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Distributions in CFT I. Cross-Ratio Space

Petr Kravchuk, Jiaxin Qiao, Slava Rychkov

TL;DR

This paper develops a distributional framework for the conformal block expansion in conformal field theory, proving that four-point functions and conformal blocks converge on the boundary of their Euclidean region as tempered distributions. Using Vladimirov's theorem, the authors show boundary values exist and that the expansion converges term-by-term against suitable test functions, enabling dispersion relations and a functional-analytic view of crossing. They unify the 1D and higher-dimensional scalar cases, derive explicit power-law bounds for correlators, and extend the analytic-functional bootstrap program to boundary functionals. While the current work is grounded in Euclidean bootstrap axioms, it lays groundwork toward Lorentzian Wightman properties and a uniform understanding of crossing via distribution theory.

Abstract

We show that the four-point functions in conformal field theory are defined as distributions on the boundary of the region of convergence of the conformal block expansion. The conformal block expansion converges in the sense of distributions on this boundary, i.e. it can be integrated term by term against appropriate test functions. This can be interpreted as a giving a new class of functionals that satisfy the swapping property when applied to the crossing equation, and we comment on the relation of our construction to other types of functionals. Our language is useful in all considerations involving the boundary of the region of convergence, e.g. for deriving the dispersion relations. We establish our results by elementary methods, relying only on crossing symmetry and the standard convergence properties of the conformal block expansion. This is the first in a series of papers on distributional properties of correlation functions in conformal field theory.

Distributions in CFT I. Cross-Ratio Space

TL;DR

This paper develops a distributional framework for the conformal block expansion in conformal field theory, proving that four-point functions and conformal blocks converge on the boundary of their Euclidean region as tempered distributions. Using Vladimirov's theorem, the authors show boundary values exist and that the expansion converges term-by-term against suitable test functions, enabling dispersion relations and a functional-analytic view of crossing. They unify the 1D and higher-dimensional scalar cases, derive explicit power-law bounds for correlators, and extend the analytic-functional bootstrap program to boundary functionals. While the current work is grounded in Euclidean bootstrap axioms, it lays groundwork toward Lorentzian Wightman properties and a uniform understanding of crossing via distribution theory.

Abstract

We show that the four-point functions in conformal field theory are defined as distributions on the boundary of the region of convergence of the conformal block expansion. The conformal block expansion converges in the sense of distributions on this boundary, i.e. it can be integrated term by term against appropriate test functions. This can be interpreted as a giving a new class of functionals that satisfy the swapping property when applied to the crossing equation, and we comment on the relation of our construction to other types of functionals. Our language is useful in all considerations involving the boundary of the region of convergence, e.g. for deriving the dispersion relations. We establish our results by elementary methods, relying only on crossing symmetry and the standard convergence properties of the conformal block expansion. This is the first in a series of papers on distributional properties of correlation functions in conformal field theory.

Paper Structure

This paper contains 21 sections, 7 theorems, 80 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Conformal block expansion
  3. One-dimensional case
  4. A toy problem
  5. Vladimirov's theorem
  6. Proof of Vladimirov's theorem \ref{['thm:vlad']}
  7. Distributional convergence of conformal block expansion
  8. Convergence for other normalizations and on other boundaries
  9. Analytic functionals
  10. Dispersion relation in cross-ratio space and the discontinuity
  11. Scalar four-point functions in higher dimensions
  12. Conformal block expansion
  13. Bounds on $g(\rho,\overline{\rho})$ and partial sums of the conformal block expansion
  14. Vladimirov's theorem
  15. Analytic functionals
  16. ...and 6 more sections

Key Result

Theorem 3.1

Let $g(\tau)$ be a function holomorphic for $0<\mathrm{Im}\,\tau<a$ for some $a>0$, satisfying the slow-growth condition near $\mathbb{R}$ as defined above. Then the boundary value $\mathrm{bv}\,g$ of $g$ on $\mathbb{R}$ exists in $\mathcal{S}'(\mathbb{R})$. Furthermore, if a sequence of functions $

Figures (7)

  • Figure 1: Regge kinematics.
  • Figure 2: Transformation from the $z$ cut plane to the $\rho$ disk to the $\tau$ upper-half plane, see the text.
  • Figure 3: A numerical check of the existence of the limit \ref{['eq:test-limit']}, for $f(\tau)$ given in the text. The curve is the integral under the limit sign, and the red dots are the partials sums of Fourier coefficients in the r.h.s. of \ref{['eq:test-limit']} up to $n=N$.
  • Figure 4: The setting of theorem \ref{['thm:restrictedboundary']}. We give one particular example of a possible region $S$. In practical applications discussed below $S$ will be either all of $\mathbb{D}\setminus(-1,0]$ or an upper or lower half.
  • Figure 5: The crossing region $\mathcal{C}^{st}$ and its parametrization using the $\rho$-coordinate and the Zhukovsky $y$-coordinate.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3