Final Report Summary - SPECTRUM OF ADS/CFT (Spectrum of the planar AdS/CFT correspondence)
The AdS/CFT correspondence, proposed by Juan Maldacena in 1997, stating that the IIB superstring theory on the AdS5×S5 target space is dual to the N = 4 supersymmetric Yang-Mills (SYM) theory in four dimensions has become one of the prime subjects of interest in gauge and string theory. Since it is a strong/weak coupling duality, it offers the unique possibility to investigate a four-dimensional interacting gauge theory non- perturbatively. Moreover, the N = 4 gauge theory shares many common features with QCD and one may hope to understand, at least qualitatively, the high-energy physics of the latter. Currently, several examples of such dualities have been successfully identified and the strong/weak coupling correspondences have become an essential tool while studying the non-perturbative aspects of gauge theories.
Although Maldacena’s duality has become a vital ingredient of modern theoretical physics and it was successfully tested in some cases, only recently progress has been made towards proving it. The main obstacles are the complications arising while quantizing both constituent theories. No non-perturbative quantization of the N = 4 SYM theory is known and the full quantization of string theory on curved backgrounds is currently also beyond reach. In the planar limit, i.e. when the rank of the symmetry group goes to infinity, these constituent theories become much simpler, as suggested by ‘t Hooft. This is due to the fact that only planar Feynman diagrams need to be taken into account on the gauge theory side, while the string theory reduces to a supersymmetric non-linear sigma model.
Remarkably, the simplification of the dynamics is much more far-reaching than expected. Rich structures emerge for different “observables” of the planar string/gauge correspondence: spectral problem, scattering amplitudes, correlation functions, etc.
Arguably, the best studied aspect is the spectrum of the duality. The dilatation operator of the planar N = 4 gauge theory, which “measures” the scaling dimensions of the operators, has been found to be asymptotically integrable. This implies that the computation of scaling dimensions of long operators is reduced to computing the energies of states of the corresponding integrable spin chain. On the other hand, the scattering processes in integrable spin chains are known to factorize into sequences of two-body scattering and the solution to the spectral problem is provided by a closed set of algebraic equations, the Bethe equations. Consequently, the computation of the scaling dimensions of long operators boils down to solving a set of coupled algebraic equations, termed asymptotic Bethe equations (ABA). A significant simplification occurs also on the string theory side. The classical supersymmetric sigma-model on the AdS5×S5 product space is integrable and there are indications that this remains true also at the quantum level.
Short strings/operators are unfortunately not accessible by these techniques due to the so- called wrapping diagrams, which typically have to be included at the order O(g^2L). These extra diagrams correspond to highly non-local interactions from the spin chain point of view. Recently, there has been a breakthrough in computing anomalous dimensions of composite operator of arbitrary length. Different groups have conjectured the Thermodynamic Bethe Ansatz (TBA) for the ground state (BPS state) and the sl(2) subsector of the theory. If supplemented with additional analytic information, these equations might be used to compute non-perturbatively the energy/dimension of any string state/composite operator of the correspondence thus accounting for the neglected wrapping diagrams. Very recently this infinite set of coupled integral equations has been written in a very compact form, raising hopes that they may be derived from the first principles on both sides of the correspondence.

The objectives of this project revolve around the following aspects:
1) Testing the veracity of the proposed TBA equations
2) Extracting from them insights into non-perturbative dynamics of AdS/CFT correspondence.
3) The leading order (LO) and next-to-leading (NLO) BFKL equations and their origins in AdS/CFT correspondence
4) Application of the integrability methods to cousin theories : AdS4/CFT3 and AdS3/CFT2.
The objectives 1),2) and 4) have been fully met and encouraging partial results towards fulfillment of 2) have been obtained. The experienced researcher has published two long (50+ pages), comprehensive and already well-cited papers in Nuclear Physics B. Both articles are interdisciplinary with cross-connections between the aforementioned objectives. With these two articles a new conceptual framework has been created that allows for a non-perturbative test of spectral equations for strong/weak coupling dualities. The underlying idea is to look at the dynamics of excitations on top of a dynamic background. Such a background for AdS/CFT is furnished by the GKP solution (this includes all currently known cases: AdS5/CFT4, AdS4/CFT3 and AdS3/CFT2!). It corresponds to long twist-two operators on the gauge theory side. It turns out that the physics of the low-energy excitations on this “vacuum state” is described by a sigma model, which is a consistent truncation of the string theory sigma model. This approach allowed to conduct non-perturbative tests of spectral equations for AdS5/CFT4 and AdS4/CFT3. This included a non-perturbative test of the dressing factor for AdS4/CFT3, which was not extensively tested so far. Furthermore, the framework developed allows to analyze how the interpolation between weak- and strong-coupling limit works in practice! Finally, the S-matrix constructed in one of these papers has found an important application in the non-perturbative computation of scattering amplitudes for AdS5/CFT4 (see arXiv:1303.1396 and follow-up articles). Thus, the results obtained during this fellowship have significantly contributed to the advancement of AdS/CFT correspondence.
Although Maldacena’s duality has become a vital ingredient of modern theoretical physics and it was successfully tested in some cases, only recently progress has been made towards proving it. The main obstacles are the complications arising while quantizing both constituent theories. No non-perturbative quantization of the N = 4 SYM theory is known and the full quantization of string theory on curved backgrounds is currently also beyond reach. In the planar limit, i.e. when the rank of the symmetry group goes to infinity, these constituent theories become much simpler, as suggested by ‘t Hooft. This is due to the fact that only planar Feynman diagrams need to be taken into account on the gauge theory side, while the string theory reduces to a supersymmetric non-linear sigma model.
Remarkably, the simplification of the dynamics is much more far-reaching than expected. Rich structures emerge for different “observables” of the planar string/gauge correspondence: spectral problem, scattering amplitudes, correlation functions, etc.
Arguably, the best studied aspect is the spectrum of the duality. The dilatation operator of the planar N = 4 gauge theory, which “measures” the scaling dimensions of the operators, has been found to be asymptotically integrable. This implies that the computation of scaling dimensions of long operators is reduced to computing the energies of states of the corresponding integrable spin chain. On the other hand, the scattering processes in integrable spin chains are known to factorize into sequences of two-body scattering and the solution to the spectral problem is provided by a closed set of algebraic equations, the Bethe equations. Consequently, the computation of the scaling dimensions of long operators boils down to solving a set of coupled algebraic equations, termed asymptotic Bethe equations (ABA). A significant simplification occurs also on the string theory side. The classical supersymmetric sigma-model on the AdS5×S5 product space is integrable and there are indications that this remains true also at the quantum level.
Short strings/operators are unfortunately not accessible by these techniques due to the so- called wrapping diagrams, which typically have to be included at the order O(g^2L). These extra diagrams correspond to highly non-local interactions from the spin chain point of view. Recently, there has been a breakthrough in computing anomalous dimensions of composite operator of arbitrary length. Different groups have conjectured the Thermodynamic Bethe Ansatz (TBA) for the ground state (BPS state) and the sl(2) subsector of the theory. If supplemented with additional analytic information, these equations might be used to compute non-perturbatively the energy/dimension of any string state/composite operator of the correspondence thus accounting for the neglected wrapping diagrams. Very recently this infinite set of coupled integral equations has been written in a very compact form, raising hopes that they may be derived from the first principles on both sides of the correspondence.

The objectives of this project revolve around the following aspects:
1) Testing the veracity of the proposed TBA equations
2) Extracting from them insights into non-perturbative dynamics of AdS/CFT correspondence.
3) The leading order (LO) and next-to-leading (NLO) BFKL equations and their origins in AdS/CFT correspondence
4) Application of the integrability methods to cousin theories : AdS4/CFT3 and AdS3/CFT2.
The objectives 1),2) and 4) have been fully met and encouraging partial results towards fulfillment of 2) have been obtained. The experienced researcher has published two long (50+ pages), comprehensive and already well-cited papers in Nuclear Physics B. Both articles are interdisciplinary with cross-connections between the aforementioned objectives. With these two articles a new conceptual framework has been created that allows for a non-perturbative test of spectral equations for strong/weak coupling dualities. The underlying idea is to look at the dynamics of excitations on top of a dynamic background. Such a background for AdS/CFT is furnished by the GKP solution (this includes all currently known cases: AdS5/CFT4, AdS4/CFT3 and AdS3/CFT2!). It corresponds to long twist-two operators on the gauge theory side. It turns out that the physics of the low-energy excitations on this “vacuum state” is described by a sigma model, which is a consistent truncation of the string theory sigma model. This approach allowed to conduct non-perturbative tests of spectral equations for AdS5/CFT4 and AdS4/CFT3. This included a non-perturbative test of the dressing factor for AdS4/CFT3, which was not extensively tested so far. Furthermore, the framework developed allows to analyze how the interpolation between weak- and strong-coupling limit works in practice! Finally, the S-matrix constructed in one of these papers has found an important application in the non-perturbative computation of scattering amplitudes for AdS5/CFT4 (see arXiv:1303.1396 and follow-up articles). Thus, the results obtained during this fellowship have significantly contributed to the advancement of AdS/CFT correspondence.