Non-perturbative aspects of gauge theories

Abstract

The purpose of this thesis is to establish some non-perturbative results in gauge theories in d<=4 spacetime dimensions. We generically refer to such theories as Quantum Chromodynamics (QCD). Gauge theories are asymptotically free, which means that at short distances interactions become weak, and one can reliably use perturbation theory in order to solve the theory. Most of our current understanding of QCD stems from studying the theory at short distances; for example, this is the regime where the bulk of experimental tests of the Standard Model of particle physics has been performed. By contrast, the long distance regime is characterized by strong interactions, and here perturbation theory generically stops working. Hence, the dynamics of QCD at macroscopic scales is to a large extent still a mystery. For example, our ab initio understanding of the spectrum of hadrons is still very limited, most data coming from numerics or phenomenological models.

In order to study gauge theories in their strongly-interacting regime, the perturbative approach is mostly useless, and one must develop new tools. In this work we study some quantitative properties of QCD in its strongly-coupled phase. Some of our main tools are supersymmetry, 't Hooft anomalies, dualities, topological fields theories, and two-dimensional chiral algebras.