The Conformal Bootstrap: Theory, Numerical Techniques, and Applications

TL;DR

The review surveys the conformal bootstrap as a nonperturbative framework for solving and constraining conformal field theories across dimensions, emphasizing how crossing symmetry, unitarity, and conformal blocks determine the CFT data. It details the theoretical machinery (conformal transformations, OPE, and blocks), numerical methods (linear programming and semidefinite programming), and a broad spectrum of applications in 3d and 4d including Ising, O(N) models, fermionic theories, QED3, and supersymmetric CFTs. Key achievements include precise Ising exponents and the emergence of islands and kinks that isolate physical theories, as well as extensive bounds on operator dimensions and OPE coefficients that illuminate the structure of strongly coupled fixed points. The work highlights both the practical power of bootstrap techniques and the rich set of open questions, such as spectrum rearrangements, symmetry enhancements, and extending results to nonunitary or nonrelativistic CFTs, with significant implications for critical phenomena and beyond-the-Standard-Model physics.

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.

Figures (58)

• Figure 1: Crossing relation for the 4pt function $\langle{\cal O}_1 {\cal O}_2 {\cal O}_3 {\cal O}_4\rangle$.
• Figure 2: (Color online) Conformal frame defining the $z$ coordinate. Figure from Hogervorst:2013sma.
• Figure 3: Positivity of this 6pt function implies reality of the 3pt function coefficient $\lambda_{123}$, see the text.
• Figure 4: (Color online) Conformal frame defining the radial coordinate. Figure from Hogervorst:2013sma.
• Figure 5: Left: A case where vectors can sum to zero with positive coefficients. Right: A case where vectors cannot sum to zero with positive coefficients and there exists a separating plane $\alpha$ such that all vectors point on one side of the plane. Figure from Poland:2016chs.