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Conformal bootstrap: from Polyakov to our times

Slava Rychkov

TL;DR

The paper chronicles the conformal bootstrap’s trajectory from its 1960s–70s origins in critical phenomena and strong interactions to the modern numerical bootstrap that yielded precise 3D Ising exponents, highlighting the unifying role of conformal invariance across disciplines. It weaves together historical developments (old bootstrap, Rome and Sofia contributions, the BPZ 2D CFT) with technical advances (Dolan–Osborn blocks, extremal functional method, semidefinite programming) and showcases the decisive impact of numerical methods on constraining CFT data. The authors emphasize that conformal symmetry provides a powerful, non-perturbative framework, evidenced by the agreement between bootstrap results, RG analyses, and experiments for critical phenomena, and point to future challenges including uniqueness proofs, nonexistence results, and extending bootstrap to 3D gauge theories using analytic functionals. Overall, the work illustrates how the bootstrap embodies a deep unity of physics and continues to drive forward our understanding of conformal field theories in diverse dimensions.

Abstract

We trace the history of conformal bootstrap from its early days to our times - a great example of unity of physics. We start by describing little-known details about the origins of conformal field theory in the study of strong interactions and critical phenomena in the 1960s and 1970s. We describe similarities and differences between approaches and results of the main groups in Moscow, Rome, and Sofia. Then come the breakthroughs in the 1980s and the 1990s, in particular 2D CFT and holography. Finally, we describe the genesis of the numerical conformal bootstrap, from the conformal technicolor bounds in the 2000s, to the determination of the 3D Ising critical exponents in the 2010s. We conclude with some outstanding challenges. We stress that conformal invariance is a symmetry of nature.

Conformal bootstrap: from Polyakov to our times

TL;DR

The paper chronicles the conformal bootstrap’s trajectory from its 1960s–70s origins in critical phenomena and strong interactions to the modern numerical bootstrap that yielded precise 3D Ising exponents, highlighting the unifying role of conformal invariance across disciplines. It weaves together historical developments (old bootstrap, Rome and Sofia contributions, the BPZ 2D CFT) with technical advances (Dolan–Osborn blocks, extremal functional method, semidefinite programming) and showcases the decisive impact of numerical methods on constraining CFT data. The authors emphasize that conformal symmetry provides a powerful, non-perturbative framework, evidenced by the agreement between bootstrap results, RG analyses, and experiments for critical phenomena, and point to future challenges including uniqueness proofs, nonexistence results, and extending bootstrap to 3D gauge theories using analytic functionals. Overall, the work illustrates how the bootstrap embodies a deep unity of physics and continues to drive forward our understanding of conformal field theories in diverse dimensions.

Abstract

We trace the history of conformal bootstrap from its early days to our times - a great example of unity of physics. We start by describing little-known details about the origins of conformal field theory in the study of strong interactions and critical phenomena in the 1960s and 1970s. We describe similarities and differences between approaches and results of the main groups in Moscow, Rome, and Sofia. Then come the breakthroughs in the 1980s and the 1990s, in particular 2D CFT and holography. Finally, we describe the genesis of the numerical conformal bootstrap, from the conformal technicolor bounds in the 2000s, to the determination of the 3D Ising critical exponents in the 2010s. We conclude with some outstanding challenges. We stress that conformal invariance is a symmetry of nature.

Paper Structure

This paper contains 20 sections, 20 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. 1970 - Migdal and Polyakov meet Kastrup in Kyiv
  3. 1970 - Polyakov argues that critical phenomena are conformally invariant
  4. In search of a dynamical principle - "old bootstrap"
  5. 1974 - Polyakov's "Non-Hamiltonian approach" - the birth of modern conformal bootstrap
  6. The Rome group
  7. Mack and the Sofia group
  8. 1984 - Belavin, Polyakov, Zamolodchikov
  9. 1980s and 1990s developments in CFTs in $d>2$ dimensions
  10. 2006 - Conformal technicolor and the numerical conformal bootstrap
  11. Personal background
  12. Conformal Technicolor
  13. Numerical conformal bootstrap
  14. 2011-2014 - Attack on the 3D Ising Model
  15. 2025 - Selected challenges for the conformal bootstrap
  16. ...and 5 more sections

Figures (10)

  • Figure 1: The bound from Rattazzi:2008pe. Figure adapted from Rychkov:2009ij where this bound was further improved.
  • Figure 2: The $d=2$ bound from Rychkov:2009ij, with a kink at the 2D Ising CFT.
  • Figure 3: The bound on the second scalar ($\epsilon'$) with the first one ($\epsilon$) at the gap Rychkov:2011et.
  • Figure 4: The very first 3D bound, with the red dot at the 3D Ising CFT location expected from the $\epsilon$-expansion. Email by Alessandro Vichi, November 11, 2011.
  • Figure 5: The 3D bound from the paper El-Showk:2012cjh.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Remark 3.1