Brane Tilings, M2-branes and Orbifolds

Abstract

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 C³/Γ can be counted according to the size of the group |Γ|. 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.