UC San Diego

This dissertation consists of two parts, each of which improves our understanding of supersymmetric field theories. In the first part I use the technique of "a- maximization" to study the RG flows of these theories. In doing so I find evidence for the strongest form of Cardy's a-theorem. In doing so we move closer to a proof of Cardy's a-theorem by the removal of a loophole in the argument for the theorem. I also examine the remaining loopholes in the argument. I then apply a-maximization in the case where the superconformal field theory has a product gauge group. In these situations I find that the dynamics of one of the gauge groups can dramatically alter the behavior of the other. I give a detailed analysis of the possible RG flows and fixed points. The focus of the second part of the dissertation is on extending and furthering our understanding of a-maximization. I construct an alternative to a-maximization, called $\ tau_}RR}$-minimization, which also determines the anomalous dimensions of chiral operators. This technique is not as powerful as a-maximization because $\tau_}RR}$ receives quantum corrections. This method is extendable beyond four dimensions. I find the the geometrical analog of a-maximization, Z-minimization, gives the same results as $\tau_}RR}$-minimization, and show how the two techniques are related. This allows for the computation of exact anomalous dimensions, in theories with known AdS/CFT duals, outside of four dimensions