title: Quiver vacua geometry and fermions on branes
Creators: name: Rogers, Jamie
Publisher: name: University of Liverpool Repository
Year: 2022

description: This thesis explores two topics from the intersection between string theory, quantum field theory, and geometry. While the two topics share some overarching themes, for practical purposes they are distinct areas of research. Therefore this work is presented in two parts, treated separately. Quiver vacua geometry Supersymmetric quantum fields theories with eight supercharges are intimately related with numerous topics in geometry. Theories living in three dimensions and enjoying N=4 supersymmetry are perhaps the most accessible such theories. The fields in these theories are arranged into hyper- and vector multiplets, both containing scalar fields. The field-content and symmetries of these theories may be denoted using a quiver, a type of graph in which edges and nodes have specific interpretations in terms of multiplets and symmetries. The possible lowest energy states (vacua) for these theories are associated to vacuum expectation values (VEVs) of the scalar fields. These possible vacua can be organised as an affine variety wherein each point corresponds to a particular set of scalar VEVs. This variety is called the moduli space of vacua for the theory in question. For the class of linear quivers, these varieties are known to be nilpotent varieties in sl_n. In Part I of this thesis, after introducing the necessary machinery, we explore the geometry of the moduli space of vacua of two sets of quiver gauge theories which appear as natural generalisations of the linear case. These are circular quiver gauge theories and D-type Dynkin quiver gauge theories. A central tool in our work is the Kraft-Procesi transition which can be thought of as a physical interpretation of the singularity structure of the affine variety associated to the possible vacua for a given theory. Fermions on branes In Part II we make a detailed study of the fermionic fields living on brane worldvolumes. In string phenomenology many of the most promising candidates for the construction of a small, positive cosmological constant in a string theoretic setting rely on ingredients in which the fermionic couplings of fields living on brane worldvolumes play a critical role. We use the superspace formulation of supergravity in eleven and ten dimensions to compute fermion couplings on the M2-brane and on Dp-branes. Fermionic couplings arise naturally from the expansion of the superfields in orders of the fermionic coordinates of superspace. The techniques we use and develop can be applied to determine the fermionic couplings to background fields up to arbitrary fermionic order. We start with the superspace formulation of 11-dimensional supergravity and use a geometric technique known as the normal coordinate method to obtain the $\theta$-expansion of the M2-brane action. We then present a method which allows the translation of knowledge of fermionic couplings on the M2-brane to knowledge of such couplings on the D2-brane, and then to any Dp-brane. This method is based on superspace generalizations of both the compactification taking 11-dimensional supergravity to type IIA supergravity and the T-duality rules connecting the type IIA and type IIB supergravities.