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TASI Lectures on the Conformal Bootstrap

David Simmons-Duffin

TL;DR

This work surveys the conformal bootstrap program, deriving CFT structure from symmetry, unitarity, and the OPE, and then implementing numerical bootstrap to bound CFT data. It builds from basic QFT and conformal symmetry through primaries/descendants, radial quantization, and conformal blocks to the crossing equations that constrain the full CFT data set. The notes highlight both analytic results (Ward identities, unitarity bounds, OPE, blocks) and practical numerical approaches (linear and semidefinite programming) that yield precise bounds and islands, notably recovering known models like the 2d and 3d Ising CFTs. The methodology demonstrates a nonperturbative, Lagrangian-free route to characterizing critical phenomena and suggests future directions for analytic insights and more complex correlator analyses.

Abstract

These notes are from courses given at TASI and the Advanced Strings School in summer 2015. Starting from principles of quantum field theory and the assumption of a traceless stress tensor, we develop the basics of conformal field theory, including conformal Ward identities, radial quantization, reflection positivity, the operator product expansion, and conformal blocks. We end with an introduction to numerical bootstrap methods, focusing on the 2d and 3d Ising models.

TASI Lectures on the Conformal Bootstrap

TL;DR

This work surveys the conformal bootstrap program, deriving CFT structure from symmetry, unitarity, and the OPE, and then implementing numerical bootstrap to bound CFT data. It builds from basic QFT and conformal symmetry through primaries/descendants, radial quantization, and conformal blocks to the crossing equations that constrain the full CFT data set. The notes highlight both analytic results (Ward identities, unitarity bounds, OPE, blocks) and practical numerical approaches (linear and semidefinite programming) that yield precise bounds and islands, notably recovering known models like the 2d and 3d Ising CFTs. The methodology demonstrates a nonperturbative, Lagrangian-free route to characterizing critical phenomena and suggests future directions for analytic insights and more complex correlator analyses.

Abstract

These notes are from courses given at TASI and the Advanced Strings School in summer 2015. Starting from principles of quantum field theory and the assumption of a traceless stress tensor, we develop the basics of conformal field theory, including conformal Ward identities, radial quantization, reflection positivity, the operator product expansion, and conformal blocks. We end with an introduction to numerical bootstrap methods, focusing on the 2d and 3d Ising models.

Paper Structure

This paper contains 52 sections, 201 equations, 30 figures, 1 table, 1 algorithm.

Table of Contents

  1. TASI Lectures on the Conformal Bootstrap
  2. Introduction
  3. Landmarks in the Space of QFTs
  4. Critical Universality
  5. The Bootstrap Philosophy
  6. QFT Basics
  7. The Stress Tensor
  8. Quantization
  9. Topological Operators and Symmetries
  10. More Symmetries
  11. Conformal Symmetry
  12. Finite Conformal Transformations
  13. The Conformal Algebra
  14. Primaries and Descendants
  15. Poincare Representations
  16. ...and 37 more sections

Figures (30)

  • Figure 1: Many microscopic theories can flow to the same IR CFT. We say that the theories are IR equivalent, or IR dual. The UV can even be something exotic like a stack of M5-branes in M-theory.
  • Figure 2: A surface $\Sigma$ supporting the operator $P^\mu(\Sigma)$ can be freely deformed $\Sigma\to\Sigma'$ without changing the correlation function, as long as it doesn't cross any operator insertions.
  • Figure 3: Surrounding $\mathcal{O}(x)$ with $P^\mu$ gives a derivative.
  • Figure 4: In a rotationally invariant Euclidean theory, we can choose any direction as time. States live on slices orthogonal to the time direction.
  • Figure 5: The charge $P^\mu(\Sigma_t)$ can be moved from one time to another $t\to t'$ without changing the correlation function.
  • ...and 25 more figures

Theorems & Definitions (1)

  • Claim 1