Ph.D. 2016, Yale University
We use the method of conformal bootstrap to systematically study the space of allowed conformal field theories (CFT) in three spacetime dimensions. We consider the crossing symmetry equations coming from the correlators of several lowest dimension operators in a given CFT and show how to setup the semidefinite program to explore the constraints implied by the equations. Constraints lead to general bounds on dimensions and 3-point functions of the operators in CFT. Three classes of CFTs considered in this work are theories containing scalars with Z_2 symmetry, theories containing scalars with O(N) symmetry and theories containing fermions with parity symmetry. While studying the general constraints on such theories, we rediscover several previously known CFTs – Ising model, O(N) vector models and Gross-Neveu-Yukawa models. We determine the properties of the spectrum of low-lying operators in all of these cases. We show that conformal bootstrap can be a very powerful method for precise computation of CFT spectrum. In particular, our result for the dimensions of two lowest lying operators in the Ising model leads to the most precise determination of critical exponents in the model while our results for the 3-point functions in O(N) models lead to most precise determination of the high-frequency conductance in O(N) vector models.