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EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions

Slava Rychkov

TL;DR

These EPFL lecture notes present a self-contained, nonperturbative route to conformal field theory in $D\geq3$ by building from physical foundations to kinematics, radial quantization, and finally the conformal bootstrap. The embedding/projective null-cone formalism is used to derive the fixed, highly constrained forms of two-, three-, and four-point functions, including spinning operators, cross-ratios, and the role of Ward identities and the stress tensor. Radial quantization provides the state-operator correspondence and unitarity bounds, which underpin the OPE and the CPW decomposition, enabling a convergent expansion of correlators into conformal blocks. The bootstrap program is then introduced, showing how crossing symmetry constrains the CFT data (operator dimensions and OPE coefficients) and, in practice, yields rigorous bounds and islands for theories in $D\ge3$, with notable success in two dimensions and active progress in higher dimensions. The notes sketch the path from kinematics to a nonperturbative, axiomatic definition of CFTs and outline the practical bootstrap machinery that has driven recent advances in critical phenomena and beyond.

Abstract

This is a writeup of lectures given at the EPFL Lausanne in the fall of 2012. The topics covered: physical foundations of conformal symmetry, conformal kinematics, radial quantization and the OPE, and a very basic introduction to conformal bootstrap.

EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions

TL;DR

These EPFL lecture notes present a self-contained, nonperturbative route to conformal field theory in by building from physical foundations to kinematics, radial quantization, and finally the conformal bootstrap. The embedding/projective null-cone formalism is used to derive the fixed, highly constrained forms of two-, three-, and four-point functions, including spinning operators, cross-ratios, and the role of Ward identities and the stress tensor. Radial quantization provides the state-operator correspondence and unitarity bounds, which underpin the OPE and the CPW decomposition, enabling a convergent expansion of correlators into conformal blocks. The bootstrap program is then introduced, showing how crossing symmetry constrains the CFT data (operator dimensions and OPE coefficients) and, in practice, yields rigorous bounds and islands for theories in , with notable success in two dimensions and active progress in higher dimensions. The notes sketch the path from kinematics to a nonperturbative, axiomatic definition of CFTs and outline the practical bootstrap machinery that has driven recent advances in critical phenomena and beyond.

Abstract

This is a writeup of lectures given at the EPFL Lausanne in the fall of 2012. The topics covered: physical foundations of conformal symmetry, conformal kinematics, radial quantization and the OPE, and a very basic introduction to conformal bootstrap.

Paper Structure

This paper contains 41 sections, 274 equations, 18 figures.

Table of Contents

  1. Physical Foundations of Conformal Symmetry
  2. Fixed points
  3. Existing techniques
  4. Towards a nonperturbative definition
  5. Operator spectrum
  6. Stress tensor and currents
  7. Ward Identities
  8. The trace of the stress tensor
  9. Conformal invariance
  10. Transformation rule for operators
  11. Conformal kinematics
  12. Projective null cone
  13. Simple applications
  14. Primary scalar 2pt function
  15. Primary scalar 3pt function
  16. ...and 26 more sections

Figures (18)

  • Figure 1: Red: section, blue: light ray and light ray$\,'$.
  • Figure 2: How the infinitesimal interval transforms under the defined $SO(D+1,1)$ action.
  • Figure 3: $D$-dimensional spacetime foliated by $(D-1)$-dimensional surfaces (leafs).
  • Figure 4: Operator insertions in the past and present create Hilbert space states, whose overlap is equal to the correlation function involving these operators.
  • Figure 5: If we choose to consider states living on different surfaces, then we must insert the evolution operator to evaluate the correlation function.
  • ...and 13 more figures