Solving Conformal Field Theories with the Functional Bootstrap

Introduction

Conformal Field Theories (CFTs) have a wide range of experimental and theoretical applications. They are used for:

• Describing classical and quantum critical phenomena, where they determine critical exponents.
• Acting as low (or high) energy limits of ordinary quantum field theories.
• Serving as theories of quantum gravity in disguise via the AdS/CFT correspondence.

Challenges in Analyzing CFTs

Unfortunately, most interesting CFTs are strongly interacting and difficult to analyze.

On one hand, perturbative and renormalization group methods usually involve approximations that are hard to control and require difficult resummations.

On the other hand, numerical simulations of the underlying systems are challenging near the critical point and can access only a limited set of observables.

The Conformal Bootstrap Program

The conformal bootstrap program is a new approach that exploits basic consistency conditions encoded into a formidable set of bootstrap equations. This program aims to map out and determine the space of CFTs.

A longstanding conjecture states that these equations provide a fully non-perturbative definition of CFTs.

Project Goals

In this project, we will develop a groundbreaking set of tools—analytic extremal functionals—to extract information from the bootstrap equations. This Functional Bootstrap has the potential to greatly deepen our understanding of CFTs and determine incredibly precise bounds on the space of theories.

Our main goals are:

1. To fully develop the functional bootstrap for the simpler and mostly unexplored one-dimensional setting, relevant for critical systems such as spin models with long-range interactions and line defects in conformal gauge theories. This will lead to analytic insights and effective numerical solutions of these systems.

2. To establish functionals as the default technique for higher-dimensional applications by developing the formalism, obtaining general analytic bounds, and integrating into existing numerical workflows to achieve highly accurate determinations of critical exponents.