Realizations of four-dimensional gravity in field and string theory

Since the beginning of the 20th century modern theoretical physics is based on the two pillars of quantum mechanics and general relativity. Our hope is to reconcile these two concepts or in other words we need a quantum theory of gravity. Today the most promising candidate seems to be string theory. The good thing of string theory is that it only has one dimensionful constant -- the string tension. In principle then all quantities are computable and finite. One of the main problems of string theory is that it is defined in ten dimensions whereas we observe four macroscopic space-time dimensions. The problem is how to get four-dimensional gravity that is similar to Einstein`s theory at sufficiently large distances and weak curvature. One can rephrase the problem by saying that we do not know how string theory chooses its vacuum state. There are several ideas or scenarios how four-dimensional gravity could be realized that are known today. First the extra dimensions, i.e. the internal manifold, could be compactified. Second the internal manifold could be non-compact but be so that the Laplacian has a normalizable zero-mode (thus the manifold must be of finite volume) and a gap. Third the internal manifold could be non- compact with finite volume so that the four-dimensional space-time is warped. As a result there is no gap in the Laplacian, however the density of eigenvalues around zero must scale appropriately so that the four-dimensional Newton`s law is reproduced at large distances. This is the Randall-Sundrum mechanism. Forth the four-dimensional universe could be a 3-brane (a four-dimensional hypersurface) embedded in a higher dimensional bulk space. Only gravity can propagate in the bulk. The four-dimensional Einstein term is then generated due to loop corrections of the fields localized on the brane. The aims of the research project are: to analyze how the quantum corrections for various effective couplings in string theory differ in the various scenarios, if these scenarios can explicitely be realized in string theory, what are their respective implications in particle physics and cosmology, and if a certain combination of these ideas leads to solutions of the problem of moduli stabilization (the choice of the vacuum state), the cosmological constant problem and the question of why gravity is so weak compared to the other fundamental forces in nature.