Topics in four-dimensional supersymmetric gauge theories

This dissertation is directed toward an improvement in our understanding of four-dimensional superconformal field theories. In particular, we will focus on the information about the RG flows and fixed points of such theories that can be gleaned from the application of a technique known as a-maximization [14]. We will first apply this technique to obtain evidence that the strongest version of Cardy's a-theorem [3] is realized for superconformal theories. We will point out how a-maximization almost proves the theorem for these types of theories and give a careful analysis of loopholes that persist in the proof. We then apply a-maximization to an analysis of the RG flows of superconformal field theories with product gauge groups. We find that as we turn on a second gauge coupling in the presence of a first, the IR theory exhibits a number of different phases depending on the gauge group ranks and matter content.; The remainder of the dissertation is devoted to better understanding and extending a-maximization itself. We find an alternative method for computing the anomalous dimensions of chiral operators, which we call tauRR-minimization. This technique is not as powerful as a-maximization as tauRR receives perturbative corrections. However, it allows us to construct a physical proof that the proposed AdS/CFT dual of a-maximization correctly determines the superconformal R charges. This dual is known as Z-minimization [58], and the essence of the proof is a demonstration that Z-minimization implements tauRR-minimization. In the case where the superconformal field theory admits an AdS dual, the correspondence enables us to compute tauRR even for strongly coupled theories, and thus obtain exact R charges. The advantage of tau RR-minimization is that, unlike a-maximization, it can also be applied to three- and two-dimensional theories.