Final Activity Report Summary - ADSCFT (Interconnections between superstring and Yang-Mills theories) The two building blocks at the basis of our present understanding of the fundamental interactions are General Relativity, describing the gravitational force, and the Standard Model of particle physics, describing the electromagnetic, nuclear and sub-nuclear forces. String theory provides a framework for combining these two building blocks into a unified theory. The quest for a unified description of all the elementary interactions in Nature has characterised the evolution of theoretical physics since the end of the 19th century. The possibility that string theory may achieve such a unification makes research in this field particularly fascinating. A remarkable property of string theory is that it naturally includes both gravity and the gauge theories that form the basis for the Standard Model. The investigation of the connections between gravity and gauge interactions in the context of string theory has led to important insights in many areas of theoretical physics and mathematics. The objective of this research project has been to better understand various aspects of these interconnections in the framework of the so-called AdS / CFT correspondence. This is a duality relating string theory in specific curved spaces to gauge theories living on the boundary of such spaces. A similar relation is referred to as a 'holographic duality' since it relates the physics in a given volume to its 'projection' on a lower dimensional 'screen', the boundary of the space. In the AdS / CFT correspondence this bulk-to-boundary relation finds a precise formulation. The AdS / CFT correspondence in its original form involves the so-called N=4 supersymmetric Yang-Mills theory. A large amount of work has been done in recent years to generalise the duality to other theories. Part of this project has focussed on this aspect. In particular, a class of deformations of the N = 4 supersymmetric Yang-Mills theory have been analysed. A formulation based on the so-called Light-cone Superspace has been developed for these theories and this has allowed to study their quantum properties. Specifically, the exact scale invariance of these models has been proven to all orders in a perturbative expansion, thus confirming a prediction of the AdS / CFT duality. The light-cone formalism developed in this context has later been applied to a novel class of three-dimensional quantum field theories which have recently been extensively investigated. While many tests of the validity of the AdS / CFT correspondence have been carried out in the past few years, there remain various open questions concerning some of the fundamental structures underlying the duality. Addressing some of these issues has been an important objective of this work. The holographic nature of the duality remains largely unclear as it is not understood how local properties of the bulk gravitational theory are encoded in the boundary gauge theory. Progress in this direction has been made as part of this project through the study of non-perturbative effects in the AdS / CFT duality. It has been shown that the study of instanton contributions to observables in the gauge theory allows to reconstruct various details of the geometry and of other local properties of the dual gravitational theory. A notable feature of the gauge / gravity duality is that it relates the weak coupling regime of string theory, which is best understood, to the strong coupling regime of the dual gauge theory, where the most interesting and least understood dynamical phenomena take place. This property makes the AdS / CFT correspondence a very powerful tool for the investigation of non-perturbative aspects of gauge theories. One area in which this feature of the duality has been employed in this project is in the calculation of gauge theory scattering amplitudes. The study of these quantities has revealed rich structures and a number of remarkable properties, which have been investigated utilising the previously mentioned light-cone superspace formalism. The AdS / CFT correspondence remains a very dynamic field of research and various aspects of this research project are currently being further developed.
