Robert Moscrop: Quivers, superconformal field theories and higher symmetries

Abstract

Quantum field theory (QFT) currently provides our best description of the natural world. Over the past century, it has been able to accurately reproduce many experimental results to an astonishing degree of accuracy and predict the existence of new particles. Despite this, QFT suffers from several mathematical problems. One such problem is the fact that it is often only possible to obtain perturbative results in a weak coupling regime rather than exact results. However, if one constrains the system by introducing supersymmetry, exact results become more tractable. In this thesis we aim to explore various non-perturbative features of QFT by working with theories with eight supercharges. 

Much of the material of this thesis aims to understand QFTs from the point of view of BPS quivers-- graphical objects that govern the worldlines of BPS states. By understanding the geometric engineering of these objects, we then use them to extract the higher symmetries of many theories. We also relate disconnected quivers in the orbifold case to the decomposition of two dimensional sigma models.

A second problem tackled in this thesis is that of the classification problem of four dimensional conformal theories with eight supercharges. More specifically, we define a quantity which allows us to single out theories whose Donagi-Witten integrable system simplifies tremendously. The simplifications make the classification problem more tractable for this subset of theories and we initiate the study of them by presenting two new putative rank-2 theories.