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Celestial OPE blocks

Alfredo Guevara

TL;DR

The paper develops celestial OPE blocks within CCFTs, linking two- and three-point data to the OPE of massless primaries into massive exchanges in Euclidean signature and then analytically continuing to Lorentzian signature on the 1+1 cylinder. It shows the blocks encode contributions from massive primaries, shadows, and light-ray transforms, with kernels fixed by Poincaré invariance and three-point data, and reveals a KLT-like relation between Euclidean and Lorentzian blocks. The Lorentzian block naturally hosts light-ray operators, and the analysis connects to recent four-point results, clarifying how a complete OPE in CCFT should include primaries, shadows, and light-ray states. The work highlights implications for the structure of CCFT, potential unitary representations on the celestial torus, and future extensions to higher-point functions and gauge/gravity sectors.

Abstract

Starting from the defining two-point and three-point functions of Celestial CFTs, Euclidean integral blocks are constructed for the OPE of scalar primaries. In their integral form they can alternatively be fixed using Poincaré symmetry acting on both massless and massive states. Subsequently, an analytic continuation is done to define the Lorentzian version of the correlation functions and the OPE blocks as valued on the $(1,1)$ cylinder, the universal cover of the recently studied celestial torus. The continuation is essentially the same that is used in the derivation of the KLT relations for string amplitudes. It is shown explicitly that the continued OPE blocks encode the contributions from massive primary states as well as their shadow and light-transformed partners of continuous spin. The corresponding pairings are also studied, thus the construction provides the fundamental relation between correlation functions and OPE coefficients in the scalar CCFT.

Celestial OPE blocks

TL;DR

The paper develops celestial OPE blocks within CCFTs, linking two- and three-point data to the OPE of massless primaries into massive exchanges in Euclidean signature and then analytically continuing to Lorentzian signature on the 1+1 cylinder. It shows the blocks encode contributions from massive primaries, shadows, and light-ray transforms, with kernels fixed by Poincaré invariance and three-point data, and reveals a KLT-like relation between Euclidean and Lorentzian blocks. The Lorentzian block naturally hosts light-ray operators, and the analysis connects to recent four-point results, clarifying how a complete OPE in CCFT should include primaries, shadows, and light-ray states. The work highlights implications for the structure of CCFT, potential unitary representations on the celestial torus, and future extensions to higher-point functions and gauge/gravity sectors.

Abstract

Starting from the defining two-point and three-point functions of Celestial CFTs, Euclidean integral blocks are constructed for the OPE of scalar primaries. In their integral form they can alternatively be fixed using Poincaré symmetry acting on both massless and massive states. Subsequently, an analytic continuation is done to define the Lorentzian version of the correlation functions and the OPE blocks as valued on the cylinder, the universal cover of the recently studied celestial torus. The continuation is essentially the same that is used in the derivation of the KLT relations for string amplitudes. It is shown explicitly that the continued OPE blocks encode the contributions from massive primary states as well as their shadow and light-transformed partners of continuous spin. The corresponding pairings are also studied, thus the construction provides the fundamental relation between correlation functions and OPE coefficients in the scalar CCFT.

Paper Structure

This paper contains 11 sections, 84 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Euclidean block in CCFT
  3. Singling out contributions in the OPE block
  4. Analytic continuation and Poincare patches
  5. Light-ray transform
  6. Lorentzian block in CCFT
  7. Singling out the light-ray operators
  8. Discussion
  9. Relation between primaries and their shadow transforms
  10. Distinct Weights and Poincare Symmetry
  11. Pairings involving light-ray operators

Figures (2)

  • Figure 1: The Lorentzian cylinder and a colored Poincare patch in the real $z,\bar{z}$ slice. Dashed lines are identified, see Kravchuk:2018htv for details on the construction. Further identifying the dotted lines leads to the Lorentzian torus.
  • Figure 2: Left: The integration contour over a projective real line on the Riemann sphere, starting and ending at $z_{3},z_{3}^{+}$ respectively. Right: Deformation of the contour towards the left cut. Note that the contour does not intersect any branch cut nor branch point.