Phases of N = 1 theories in d = 2 + 1 and non-supersymmetric conformal manifolds, or Is there life beyond holomorphy?

In this thesis we mainly review two works, one regarding three-dimensional N = 1 field theories and the other about an interesting structure which may exist in conformal field theories, known as conformal manifolds. After an Introduction in which we put things into context, we discuss in great detail in chapter 2 the dynamics of a special class of three-dimensional theories, that is N = 1 QCD with non-vanishing Chern-Simons level coupled to one adjoint matter multiplet. The important feature of N = 1 supersymmetric theories in 2 + 1 dimensions is that the Witten index can jump on co-dimension one walls in parameter space, where new vacua come from infinity of field space. We demonstrate that this physics is captured by the two-loop effective potential. Together with the decoupling limit at large masses for matter fields, it allows to formulate a robust conjecture regarding the phase diagram of the theory. Another interesting result is the appearance of metastable supersymmetry breaking vacua for sufficiently small values of Chern-Simons level. The third chapter focuses on the constraints that a conformal field theory should enjoy to admit exactly marginal deformations, i.e. to be part of a conformal manifold. While in two spacetime dimensions conformal manifolds are rather common, their existence in d > 2 is absolutely non-trivial. In fact, in absence of supersymmetry, no single example of a conformal manifold is known in d > 2 dimensions. Using tools from conformal perturbation theory, we derive a sum rule from which one can extract restrictions on the spectrum of low spin and low dimension operators. We then focus on conformal field theories admitting a gravity dual description, and as such a large-N expansion. We discuss the relation between conformal perturbation theory and loop expansion in the bulk, and show how such connection could help in the search for conformal manifolds beyond the planar limit. Our results do not rely on supersymmetry, here, and therefore apply also outside the realm of superconformal field theories. Both chapters end with conclusion and outlook sections.