The Conformal Bootstrap: Theory, Numerical Techniques, and Applications (1805.04405v3)

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.

• The paper's main contribution is its comprehensive review of conformal bootstrap theory and numerical techniques for solving CFT constraints.
• It explains the use of convex optimization methods like linear and semidefinite programming to efficiently compute conformal blocks and extract operator spectra.
• Applications demonstrated include precise predictions in 3D models, such as the Ising model and O(N) systems, with implications for higher-dimensional theories.

Insights into "The Conformal Bootstrap: Theory, Numerical Techniques, and Applications"

The paper "The Conformal Bootstrap: Theory, Numerical Techniques, and Applications" by David Poland, Slava Rychkov, and Alessandro Vichi provides a comprehensive review of the conformal bootstrap technique and its applications in theoretical physics. Conformal field theories (CFTs) are pivotal for understanding critical phenomena and quantum field theories. The conformal bootstrap method, which utilizes symmetries and consistency conditions, has advanced significantly, particularly in three dimensions and beyond.

Key Insights and Results

1. Conformal Symmetry and the Bootstrap: The conformal bootstrap leverages the feature of conformal symmetry in CFTs. This symmetry, which extends scale invariance, imposes strong constraints on correlation functions, uniquely fixing two-point and three-point functions up to a few constants and allowing the construction of four-point functions through conformal blocks.
2. Numerical Techniques: The paper discusses several numerical methodologies for solving the bootstrap equations:
   • Linear Programming and Semidefinite Programming (SDP): These techniques transform the problem of checking whether a set of CFT data is consistent with crossing symmetry into a convex optimization problem.
   • Rational Approximations: Instead of computing conformal blocks exactly, which is often impractical, rational approximations provide efficient means to compute these blocks numerically.
   • Spectrum Extraction: By analyzing the solutions on the boundary of allowed regions, one can infer the spectrum and OPE coefficients of operators, providing insights into the CFT's dynamics.
3. Application in 3D CFTs: The numerical bootstrap is particularly prominent in three dimensions, yielding precise predictions for critical phenomena:
   • Ising Model: Results for the 3D Ising model demonstrate the bootstrap's power by determining the critical exponents with unprecedented precision.
   • $O(N)$ Models: The method extends to $O(N)$ symmetric models, elucidating various universality classes and guiding experimental verifications.
4. 4D CFTs and Beyond: While many results pertain to 3D, the bootstrap also constrains CFTs in higher dimensions:
   • 4D Gauge Theories: Investigations into QCD-like theories in four dimensions reveal potential constraints on the conformal window and implications for particle physics.
   • Beyond Standard Model Physics: The conformal bootstrap provides tests for scenarios addressing the hierarchy problem, exploring whether scalar dimensions exceed certain thresholds that would support new physics.
5. Supersymmetric CFTs: The presence of supersymmetry introduces additional structures and constraints, enabling more precise predictions and opening avenues for exploring non-Lagrangian theories.

Theoretical and Practical Implications

The theoretical implications of this paper center around its contribution to our understanding of critical points in statistical mechanics and their counterparts in high-energy physics. Practically, the results from bootstrap studies serve as benchmarks for both perturbative and lattice calculations, offering independent tests of theoretical models.

Future Directions

The paper highlights several avenues for future exploration:

• Further refinement of numerical techniques and their application to increasingly complex CFTs.
• Application to CFTs with extended supersymmetry and exploration of their non-trivial dualities.
• Investigation of the connection between bootstrap results and integrability in two-dimensional systems.

In summary, "The Conformal Bootstrap: Theory, Numerical Techniques, and Applications" showcases the power of the conformal bootstrap in addressing fundamental problems in theoretical physics. The potential for discovering new theories through this technique suggests exciting developments in the coming years.