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A Basis of Analytic Functionals for CFTs in General Dimension

Dalimil Mazac, Leonardo Rastelli, Xinan Zhou

TL;DR

This work advances analytic control of the conformal bootstrap in general dimension by constructing a dual basis of linear functionals for two-variable four-point functions. The authors propose a primal basis formed from s- and t-channel double-trace blocks and their Delta-derivatives, and develop two complementary methods to build the dual functionals, tying the construction to Polyakov-Regge blocks and Witten diagrams. They show the dual functionals yield exact, nonperturbative sum rules and establish deep links to the Carmi–Caron-Huot dispersion relation through a universal generating kernel. The framework paves the way for systematic, nonperturbative constraints on CFT data and provides a bridge between analytic functionals and holographic (Mellin-space) descriptions. Potential applications include perturbative expansions around mean-field theory and nonperturbative bounds across dimensions, with extensions to mixed correlators and spinning operators on the horizon.

Abstract

We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition in the u-channel Regge limit. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.

A Basis of Analytic Functionals for CFTs in General Dimension

TL;DR

This work advances analytic control of the conformal bootstrap in general dimension by constructing a dual basis of linear functionals for two-variable four-point functions. The authors propose a primal basis formed from s- and t-channel double-trace blocks and their Delta-derivatives, and develop two complementary methods to build the dual functionals, tying the construction to Polyakov-Regge blocks and Witten diagrams. They show the dual functionals yield exact, nonperturbative sum rules and establish deep links to the Carmi–Caron-Huot dispersion relation through a universal generating kernel. The framework paves the way for systematic, nonperturbative constraints on CFT data and provides a bridge between analytic functionals and holographic (Mellin-space) descriptions. Potential applications include perturbative expansions around mean-field theory and nonperturbative bounds across dimensions, with extensions to mixed correlators and spinning operators on the horizon.

Abstract

We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition in the u-channel Regge limit. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.

Paper Structure

This paper contains 29 sections, 220 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Warm-up: Single Variable
  3. The space of functions
  4. Dispersion relation and a function basis
  5. The dual basis: the functionals
  6. The Function Space of Four-Point Correlators
  7. Preliminaries
  8. Spaces $\mathcal{U}$ and $\mathcal{V}$
  9. A first attempt at a double-trace expansion
  10. A new look at the double discontinuity
  11. The Primal and Dual Basis
  12. Our proposal
  13. Linear independence of the primal basis
  14. Polyakov-Regge blocks
  15. General external dimensions
  16. ...and 14 more sections

Figures (2)

  • Figure 1: An illustration of the contours $C_\pm$.
  • Figure 2: Contour prescription for the general functional \ref{['eq:FunctionalGeneral']}. The $w$ variable is integrated over contour $C_-$ and $\bar{w}$ over $C_+$. All the dual basis functionals $\alpha^{s}_{n,\ell}$, $\beta^{s}_{n,\ell}$, $\alpha^{t}_{n,\ell}$ and $\beta^{t}_{n,\ell}$ take this form for suitable integral kernel $\mathcal{H}(w,\bar{w})$.