D-branes on Calabi-Yau manifolds

Already a few years after Einstein`s general theory of relativity uncovered the relation between gravitational fields and the curvature of space-time Kaluza and Klein found a geometrical interpretation for the electromagnetic fields as curvature components in a (tiny) fifth dimension. But it took several decades with the development of quantum field theory and the discovery of the other particle interactions untill that idea eventually found its natural home in string theory, which provides a framework for a unified description of all forces of Nature in a 10-dimensional space-time structure. The discovery of D-branes and of their implications for dualities and for nonperturbative quantum effects in string theory recently brought about a drastic modification of our picture of these small extra dimensions. D-branes are spatially extended solutions to the nonlinear background field equations with finite energy density and they support physical degrees of freedom so that our familiar forces only act on a subspace of the 10-dimensional space-time, while physics could be quite different on other branes. Obviously, a good understanding of the properties of D- branes is then essential for further progress in our perception of particle interactions. Consistency conditions imply that the internal space is a so-called Calabi--Yau manifold. In the proposed research we want to study D-branes on such backgrounds. While concepts of classical geometry are sufficient for weakly curved spaces, quantum effects become essential at small volumes. In particular, particle spectra, nonperturbative contributions to the effective potential of supersymmetric gauge theories and walls of marginal stability are of utmost physical interest. In order to investigate these problems we will use string dualities like mirror symmetry, which allow to reduce the exact quantum geometry to classical results in the dual model. A second important tool will be exact results from conformal field theory, which give important hints at the structure of the theory at certain values in the deep quantum regime and thus provide clues and tests for general concepts. The aim of our work is an improved understanding of the physics of D-branes as building blocks of a unified string model of fundamental interactions.

This project dealt with a number of mathematical aspects concerning the classification of string theory compactifications with a given number of supersymmetries, and physical implications for issues like their vacuum structure and black hole entropy. String theory is currently the most promising candidate for a unifying theory of all the fundamental forces of Nature. In its most symmetric version, the so-called supersymmetric string theory, quantum mechanical consistency requirements yield a ten-dimensional space-time. In order to arrive at the observed four-dimensional space-time, the theory has to be compactified on a six-dimensional space. The four-dimensional theory is still required to be supersymmetric, mostly for mathematical simplicity and solvability. This requirement narrows down the possibilities for the compact six-dimensional space to the so-called Calabi-Yau spaces. Among the questions we answered in this project is the partial classification of such spaces from a mathematical point of view and its relation to black hole states in the corresponding physical theory. Such spaces can be studied by looking at the different ways a two-dimensional surface, a so-called Riemann surface, without boundary can be embedded into it. For this study we do not need the full superstring theory, only a subsector of it, the topological string theory needs to be considered. This theory is simpler in that it is in principle exactly solvable. It does not have the full physical content, but most of it is determined, in particular the number of physical states in this theory is related to the number of ways of embedding a Riemann surface into a Calabi-Yau space. These numbers are topological invariants which are of great interest in mathematics. In physics, they are related to the entropy of supersymmetric black holes. Therefore, the study of the topological string theory yields mathematical information about these spaces, and vice versa. We have both enhanced existing techniques and extended the class of spaces for which these invariants can explicitly be computed. The same can in principle be done for embeddings of Riemann surfaces with a boundary. This is considerably more difficult because on one hand the number of supersymmetries gets reduced by at least one half, and on the other hand, the boundary carries additional information that has to be included. Namely, it is fixed on a particular subspace of the Calabi-Yau space, a so-called D-brane. In a simple approach we investigated how the properties of the D-brane depend on the configuration on the boundary as well as on the change of the size of the Calabi-Yau space. Using the techniques developed for the first question we obtained predictions on the number of such embeddings of a Riemann surface with boundaries.