## Final Activity Report Summary - N.S.M.S.T. (Nonlinear Sigma Models in String Theory)

At present we have a well-confirmed theory (QCD) of the strong interactions which bind the quarks into neutrons and protons. This theory fits into a larger class of quantum field theories called gauge theories. We also have a promising candidate for a unified theory of all matter together with their interactions - the string theory. The main problem in the study of gauge theories is to understand what happens at strong coupling where the qualitative behaviour is expected to be radically different from the well-understood situation at weak coupling. An important problem in our theoretical understanding of string theory is to understand what happens when space-time is strongly curved. This is the regime where string theory will differ drastically from any point-particle theory. The resulting effects will be crucial for the physics of black holes, and may modify our ideas about the big bang decisively.

There is growing evidence that the two problems are often closely related. A class of conjectures of this type is referred to as the AdS-CFT-correspondence. In an interesting particular case we even expect that both sides of the correspondence are exactly soluble. If this expectation, usually referred to as the integrability of the theories involved in this correspondence, is actually realised we could prove the AdS-CFT correspondence in this case. This would also allow us to analyse both gauge theory and string theory quantitatively in regimes which were believed to be inaccessible for theoretical studies until recently. However, despite much effort it was not shown yet that the theories in question are indeed integrable. The reason was a lack of sufficiently powerful mathematical tools for the analysis of such theories.

The project at hand was aimed at the development of mathematical methods that would allow us to demonstrate that the string theory on certain Anti-de Sitter (AdS-) spaces is integrable. This would follow from the integrability of the two-dimensional quantum field theories called Nonlinear Sigma Models that represent the main ingredients used in the theoretical analysis of the spectrum of excitations and of the interactions of strings on AdS-spaces.

We have made substantial progress towards this aim by developing a systematic mathematical procedure for approximating large classes of two-dimensional quantum field theories in such a way that the integrability of the theories is always under full control. The method has been successfully applied to first nontrivial examples relevant for string theory on curved spaces, demonstrating the integrability of these examples. This includes a gauge-fixed version for the non-linear sigma model on a space which contains the two-dimensional Anti-De Sitter space as a part. Such a space-time can be seen as a simplified relative of the Anti-de Sitter backgrounds of current interest, which nevertheless captures many important qualitative features of the more general cases.

Even though we have not yet applied our techniques to the cases of three- and five-dimensional AdS-spaces which motivate this line of research, we believe that our methods can be generalised to solve these cases too. Having developed the first systematic method for demonstrating the integrability of large classes of such theories is one of the main achievements of our project. No such method was known before. Important progress was also made in the development of theoretical tools for the analysis of the spectrum of excitations in these theories. We believe that our work initiates a new line of research on the subject that will lead to the solution of the problems above.

There is growing evidence that the two problems are often closely related. A class of conjectures of this type is referred to as the AdS-CFT-correspondence. In an interesting particular case we even expect that both sides of the correspondence are exactly soluble. If this expectation, usually referred to as the integrability of the theories involved in this correspondence, is actually realised we could prove the AdS-CFT correspondence in this case. This would also allow us to analyse both gauge theory and string theory quantitatively in regimes which were believed to be inaccessible for theoretical studies until recently. However, despite much effort it was not shown yet that the theories in question are indeed integrable. The reason was a lack of sufficiently powerful mathematical tools for the analysis of such theories.

The project at hand was aimed at the development of mathematical methods that would allow us to demonstrate that the string theory on certain Anti-de Sitter (AdS-) spaces is integrable. This would follow from the integrability of the two-dimensional quantum field theories called Nonlinear Sigma Models that represent the main ingredients used in the theoretical analysis of the spectrum of excitations and of the interactions of strings on AdS-spaces.

We have made substantial progress towards this aim by developing a systematic mathematical procedure for approximating large classes of two-dimensional quantum field theories in such a way that the integrability of the theories is always under full control. The method has been successfully applied to first nontrivial examples relevant for string theory on curved spaces, demonstrating the integrability of these examples. This includes a gauge-fixed version for the non-linear sigma model on a space which contains the two-dimensional Anti-De Sitter space as a part. Such a space-time can be seen as a simplified relative of the Anti-de Sitter backgrounds of current interest, which nevertheless captures many important qualitative features of the more general cases.

Even though we have not yet applied our techniques to the cases of three- and five-dimensional AdS-spaces which motivate this line of research, we believe that our methods can be generalised to solve these cases too. Having developed the first systematic method for demonstrating the integrability of large classes of such theories is one of the main achievements of our project. No such method was known before. Important progress was also made in the development of theoretical tools for the analysis of the spectrum of excitations in these theories. We believe that our work initiates a new line of research on the subject that will lead to the solution of the problems above.