Dualities in Modern String Theory and Applications

The theory of quantized gauge fields describes the most fundamental constituents of matter and interactions among them with an unprecedented accuracy. When the interactions are weak, perturbation theory predicts observable quantities with a remarkable precision. On the contrary, non-perturbative phenomena in strongly-interacting gauge theories are notoriously difficult to understand. It has been anticipated for a long time that string theory may give an effective description of strongly interacting gauge fields. Gauge theories often have solitons - solutions of the classical field equations with finite energy. In modified theories of weak interactions there are, for example, magnetic monopoles. The presence of such objects can imply that there is a duality between electric and magnetic charges. Such dualities are powerful symmetries, since they often relate separate regimes of a given theory and can generate new solutions. String theory has higher- dimensional solitonic solutions called branes. In string theory, there is also a number of dualities, such as dualities between strongly and weakly coupled regimes of different versions of string theory and even more - duality between string and gauge theories. The strings propagate in the bulk of a higher-dimensional space of which the four-dimensional space-time is the boundary. The string is in some sense "holographically" projected onto the lower-dimensional "screen". Whether all gauge theories have holographic duals or such theories are exceptional is unclear at the moment. However, once duality is established, the knowledge from weak coupling regime in one theory can be extended to the strongly coupled dual theory and thus the importance of investigating dualities convincingly follows. In the present project we study various aspects of holographic string/gauge duality and other geometrical topics that are required for the analysis of phenomenological implications of string unification. Particular emphasis is given to AdS/CFT duality, Integrability, Topological Strings, Seiberg-Witten theory, Matrix Models and to Calabi- Yau compactifications and model building. Pure spinors are used, on the one hand, for the study of quantum effects in backgrounds with fluxes, and on the other hand for the analysis of the resulting unconventional geometries. For all of our major topics there exists complementary expertise in the research groups at ITEP (Moscow) and at the Univ. of Technology (Vienna). The joint project will hence lead to an intensive transfer of knowledge that will enhance the compentences of both groups and we expect a very productive collaboration and developments beyond the original scope of this project.

The newest theoretical developments in the area of superstrings build upon the discovery and analysis of so-called dualities, which permit complementary descriptions of string theories and quantum field theories as well as higher-dimensional gravitational theories and corresponding techniques for their treatment. The so-called string-gauge-duality has turned out to be particularly fruitful. This was originally conjectured for the case of a duality between a certain highly symmetric strongly coupled gauge theory and string theory in a five-dimensional curved (anti-de Sitter) space, and subjected successfully to many tests. Over the years this has found many generalizations. One of the main topics of this project has been the study of dualities with less supersymmetry, which are of interest to more realistic applications, as well as the calculation of more complicated correlation functions in supersymmetric string-gauge duality. A second focus of this project was on the internal geometry of superstring theories, which are governed by so-called Calabi-Yau manifolds, and the development of new methods for constructing novel models for the unification of all fundamental interactions of elementary particles, partly made possible by massive computer assistance. Motivated by stringtheoretical dualities, the so-called F-theory opens new possibilities. Thanks to the specific know-how in toric geometry available to this project, new global constructions were obtained.