Gauge Theories, D-Branes and Strings 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 quark- antiquark potential is approximately Coulombic, due to the Asymptotic Freedom. At short distances, must smaller than 1 fermi, the quark- antiquark 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. 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) 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). 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: 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. 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. 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. 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) 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 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. 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.

Super-Conformal Invariance 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. 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 x  -> a x   It is also invariant under space-time inversions. The full super-conformal group is SU(2,2|4). 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 x  -> a x   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 near- horizon form is a black hole in AdS 5 Thermal CFT is described by a near- extremal 3-brane background whose near- horizon form is a black hole in AdS 5 The CFT temperature is identified with the Hawking T of the horizon located at z h The CFT temperature is identified with the Hawking T of the horizon located at z h Any event horizon contains Bekenstein- Hawking entropy Any event horizon contains Bekenstein- Hawking entropy A brief calculation gives the entropy density Gubser, IK, Peet A brief calculation gives the entropy density Gubser, 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 interpolating function is determined by Feynman graph calculations in the N =4 SYM theory 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 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 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 higher- derivative terms in the type IIB effective action: Corrections to the interpolating function at strong coupling come from the higher- derivative terms in the type IIB effective action: Gubser, IK, Tseytlin 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. 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. 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/ ). Karsch (hep-lat/ ). N-dependence in the pure glue theory enters largely through the overall normalization. N-dependence in the pure glue theory enters largely through the overall normalization. Bringoltz and Teper (hep-lat/ ) Bringoltz and Teper (hep-lat/ )

The AdS/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. 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 AdS 5 and of the 5-d compact space becomes large: When a gauge theory is strongly coupled, the radius of curvature of the dual AdS 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 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  At weak coupling the dual string theory becomes difficult. Feynman graphs instead develop a weak coupling expansion in powers of  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 AdS 5 Gauge invariant operators in the CFT 4 are in one-to-one correspondence with fields (or extended objects) in AdS 5 Operator dimension is determined by the mass of the dual field; e.g. for scalar operators GKPW 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 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 l 1/4. 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 l 1/4.

While the above arguments provide a solid motivation for the AdS/CFT correspondence, its proof has not yet been found. While the above arguments provide a solid motivation for the AdS/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. 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.'' 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 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 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 AdS 5. Gubser, IK, Polyakov is a folded string spinning around the center of AdS 5. Gubser, IK, Polyakov

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

At strong coupling, the AdS/CFT corresponds predicts via the spinning string energy calculations Gubser, IK, Polyakov; Frolov, Tseytlin 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 At weak coupling the expansion of the universal function f(g) up to 3 loops is Kotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, Smirnov Kotikov, Lipatov, Onishchenko, Velizhanin; Bern, Dixon, Smirnov

Exact Integrability Perturbative calculations of anomalous dimensions are Perturbative calculations of anomalous dimensions are mapped to integrable spin chains, suggesting exact mapped to integrable spin chains, suggesting exact integrability of the N =4 SYM theory. Minahan, Zarembo; Beisert, Kristjansen, Staudacher integrability of the N =4 SYM theory. Minahan, Zarembo; Beisert, Kristjansen, Staudacher For example, for the `SU(2) sector’ operators 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. 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 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 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 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. Recently, great progress has been achieved on finding f(g) at 4 loops and beyond. Using the the spin chain symmetries, the Bethe ansatz equations were restricted to the form Staudacher, Beisert Using the 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. 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 AdS/CFT correspondence which implies that it is non-trivial at strong coupling. Arutyunov, Frolov, Staudacher A simple assumption, initially advocated by some physicists, is that the phase is trivial. The only problem is that this contradicts the AdS/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. 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 AdS/CFT correspondence seemed to be hanging in the balance! 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 AdS/CFT correspondence seemed to be hanging in the balance!

The monumental BCDKS 4-loop calculation took many months to complete. In the meantime, Beisert, Hernandez and Lopez decided to assume the strong coupling behavior of the phase factor predicted by AdS/CFT and to use Janik's crossing symmetry assumption for developing the strong coupling expansion of the phase factor. The monumental BCDKS 4-loop calculation took many months to complete. In the meantime, Beisert, Hernandez and Lopez decided to assume the strong coupling behavior of the phase factor predicted by AdS/CFT and to use Janik's crossing symmetry assumption for developing the strong coupling expansion of the phase factor. Finally, the different approaches converged late in 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. Finally, the different approaches converged late in 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. 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 f(g) is determined through solving an integral equation The BES kernel is The BES kernel is The first term is the ES kernel The first term is the ES kernel while the second one is due to the dressing phase in the magnon S-matrix 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 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%. 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 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 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 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 AdS/CFT predictions, one needs to resum the perturbative expansion. One approach to this is to look for the solution of the BES equation for all g. This is hard, but a simple and very accurate numerical approach was found. Benna, Benvenuti, Klebanov, Scardicchio To compare the large g behavior of f(g) directly with the AdS/CFT predictions, one needs to resum the perturbative expansion. One approach to this is to look for the solution of the BES equation for all g. This is hard, but a simple and 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 To solve the equation at finite coupling, we use a basis of linearly independent functions Determination of is tantamount to inverting an infinite matrix. Determination of is tantamount to inverting an infinite matrix. Truncation to finite matrices converges very rapidly. Benna, Benvenuti, IK, Scardicchio 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 blue line refers to the BES equation, red line to the ES, green line to the equation where the dressing kernel is divided by 2. The first two terms of the numerical large g asymptotics are in very precise agreement with the AdS/CFT spinning string predictions. The third is an approximate prediction. The first two terms of the numerical large g asymptotics are in very precise agreement with the AdS/CFT spinning string predictions. The third is an approximate prediction.

Expanding at strong coupling, Expanding at strong coupling, The leading solution is The leading solution is Alday, Arutyunov, Benna, IK 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 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 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. 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 AdS is dual to the energy scale of the gauge theory: small z is the UV; large z is the IR. 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. Because of the 5-th dimension z, the string picture applies even to theories that are conformal. 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) Because of the 5-th dimension z, the string picture applies even to theories that are conformal. 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)

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 but nearly Coulombic at small distance. It is possible to generalize the AdS/CFT correspondence in such a way that the quark- antiquark potential is linear at large distance but nearly Coulombic at small distance. The 5-d metric should have a warped form (Polyakov): 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 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. Introduction of minimal supersymmetry ( N =1) facilitates construction of gauge/string dualities. A useful tool is to place D3-branes and wrapped D5-branes at the tip of a 6-d cone, e.g. the conifold. A useful tool is to place D3-branes and wrapped D5-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) 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 is the metric of the deformed conifold, a simple Calabi-Yau space defined by the following constraint on 4 complex variables: defined by the following constraint on 4 complex variables:

Comparison of warp factors in the AdS, 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. Comparison of warp factors in the AdS, 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 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. 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. 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. 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 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 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? Could there be applications of these models to new physics?

Conclusions The AdS/CFT correspondence makes a multitude of dynamical predictions about strongly coupled conformal gauge theories. They always appear to make sense, but are often difficult to check quantitatively (e.g., the ¾ in the entropy). The AdS/CFT correspondence makes a multitude of dynamical predictions about strongly coupled conformal gauge theories. They always appear to make sense, but are often difficult to check 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 AdS/CFT duality. 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 AdS/CFT duality. Gauge/string duality gives a new geometrical view of such important phenomena as confinement, dimensional transmutation and chiral symmetry breaking. 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) 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 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 which for small r approaches

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