one conjectured by BCDKS.

The BES Equation • f(g) is determined through 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 S-matrix

• This equation yields analytic predictions for all planar perturbative coefficients • The gauge theory 4 -loop 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 • 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 • To compare the large g behavior of f(g) directly with the Ad. S/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 very 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 Ad. S/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 of 1/g is -K/(4 p 2), in agreement with the numerical result. The expression containing the Catalan constant K is in exact agreement with the string sigma model 2 -loop 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.

The quark anti-quark potential • The z-direction of Ad. S is dual • to the energy scale of the gauge theory: small z is the UV; large z is the IR. Because of the 5 -th dimension z, the string picture applies even to theories that are conformal. The quark and antiquark 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 Ad. S space, this gives Coulomb potential (Maldacena; Rey, Yee)

String Theoretic Approaches to Confinement • It is possible to generalize • the Ad. S/CFT correspondence in such a way that the quarkantiquark potential is linear at large distance but nearly Coulombic at small 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

Confinement in SYM theories • Introduction of minimal • • • supersymmetry (N=1) facilitates construction of gauge/string dualities. A useful tool is to place D 3 -branes and wrapped D 5 -branes at the tip of a 6 -d cone, e. g. the conifold. The 10 -d geometry dual to the gauge theory on these branes is the warped deformed conifold (IK, Strassler) is the metric of the deformed conifold, a simple Calabi-Yau space defined by the following constraint on 4 complex variables:

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

• The graph of quark anti- quark potential is qualitatively similar to that found in numerical simulations of QCD. The upper graph, from the recent Senior Thesis of V. Cvicek shows the string theory result for the warped deformed conifold. • The lower graph shows lattice QCD results by G. Bali et al with r 0 ~ 0. 5 fm.

• All of this provides us with an exact solution of a • • class of 4 -d large N confining supersymmetric gauge theories. This should be a good playground for testing various ideas about strongly coupled gauge theory. Some results on glueball spectra are already available, and further calculations are ongoing. Krasnitz; Caceres, Hernandez; Dymarsky, Melnikov; Berg, Haack, Muck • High energy scattering of bound states in • confining gauge/gravity models has also been studied successfully, e. g. the recent work on BFKL pomeron. Brower, Polchinski, Strassler, Tan Could there be applications of these models to new physics?

Conclusions • The Ad. S/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 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 and other constraints led to determination of the cusp anomaly function. This provides strong new evidence for the validity of the Ad. S/CFT duality. Gauge/string duality gives a new geometrical view of such important phenomena as confinement, dimensional transmutation and chiral symmetry breaking.

Breaking the Ice • Dirichlet branes (Polchinski) led string theory back to gauge theory in the mid 90’s (artwork by E. Imeroni) • 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 Horowitz, Strominger; Duff, Lu which for small r approaches

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

• 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.