Brane Tilings, M2-branes and Orbifolds

Brane Tilings represent one of the largest classes of superconformal theories with known gravity duals in 3+1 and also 2+1 dimensions. They provide a useful link between a large class of quiver gauge theories and their moduli spaces, which are the toric Calabi-Yau (CY) singularities. This thesis includes a discussion of an algorithm that can be used to generate all brane tilings with any given number of superpotential terms. All tilings with at most 8 superpotential terms have been generated using an implementation of this method. Orbifolds are a subject of central importance in string theory. It is widely known that there may be two or more orbifolds of a space by a finite group. Abelian Calabi-Yau orbifolds of the form $\BC^3 / \Gamma$ can be counted according to the size of the group $|\Gamma|$. Three methods of counting these orbifolds will be given. A brane tiling together with a set of Chern Simons levels is sufficient to define a quiver Chern-Simons theory which describes the worldvolume theory of the M2-brane probe. A forward algorithm exists which allows us to easily compute the toric data associated to the moduli space of the quiver Chern-Simons theory from knowledge of the tiling and Chern-Simons levels. This forward algorithm will be discussed and illustrated with a few examples. It is possible that two different Chern-Simons theories have the same moduli-space. This effect, sometimes known as `toric duality' will be described further. We will explore how two Chern--Simons theories (corresponding to brane tilings) can be related to each other by the Higgs mechanism and how brane tilings (with CS levels) that correspond to 14 fano 3-folds have been constructed. The idea of `child' and `parent' brane tilings will be introduced and we will discuss how it has been possible to count `children' using the symmetry of the `parent' tiling.