While motivated in part by the hopes of building phenomenologically successful models of particle physics, we have uncovered in our study of supersymmetric theories a rich trove of field theory phenomena. Supersymmetry provides powerful constraints on dynamics. In this chapter, we will discover more remarkable features of supersymmetric field theories. We will first study classes of (super)conformally invariant field theories. Then we will turn to the dynamics of supersymmetric QCD with Nf ≥ Nc, where we will encounter new, and rather unfamiliar, types of behavior.

Conformally invariant field theories

In quantum field theory, theories which are classically scale-invariant typically are not scale invariant at the quantum level. QCD is a familiar example. In the absence of quark masses, we believe the theory confines and has a mass gap. The CPN models are an example where we were able to show systematically how a mass gap can arise in a scale-invariant theory. The breaking of scale invariance in all of these cases is associated with the need to impose a cutoff on the high-energy behavior of the theory. In a more Wilsonian language, one needs to specify a scale where the theory is defined, and this requirement breaks the scale invariance.

There is, however, a subset of field theories which are scale invariant. We have seen this in the case of N = 4 supersymmetric field theories in four dimensions. In this section, we will see that this phenomenon can occur in N = 1 theories, and explore some of its consequences. In the next section we will discuss a set of dualities among N = 1 supersymmetric field theories, in which conformal invariance plays a crucial role.