## Final Report Summary - ADS CFT (AdS/CFT correspondence: extrapolation techniques and space-time geometry from gauge theories)

The project studied several aspects of a conjectured equivalence between string theories and gauge field theories, known as AdS/CFT correspondence. The potential applications of this duality are vast: it could be used to describe diverse strongly coupled systems, which would be otherwise difficult to characterise. Understanding how strongly coupled gauge theories -inaccessible by perturbative means- can be recast into classical gravity or weakly coupled string theory is central in the AdS/CFT correspondence. The study of methods that provides some description of AdS/CFT systems for any value of the coupling constant, interpolating between weak and strong coupling results, was targeted in the project.

In the planar limit of the most studied realisation of this duality (when the string theory is formulated in AdS5xS5) the spectral problem was discovered to be integrable, which means it can be solved by some algebraic method as the Bethe ansatz. One specific objective of the project was to explore different generalisations in the quest for new integrable cases.

Many training activities were planned to improve the researcher knowledge on integrability matters and their applications in the AdS/CFT correspondence. He participated in several scientific workshops on the subject, organised a series of seminars, hosted researchers working in the field to discuss with them aspects of the project and visited collaborators.

Cases when D-branes and open strings are added, as well as strings in some other asymptotically AdS5xS5 space-times, were thoroughly considered and their integrability was questioned. The exact coupling constant dependence of both the dispersion relation and the scattering matrices of world-sheet excitations were determined. New integrable open boundary conditions were discovered. For those cases a Bethe ansatz was formulated and a method to incorporate corrections due to finite size effects was also developed.

In some cases where integrability was ruled out, it was still possible to take a limit, like the one taken by Berenstein, Maldacena and Nastace in their renowned article, to extrapole weak coupling computations to the strong coupling regime. The existence of symmetries that determine the exact dispersion relation of world-sheet excitations was crucial. This limit allowed inferring geometrical information of the string theory backgrounds from 1-loop gauge theory computations.

An alternative mean to study these backgrounds used an effective matrix model that characterises supersymmetric states. The matrix problem is reduced to solving for wave-functions giving the probability for the distribution of the eigenvalues of the matrices associated to given supersymmetric states. For the ground state, which should correspond to AdS5xS5, the geometry of the S5 had been seen to emerge from this analysis. Previous attempts to characterise more general supersymmetric states used numerical methods. An outcome of this project was a perturbative method which provides an analytical description for the distribution of eigenvalues of some generic supersymmetric states. This constitutes a first step towards the reconstruction of the metric functions of these backgrounds by using a matrix model originated from the field theory Lagrangian.

The project has broadened the understanding of the AdS/CFT correspondence in various directions. The goals of developing methods to interpolate between weak and strong coupling regimes and using them to reconstruct geometrical information have been accomplished. The project contributed to European excellence and competitiveness in a field of research which is bound to be one of the most active ones in high energy physics for many years to come. As a tool to describe strongly coupled systems, the AdS/CFT correspondence could help to understand longstanding and diverse open problems such as confinement or even some quantum critical phenomena relevant for understanding high Tc superconductivity.

In the planar limit of the most studied realisation of this duality (when the string theory is formulated in AdS5xS5) the spectral problem was discovered to be integrable, which means it can be solved by some algebraic method as the Bethe ansatz. One specific objective of the project was to explore different generalisations in the quest for new integrable cases.

Many training activities were planned to improve the researcher knowledge on integrability matters and their applications in the AdS/CFT correspondence. He participated in several scientific workshops on the subject, organised a series of seminars, hosted researchers working in the field to discuss with them aspects of the project and visited collaborators.

Cases when D-branes and open strings are added, as well as strings in some other asymptotically AdS5xS5 space-times, were thoroughly considered and their integrability was questioned. The exact coupling constant dependence of both the dispersion relation and the scattering matrices of world-sheet excitations were determined. New integrable open boundary conditions were discovered. For those cases a Bethe ansatz was formulated and a method to incorporate corrections due to finite size effects was also developed.

In some cases where integrability was ruled out, it was still possible to take a limit, like the one taken by Berenstein, Maldacena and Nastace in their renowned article, to extrapole weak coupling computations to the strong coupling regime. The existence of symmetries that determine the exact dispersion relation of world-sheet excitations was crucial. This limit allowed inferring geometrical information of the string theory backgrounds from 1-loop gauge theory computations.

An alternative mean to study these backgrounds used an effective matrix model that characterises supersymmetric states. The matrix problem is reduced to solving for wave-functions giving the probability for the distribution of the eigenvalues of the matrices associated to given supersymmetric states. For the ground state, which should correspond to AdS5xS5, the geometry of the S5 had been seen to emerge from this analysis. Previous attempts to characterise more general supersymmetric states used numerical methods. An outcome of this project was a perturbative method which provides an analytical description for the distribution of eigenvalues of some generic supersymmetric states. This constitutes a first step towards the reconstruction of the metric functions of these backgrounds by using a matrix model originated from the field theory Lagrangian.

The project has broadened the understanding of the AdS/CFT correspondence in various directions. The goals of developing methods to interpolate between weak and strong coupling regimes and using them to reconstruct geometrical information have been accomplished. The project contributed to European excellence and competitiveness in a field of research which is bound to be one of the most active ones in high energy physics for many years to come. As a tool to describe strongly coupled systems, the AdS/CFT correspondence could help to understand longstanding and diverse open problems such as confinement or even some quantum critical phenomena relevant for understanding high Tc superconductivity.