Geometric Decomposition of Supersymmetric Quantum Field Theories

Abstract

Quantum field theories, despite their central importance in modern physics, remain poorly understood due to a lack of techniques available for studying their strong coupling regimes. This thesis reports progress in the development of a geometric language for describing and analyzing aspects of five dimensional supersymmetric quantum field theories with interacting strongly coupled ultraviolet fixed points. It is explained in detail how various properties of these five dimensional field theories admit a mathematical interpretation in terms of the topology and complex structure of Calabi-Yau varieties, and how this correspondence between field theory and geometry has advanced the understanding of the landscape, dualities, and renormalization group flows of these theories. In particular, it shown how the geometric framework sets the stage for a potentially exhaustive classification of strongly coupled five dimensional fixed points by studying renormalization group flows from six dimensional supersymmetric quantum field theories compactified on a circle.