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A tauberian theorem for the conformal bootstrap

Jiaxin Qiao, Slava Rychkov

TL;DR

The paper provides a rigorous bridge between crossing symmetry in 1d CFTs and the high‑dimensional tail of the exchanged spectrum by proving a tauberian theorem for conformal bootstrap. By replacing conformal blocks with their Bessel‑function asymptotics and recasting the problem as a weighted spectral density average, the authors derive precise power‑law growth for the integrated spectrum with exponent γ = 4Δφ, subject to a unit‑operator gap. They give a self‑contained Fourier‑analytic proof, show two implications between carefully chosen weight functions, and extend the framework to unequal external dimensions and large‑N gauge theories, thereby grounding earlier lightcone bootstrap intuitions. The work also clarifies the role of positivity, the Hardy–Littlewood tauberian connection, and sets the stage for extending these ideas to higher dimensions via analytic structure and inversion formulas. Overall, it provides a robust mathematical foundation for extracting high‑dimension CFT data from crossing, with implications for both analytic bootstrap and numerical bounds.

Abstract

For expansions in one-dimensional conformal blocks, we provide a rigorous link between the asymptotics of the spectral density of exchanged primaries and the leading singularity in the crossed channel. Our result has a direct application to systems of SL(2,R)-invariant correlators (also known as 1d CFTs). It also puts on solid ground a part of the lightcone bootstrap analysis of the spectrum of operators of high spin and bounded twist in CFTs in d>2. In addition, a similar argument controls the spectral density asymptotics in large N gauge theories.

A tauberian theorem for the conformal bootstrap

TL;DR

The paper provides a rigorous bridge between crossing symmetry in 1d CFTs and the high‑dimensional tail of the exchanged spectrum by proving a tauberian theorem for conformal bootstrap. By replacing conformal blocks with their Bessel‑function asymptotics and recasting the problem as a weighted spectral density average, the authors derive precise power‑law growth for the integrated spectrum with exponent γ = 4Δφ, subject to a unit‑operator gap. They give a self‑contained Fourier‑analytic proof, show two implications between carefully chosen weight functions, and extend the framework to unequal external dimensions and large‑N gauge theories, thereby grounding earlier lightcone bootstrap intuitions. The work also clarifies the role of positivity, the Hardy–Littlewood tauberian connection, and sets the stage for extending these ideas to higher dimensions via analytic structure and inversion formulas. Overall, it provides a robust mathematical foundation for extracting high‑dimension CFT data from crossing, with implications for both analytic bootstrap and numerical bounds.

Abstract

For expansions in one-dimensional conformal blocks, we provide a rigorous link between the asymptotics of the spectral density of exchanged primaries and the leading singularity in the crossed channel. Our result has a direct application to systems of SL(2,R)-invariant correlators (also known as 1d CFTs). It also puts on solid ground a part of the lightcone bootstrap analysis of the spectrum of operators of high spin and bounded twist in CFTs in d>2. In addition, a similar argument controls the spectral density asymptotics in large N gauge theories.

Paper Structure

This paper contains 23 sections, 148 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Formulation of the problem and the main result
  3. From conformal blocks to the Bessel function
  4. Large $\Delta$
  5. Small $x$
  6. Reduction to the tauberian theorem
  7. Proof of the tauberian theorem
  8. Proof that (\ref{['W2']}) implies (\ref{['W1']})
  9. Proof that (\ref{['W1']}) implies (\ref{['W2']})
  10. Construction of functions $W_2^{\pm}$
  11. Comments and extensions
  12. Nonvanishing Fourier transform
  13. Hardy-Littlewood tauberian theorem
  14. Positivity of the spectral density
  15. Generalization to unequal external dimensions
  16. ...and 8 more sections

Figures (7)

  • Figure 1: Could $Q(Y)$ oscillate between two different powerlaw asymptotics? We will show that such a behavior is impossible.
  • Figure 2: The weight functions (\ref{['W1W2']}) (for $\gamma=1$).
  • Figure 3: An intuitive illustration, see the text. Notice that $\rho(x)$ vanishes for $x<\log \Delta_0$.
  • Figure 4: Approximating the function $W_2$ from above and below by functions $W_2^\pm$.
  • Figure 5: The integrated spectral density $\int_0^Y d\Delta\,r(\Delta)$ corresponding to the series (\ref{['ex']}).
  • ...and 2 more figures