Branes, graphs and singularities

Abstract

In this thesis, we study various aspects of string theory on geometric and nongeometric backgrounds in the presence of branes. In the first part of the thesis, we study non-compact geometries. We introduce "brane tilings" which efficiently encode the gauge group, matter content and superpotential of various quiver gauge theories that arise as low-energy effective theories for D-branes probing singular non-compact Calabi-Yau spaces with toric symmetries. Brane tilings also offer a generalization of the AdS/CFT correspondence. A technique is developed which enables one to quickly compute the toric vacuum moduli space of the quiver gauge theory. The equivalence of this procedure and the earlier approach that used gauged linear sigma models is explicitly shown. As an application of brane tilings, four dimensional quiver gauge theories are constructed that are AdS/CFT dual to infinite families of Sasaki-Einstein spaces. Various checks of the correspondence are performed. We then develop a procedure that constructs the brane tiling for an arbitrary toric Calabi-Yau threefold. This solves a longstanding problem by computing superpotentials for these theories directly from the toric diagram of the singularity. A different approach to the low-energy theory of D-branes uses exceptional collections of sheaves associated to the base of the threefold. We provide a dictionary that translates between the language of brane tilings and that of exceptional collections. Geometric compactifications represent only a very small subclass of the landscape: the generic vacua are non-geometric. In the second part of the thesis, we study perturbative compactifications of string theory that rely on a fibration structure of the extra dimensions. Non-geometric spaces preserving .A = 1 supersymmetry in four dimensions are obtained by using T-dualities as monodromies. Several examples are discussed, some of which admit an asymmetric orbifold description. We explore the possibility of twisted reductions where left-moving spacetime fermion number Wilson lines are turned on in the fiber.