Conformal Field Theories and the Analytic Bootstrap

Abstract

The study of Conformal Field Theories (CFT's) is among the most important subjects in modern theoretical physics. CFT's describe critical phenomena, leading to applications in a diverse array of physical systems. From the perspective of the Renormalization Group (RG), CFT's play a fundamental role in understanding the more general space of Quantum Field Theories. The AdS/CFT correspondence makes CFT's relevant in the study of quantum gravity. CFT's have applications in many other fields including string theory, holography and physics beyond the standard model.

Despite their importance, CFT's are relatively difficult to solve mathematically. In recent years, great progress has been made by the conformal bootstrap, which uses symmetry to constrain the space of allowed CFT's. Much of the bootstrap's success has come from numerical techniques. However, numerical methods are computationally intensive, limiting what can be learned about a theory. For this reason, great interest has arisen in analytic bootstrap techniques. In this thesis, we focus on a particular analytic bootstrap technique, known as the Lorentzian inversion formula.

Discovered by Caron-Huot in 2017, the inversion formula computes s-channel CFT data by summing over t and u channel conformal blocks. Despite its potential, the inversion formula has found little application in the literature. We develop the mathematical formalism of the inversion formula, presenting formulae for the anomalous dimension and OPE coefficient corrections. We apply our results to the 3D Ising and 3D O(N) CFT's, focusing on the spectrum of double twist operators. For O(N) models, we consider N=2,3. Our results provide the best known analytic predictions for both cases. For O(2) and O(3), no numerical data exists for the double twist spectrum, so our results give new predictions for each of these theories! We conclude by offering thoughts on how to extend our results to other theories of interest.