The conformal bootstrap: Theory, numerical techniques, and applications

TLDR

Conformal field theories describe universal physics at scale‑invariant critical points, modeling continuous phase transitions in fluids, magnets, and other materials, and underpinning modern quantum field theory. The article aims to review recent nonperturbative advances in the conformal bootstrap, a method to study strongly coupled theories using symmetries and consistency conditions. It introduces conformal field theories and conformal blocks, explains numerical bootstrap techniques based on convex optimization, and details their application to fixed points in three and four dimensions with minimal or no supersymmetry. These advances have produced record‑precision determinations of critical exponents and correlation functions for the Ising and O(N) models in three dimensions.

Abstract

Conformal field theories have been long known to describe the fascinating universal physics of scale invariant critical points. They describe continuous phase transitions in fluids, magnets, and numerous other materials, while at the same time sit at the heart of our modern understanding of quantum field theory. For decades it has been a dream to study these intricate strongly coupled theories nonperturbatively using symmetries and other consistency conditions. This idea, called the conformal bootstrap, saw some successes in two dimensions but it is only in the last ten years that it has been fully realized in three, four, and other dimensions of interest. This renaissance has been possible both due to significant analytical progress in understanding how to set up the bootstrap equations and the development of numerical techniques for finding or constraining their solutions. These developments have led to a number of groundbreaking results, including world record determinations of critical exponents and correlation function coefficients in the Ising and $O(N)$ models in three dimensions. This article will review these exciting developments for newcomers to the bootstrap, giving an introduction to conformal field theories and the theory of conformal blocks, describing numerical techniques for the bootstrap based on convex optimization, and summarizing in detail their applications to fixed points in three and four dimensions with no or minimal supersymmetry.