Topics in Supersymmetric Quantum Field Theory

Abstract

This thesis describes several new tools for analyzing supersymmetric quantum field theories, focusing on theories with four supercharges in three and four dimensions. In chapter two, we discuss supercurrents, supersymmetry multiplets that include the energy-momentum tensor. Physically, different supercurrents give rise to different brane charges in the supersymmetry algebra. They also encode different ways of placing supersymmetric field theories on a curved manifold. Under certain conditions this procedure preserves some of the supersymmetry. In chapter three, we explore these conditions for the case of four-dimensional N=1 theories with a U(1)R symmetry. In particular, we find that a manifold admits a single supercharge if and only if it is Hermitian.

In chapter four, we shift the focus to three-dimensional field theories. We study Chern-Simons contact terms -- contact terms of conserved currents and the energy-momentum tensor, which are associated with Chern-Simons terms for background fields. While the integer parts of these contact terms are ambiguous, their fractional parts constitute new meaningful observables. In N=2 supersymmetric theories with a U(1)R symmetry certain Chern-Simons contact terms can lead to a novel superconformal anomaly. In chapter five, we use this understanding to elucidate the structure of the free energy F of these theories on a three sphere. In particular, we prove the F-maximization principle for N=2 superconformal theories. We also explain why computing F via localization leads to a complex answer, even though we expect it to be real in unitary theories.