Branes and the vacuum structure of supersymmetric gauge theories

Abstract

The study of quantum field theories and supersymmetric quantum field theories has thrived in the recent decades. An aspect of this study has a particularly geometrical nature and as such has also started to spearhead progress in related fields of mathematics. This aspect is the study of vacua. Traditional computations in quantum field theory understand particle states as perturbations around a vacuum state. Many theories have a plethora of candidates to be this vacuum state. Different choices of vacuum imply different properties of the particles in the theory. These physical differences can be understood by creating a moduli space: an affine variety in which each point of the variety corresponds to a different vacuum of the theory. The physical properties of the vacua are then realized as geometrical properties of the different points in the variety. In this thesis we analyse these properties and develop new techniques to further their study: the Kraft-Procesi transition and the magnetic quivers. In order to do so we employ a particular fruitful approach to this type of problems: embedding the quantum field theories at hand in a string theoretical background. By this procedure, the low energy dynamics of branes in the string theory set up correspond to motions along the moduli space of vacua of the theory. The new results that we present here aim to be a set of tools that can readily be applied to the analysis of many different supersymmetric quantum field theories. The goal is that the tools are based on manipulations of simple diagrams, which should make them easy to remember and easy to implement.