D branes, Gepner points and string geometry

Research project P 14639 D branes, Gepner points and string geometry Maximilian KREUZER 09.10.2000 The last five years have seen dramatic progress in our understanding of non-perturbative string physics due to the realization of the relevance of D-branes and the discovery of a web of dualities among the consistent theories in 10 dimensions and M-theory. Although definite predictions of string phenomenology still have to be interpreted with restraint, it is important to study realistic string models in the light of these new developements and to try to make contact to observable physics. While most of the recent studies are focused on situations with a high degree of (super) symmetry, it can be hoped that the lessons that have been learned are also applicable to more realistic situations and shed new light on phenomenological string theory scenarios. In perturbative string physics heterotic (0,2) models are the most general class of realistic models. While it is easy to construct such models with CFT techniques, we have a very limited geometrical understanding due to the great difficulties posed by the stability condition for the vector bundles that determine the gauge background. Using gauged linear s model techniques along the lines of Distler and Kachru, one finds ambiguities in the resolutions of singularities that lead to chirality changing phase transitions. For a subclass of these models an exact CFT description in terms of simple current modular invariants has been conjectured. In the proposed research we want to focus on these models and supplement our geometrical and algebraic understanding with more recent results from D-brane physics and F-theory dualities that provide new tools for the study of vector bundles in heterotic models. A central issue will be the study of chirality changing phase transitions with these methods. Moreover, boundary states in CFTs provide important tests for our geometrical understanding in deeply nonperturbative regions of moduli space. Another important issue is the investigation of aspects of mirror symmetry in this context. While dualities greatly extend the applicability of geometrical methods, we may still benefit from CFT results in testing and supplementing these results in order to improve our understanding of the physics of realistic string models.

A complete understanding of the nature of particle interactions at small distances and in the early universe requires a unification of general relativity and quantum physics, which, presumably, is accomplished by string theory. It is widely believed that supersymmetry will play an important role in stabilizing the hierarchy of fundamental interactions as an approximate symmetry near and above the energy scale that is currently accessible to experiments. It has also been clear for some time that nonperturbative effects will eventually become important even for a qualitative understanding of nature of unification. In the last decade remarkable progress in this direction has been accomplished with the advent of D-branes and nonperturbative dualities in string theory. This has shifted the focus from heterotic strings to type II models, where brane world scenarios have lead to completely new ways of thinking about the origin and the structure of our four-dimensional space-time. In the present research project we have investigated the mathematical and physical structure of an important class of models that is constructed in terms of toric Calabi--Yau spaces. We have created a program package that allows a detailed and systematic analysis of these spaces, and we have further developed and applied the tools that are available for the resolution of singularities, which is necessary for the evaluation of physical couplings and for the analysis of their duality structure. We also obtained new results on the structure of space-time in the presence of strong fields, which modify the dynamics of D-branes, and on the stability of D-branes in the deep non-perturbative realm, where the geometry of space-time becomes fuzzy and quantum geometry has to replace the tools of classical geometry.