Analytic bootstrap for perturbative conformal field theories

Thesis

Conformal field theories (CFTs) play a central role in theoretical physics with many applications ranging from condensed matter to string theory. The conformal bootstrap studies conformal field theories using mathematical consistency conditions and has seen great progress over the last decade. In this thesis we present an implementation of analytic bootstrap methods for perturbative conformal field theories in dimensions greater than two, which we achieve by combining large spin perturbation theory with the Lorentzian inversion formula. In the presence of a small expansion parameter, not necessarily the coupling constant, we develop this into a systematic framework, applicable to a wide range of theories.

The first two chapters provide the necessary background and a review of the analytic bootstrap. This is followed by a chapter which describes the method in detail, taking the form of a practical guide to large spin perturbation theory by means of a step-by-step implementation. The goal is to compute the CFT-data that define a given conformal field theory, and this is achieved by considering contributions from operators in a four-point correlator through the crossing equation. We give a general recipe for determining which operators to consider, how to find their contributions from conformal blocks and how to compute the corresponding CFT-data through the inversion formula.

The second part of the thesis presents several explicit implementations of the framework, taking examples from a number of well-studied conformal field theories. We show how many literature results can be reproduced from a purely bootstrap perspective and how a variety of new results can be derived. We consider in depth how to determine the CFT-data in the ε expansion for the Wilson-Fisher model from crossed-channel operators. All CFT-data to order ε follow from only the identity and the bilinear scalar operator, and by considering contributions from two infinite families of operators we generate new results at order ε⁴. We study in similar depth conformal gauge theories in four dimensions, where we find a five-parameter solution for the most general form of the one-loop four-point correlator of bilinear scalars. For particular parameter values this reproduces the case of the Konishi operator and the stress tensor multiplet in weakly coupled Ν=4 super Yang-Mills theory. We then present more briefly four additional examples. These include the critical O(N) model in a large N expansion, a solution for φ⁴ theory with any global symmetry, multicritical theories to order ε² near their critical dimensions, including new results for the central charge, and the four-point correlator of bilinear scalars in the ε expansion. We conclude the thesis with a discussion and some appendices.

Authors: Henriksson, J

• DOI: 10.5287/ora-pvrgbkxqj
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Conformal bootstrap; Conformal field theory
• Subjects: Mathematical physics