Monday, 16:00-18:00, Schrödinger lecture hall
starting on October 1, 2007

 
The idea of a correspondence between the large N limit of gauge theories and string theory was proposed more than thirty years ago. It has been developed over the years in many directions, but an explicit realization was only provided by the conjectured AdS/CFT correspondence, which asserts a duality between string theory in Anti-de-Sitter spaces and Yang-Mills theories. Since then this became a major research area and many fascinating discoveries were made in the last decade.

The lectures are based on the recent advances on both, the string theory and the gauge theory sides of this duality. The aim is to describe the physical ideas and to provide the mathematical tools to attack the challenges arising in this fascinating area.

The content of the lectures is as follows.

In part I we give a brief review of (super) string theory. This includes the notion of world-sheet symmetries and 2d (super) conformal field theories. Basic facts about vertex operators and representations of infinite dimensional algebras will be discussed.

Part II of the lectures deals with the supersymmetry algebra, the N = 4 superconformal algebra in four dimensions, and their representations. The properties of N = 4 supersymmetric Yang-Mills (SYM) theory will be discussed, as well as its derivation from N = 1 SYM in 10 dimensions.

In part III we consider the low-energy limit of string theory and its effective target space action. The important topics will be D-branes and their dynamics, as well as dualities between various string theories. Based on the properties of the branes and the effective geometries accounting for the back reaction, the Maldacena conjectures will be formulated and discussed. Simple examples of how the conjectured duality works, namely, how the spectrum of anomalous dimensions of gauge invariant operators and the energy spectrum of the string theory are related, will be provided.

The last part of the lectures deals with the most recent developments in the area. Many papers have indicated that type IIB string theory on AdS5 x S5 and N = 4 SYM in 4 dimensions may be integrable in the planar limit (and maybe beyond). The techniques of integrable systems have therefore become useful in studying the AdS/CFT correspondence in details. In this part we will make various reductions of string theory to integrable models. Integrability of the theory will be considered from various points of view. First we will concentrate on the formulation of the duality at the energy scales where it can be approximated with certain spin chain models. We will give a simple example of how one can solve these models by making use of the Bethe ansatz. The next point of view will be the discussion of the integrable structures appearing in string theory. These contain topics like Backlund transformations, local and non-local currents/charges, etc. We will discuss how the integrable structures can be used to find solutions and to establish duality.