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

Phases of N=1{\cal N}=1 Theories in 2+1 Dimensions

We study the dynamics of 2+1 dimensional theories with ${\cal N}=1$ supersymmetry. In these theories the supersymmetric ground states behave discontinuously at co-dimension one walls in the space of couplings, with new vacua coming in from infinity in field space. We show that the dynamics near these walls is calculable: the two-loop effective potential yields exact results about the ground states near the walls. Far away from the walls the ground states can be inferred by decoupling arguments. In this way, we are able to follow the ground states of ${\cal N}=1$ theories in 2+1 dimensions and construct the infrared phases of these theories. We study two examples in detail: Adjoint SQCD and SQCD with one fundamental quark. In Adjoint QCD we show that for sufficiently small Chern-Simons level the theory has a non-perturbative metastable supersymmetry-breaking ground state. We also briefly discuss the critical points of this theory. For SQCD with one quark we establish an infrared duality between a $U(N)$ gauge theory and an $SU(N)$ gauge theory. The duality crucially involves the vacua that appear from infinity near the walls.Comment: 53 page

Living on the walls of super-QCD

We study BPS domain walls in four-dimensional N = 1 massive SQCD with gauge group SU (N) and F < N flavors. We propose a class of three-dimensional Chern-Simons-matter theories to describe the effective dynamics on the walls. Our proposal passes several checks, including the exact matching between its vacua and the solutions to the four-dimensional BPS domain wall equations, that we solve in the small mass regime. As the flavor mass is varied, domain walls undergo a second-order phase transition, where multiple vacua coalesce into a single one. For special values of the parameters, the phase transition exhibits supersymmetry enhancement. Our proposal includes and extends previous results in the literature, providing a complete picture of BPS domain walls for F < N massive SQCD. A similar picture holds also for SQCD with gauge group Sp (N) and F < N + 1 flavors

Scalar multiplet recombination at large N and holography

We consider the coupling of a free scalar to a single-trace operator of a large N CFT in d dimensions. This is equivalent to a double-trace deformation coupling two primary operators of the CFT, in the limit when one of the two saturates the unitarity bound. At leading order, the RG-flow has a non-trivial fixed point where multiplets recombine. We show this phenomenon in field theory, and provide the holographic dual description. Free scalars correspond to singleton representations of the AdS algebra. The double-trace interaction is mapped to a boundary condition mixing the singleton with the bulk field dual to the single-trace operator. In the IR, the singleton and the bulk scalar merge, providing just one long representation of the AdS algebra. \ua9 2016, The Author(s)

On non-supersymmetric conformal manifolds: field theory and holography

We discuss the constraints that a conformal field theory should enjoy to admit exactly marginal deformations, i.e. to be part of a conformal manifold. In particular, using tools from conformal perturbation theory, we derive a sum rule from which one can extract restrictions on the spectrum of low spin operators and on the behavior of OPE coefficients involving nearly marginal operators. We then consider 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, and therefore apply also outside the realm of superconformal field theories

Phases of N=1 quivers in 2+1 dimensions

We consider the IR phases of two-node quiver theories with N = 1 supersymmetry in d = 2 + 1 dimensions. It turns out that the discussion splits into two main cases, depending on whether the Chern-Simons levels associated with the two nodes have the same sign, or the opposite signs, with the latter case being more non-trivial. The determination of the phase diagrams allows us to conjecture certain infrared dualities involving either two quiver theories, or a quiver and adjoint QCD. We also provide a short discussion on quivers possessing time reversal symmetry

Broken current anomalous dimensions, conformal manifolds, and renormalization group flows

We consider deformations of a conformal field theory that explicitly break some global symmetries of the theory. If the deformed theory is still a conformal field theory, one can exploit the constraints put by conformal symmetry to compute broken currents anomalous dimensions. We consider several instances of this scenario, using field theory techniques and also holographic ones, where necessary. Field theoretical methods suffice to discuss examples of symmetry-breaking deformations of the O(N) model in d = 4-epsilon dimensions. Holography is instrumental, instead, for computing current anomalous dimensions in beta-deformed superconformal field theories and in a class of supersymmetric renormalization group flows at large N. \ua9 2017 American Physical Societ

On the 6d origin of non-invertible symmetries in 4d

Abstract It is well-known that six-dimensional superconformal field theories can be exploited to unravel interesting features of lower-dimensional theories obtained via compactifications. In this short note we discuss a new application of 6d (2,0) theories in constructing 4d theories with Kramers-Wannier-like non-invertible symmetries. Our methods allow to recover previously known results, as well as to exhibit infinitely many new examples of four dimensional theories with “M-ality” defects (arising from operations of order M generalizing dualities). In particular, we obtain examples of order M = p k , where p > 1 is a prime number and k is a positive integer