Conformal symmetry imposes very strong constraints on quantum field theories. In two dimensions, the conformal symmetry algebra is infinite-dimensional, and two-dimensional conformal field theories can be completely solvable, in the sense that all their correlation functions may be computed. These theories have an ample range of applications, from string theory to critical phenomena in statistical physics, and they have been widely studied during the last decades.In this thesis we study two-dimensional conformal field theories with Virasoro algebra symmetry, following the conformal bootstrap approach. Under the assumption that degenerate fields exist, we provide an extension of the analytic conformal bootstrap method to theories with non-diagonal spectrums. We write the equations that determine structure constants, and find explicit solutions in terms of special functions. We validate this results by numerically computing four-point functions in diagonal and non-diagonal minimal models, and verifying that crossing symmetry is satisfied.In addition, we build a proposal for a family of non-diagonal, non-rational conformal field theories for any central charges such that Re(c)

In this thesis, we study strongly interacting superconformal field theories in two and three dimensions. In two dimensions, we investigate N = (0, 2) gauge theories using the gauged linear sigma models (GLSM). In those theories, we identify simple mechanism by which worldsheet description of H-flux satisfying Green-Schwarz Bianchi identity arises. Under the renormalization group flow, we argue that these models flow into superconformal fixed points describing string theory compactifications backgrounds with non-trivial H-flux turned on. By analyzing quantum-consistency of effective theories with such mechanism, we identify conditions under which these theories to become interacting superconformal field theories in the infrared. In three dimensions, we study maximally supersymmetric (N = 8) conformal field theories by conformal bootstrap approach. We focus on studying the four-point function of stress-tensor multiplet. The superconformal blocks for the four-point function are computed by analyzing superconformal Ward identity. Using these blocks, we study crossing symmetry constraints both numerically and analytically. Doing so, we obtain universal bounds and exact relations of N = 8 superconformal field theory data.

In this dissertation, we study bootstrap constraints on conformal field theories in two dimensions.

"In this thesis, we use three different sets of techniques to study the spectrum of Conformal Field Theories. We start by computing the conformal blocks for scalar thermal one-point functions in general dimensions. We achieve this by using three different methods: direct calculation, Casimir differential equation and Witten diagrams. These blocks are then used to find an asymptotic formula for the OPE coefficients of primary operators when two of them are heavy. The next part of the thesis uses the Lorentzian inversion formula to understand the low spin spectrum of specific CFTs. We look at the 3d Ising model and specifically describe how to obtain information about the $[\sigma\epsilon]_0$ Regge trajectory. We additionally use the same techniques to describe the low spin data of the critical $O(N)$ model at large $N$ in three dimensions. The third part of the thesis focusses on CFTs in two dimensions. We develop a new formulation for the conformal bootstrap by using the Virasoro fusion kernel. The identity kernel allows us to define Virasoro Mean Field Theory, which is the spectrum necessary in one channel to recover the identity in the other. We then use the new formulation of the crossing equation to show that this VMFT universally describes the large spin part of the CFT spectrum and we compute corrections to this universal behaviour"--

We investigate properties of various conformally invariant quantum systems, especially from the point of view of the conformal bootstrap. First, we study twist line defects in three-dimensional conformal field theories. Numerical results from lattice simulations point to the existence of such conformal defect in the critical 3D Ising model. We show that this fact is supported by both epsilon expansion and the conformal bootstrap calculations. We find that our results are in a good agreement with the numerical data. We also make new predictions for operator dimensions and OPE coefficients from the bootstrap approach. In the process we derive universal bounds on one-dimensional conformal field theories and conformal line defects. Second, we analyze the constraints imposed by the conformal bootstrap for theories with four supercharges in spacetime dimension between 2 and 4. We show how superconformal algebras with four Poincaré supercharges can be treated in a formalism applicable to any, in principle continuous, value of d and use this to construct the superconformal blocks for any dimension between 2 and 4. We then use numerical bootstrap techniques to derive upper bounds on the conformal dimension of the first unprotected operator appearing in the OPE of a chiral and an anti-chiral superconformal primary. We obtain an intriguing structure of three distinct kinks. We argue that one of the kinks smoothly interpolates between the d=2, N=(2, 2) minimal model with central charge c=1 and the theory of a free chiral multiplet in d=4, passing through the critical Wess-Zumino model with cubic superpotential in intermediate dimensions. Finally, we turn to the question of the analytic origin of the conformal bootstrap bounds. To this end, we introduce a new class of linear functionals acting on the conformal bootstrap equation. In 1D, we use the new basis to construct extremal functionals leading to the optimal upper bound on the gap above identity in the OPE of two identical primary operators of integer or half-integer scaling dimension. We also prove an upper bound on the twist gap in 2D theories with global conformal symmetry. When the external scaling dimensions are large, our functionals provide a direct point of contact between crossing in a 1D CFT and scattering of massive particles in large AdS. In particular, CFT crossing can be shown to imply that appropriate OPE coefficients exhibit an exponential suppression characteristic of massive bound states, and that the 2D flat-space S-matrix should be analytic away from the real axis.

We study conformal defects in two-dimensional conformal field theories (CFTs). These are one-dimensional objects across which the difference between the holomorphic and antiholomorphic parts of the stress-energy tensor is continuous. Such defects may exist within a CFT as well as between two different CFTs. There are two subclasses of conformal defects that are well-known: topological defects, which preserve the holomorphic and antiholomorphic parts of the stress-energy tensor separately, and factorising defects, which can be considered as products of conformal boundary conditions separating the theory along the defect. In this thesis, we call conformal defects, which do not fall into either of the aforementioned subclasses, non-trivial conformal defects. The primary focus of this thesis is studying the non-trivial conformal defect present in a unitary Virasoro minimal model which was first predicted by Kormos, Runkel, and Watts [98]. As a first step, we calculate the reflection and transmission coefficients, which were first defined in [91], of these defects using the leading-order perturbative calculation. We then consider conformal defects in the tri-critical Ising model as a concrete example. We revisit the construction of super-conformal defects proposed by Gang and Yamaguchi [94] and give a more systematic construction of such defects using super W-algebras. In addition, we propose a topological interface separating the super-conformal and bosonic theories, from which conformal defects in the latter theory can be obtained from the former one. Using the topological interfaces and superconformal defects, we obtain non-topological and non-factorising defects in the bosonic tri-critical Ising model.