Supergravity and field theory solutions in AdS/CFT correspondence

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Modern theoretical physics has made big steps in addressing longstanding, fundamental questions. Problems like the confinement of quarks in QCD, the black hole information paradox, the cosmological constant problem and dark matter are now looking for answers in a single, unified theory, string theory.The most recent, and also one of the most promising, development in approaching a set of these issues is the AdS/CFT conjecture. This conjecture relates superconformal field theories to string theory or M-theory defined on spacetimes which are products of Anti de Sitter space, of one dimension higher than the corresponding super conformal field theory is defined on, times a compact space summing up to a total number of ten or eleven dimensions. One of the most interesting aspects, and also one of the main difficulties proving the conjecture, is that it is a weak-strong coupling correspondence.Conformal field theories defined on spacetimes of various dimensions provide us with a viable description of interesting phenomena like critical phenomena, phase transitions, renormalization group flows describing where massive quantum field theories flow to under the renormalization group. The range of areas that conformal field theories have been proved to be of primary interest is vast, varying from the Quantum Hall Effect to Network Models relevant in economics studies.In order to actually take advantage of the conformal field theoretical description one should be able to perform calculations, in particular the correlation functions, at all regimes of the various coupling constants that appear. In general, the systematics of perturbation theory is applicable only in the weak coupling regime of any field theory whereas interesting phenomena, like the QCD confinement, happen at the strong coupling regime which is inaccessible with the standard perturbative techniques and methodology. On the other side of the duality, the weak coupling description of the corresponding string theory is, in principle, tractable and perturbation theory can be employed to extract quantitative answers from it. The AdS/CFT correspondence provides us with an invaluable tool in studying strongly coupled conformal field theories.The AdS/CFT conjecture has not been proved at full generality yet and the collection of evidence that would lead to proving, or disproving it, is rather crucial at this stage. My research will partially focus in various non-trivial investigations on both sides of the correspondence for which the dictionary of the conjecture suggests that we have a different description of the same physical system. One of the examples that I want to consider is the study supergravity dual of supersymmetric Wilson loop operators present in superconformal gauge theories as N=4 super Yang-Mills theory in four spacetime dimensions. These operators have been extensively studied and various properties of them have been investigated. Since AdS/CFT predicts that every operator should have a string theory dual, it is rather crucial to prove that a string theory description of them exists and that it predicts exactly the same properties with the ones that have been discovered from field theory considerations.The second part of my research will focus on the ongoing program of classification of superconformal field theories that have a string theory description.In general, conformal invariance of string theory requires that the backgrounds used to study the theory on should be solutions of the corresponding low energy limit, a supergravity theory. A special class of such backgrounds would be the ones relevant for AdS/CFT, namely backgrounds that contain a global Anti de Sitter factor in the geometry. The question then reduces to describing which compact spaces are allowed to be the second factor of the product space.