Explorations in the Conformal Bootstrap

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