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A Crossing-Symmetric OPE Inversion Formula

Dalimil Mazac

TL;DR

The paper develops a Lorentzian OPE inversion formula for the principal series of sl(2,R) in 1D CFTs with identical external operators, rendering crossing symmetry manifest. By inverting a single crossed-channel block, it identifies the resulting coefficient function as that of a crossing-symmetric sum of AdS2 exchange Witten diagrams (Polyakov blocks), and shows that these blocks encode the full OPE data while enforcing double-trace cancellations through infinite sum rules. It demonstrates that the principal-series coefficient function is meromorphic across the complex plane and connects the inversion kernels’ residues to analytic extremal functionals, unifying several analytic bootstrap strands. The work also provides explicit kernels for bosonic and fermionic cases, improves the bosonic inversion to cover all Regge-bounded correlators, and clarifies the Polyakov bootstrap’s nonperturbative content and its relation to conventional crossing. These results deepen analytic control over 1D conformal dynamics and offer pathways for AdS2 perturbation theory and nonperturbative bootstrap in simplified settings.

Abstract

We derive a Lorentzian OPE inversion formula for the principal series of $sl(2,\mathbb{R})$. Unlike the standard Lorentzian inversion formula in higher dimensions, the formula described here only applies to fully crossing-symmetric four-point functions and makes crossing symmetry manifest. In particular, inverting a single conformal block in the crossed channel returns the coefficient function of the crossing-symmetric sum of Witten exchange diagrams in AdS, including the direct-channel exchange. The inversion kernel exhibits poles at the double-trace scaling dimensions, whose contributions must cancel out in a generic solution to crossing. In this way the inversion formula leads to a derivation of the Polyakov bootstrap for $sl(2,\mathbb{R})$. The residues of the inversion kernel at the double-trace dimensions give rise to analytic bootstrap functionals discussed in recent literature, thus providing an alternative explanation for their existence. We also use the formula to give a general proof that the coefficient function of the principal series is meromorphic in the entire complex plane with poles only at the expected locations.

A Crossing-Symmetric OPE Inversion Formula

TL;DR

The paper develops a Lorentzian OPE inversion formula for the principal series of sl(2,R) in 1D CFTs with identical external operators, rendering crossing symmetry manifest. By inverting a single crossed-channel block, it identifies the resulting coefficient function as that of a crossing-symmetric sum of AdS2 exchange Witten diagrams (Polyakov blocks), and shows that these blocks encode the full OPE data while enforcing double-trace cancellations through infinite sum rules. It demonstrates that the principal-series coefficient function is meromorphic across the complex plane and connects the inversion kernels’ residues to analytic extremal functionals, unifying several analytic bootstrap strands. The work also provides explicit kernels for bosonic and fermionic cases, improves the bosonic inversion to cover all Regge-bounded correlators, and clarifies the Polyakov bootstrap’s nonperturbative content and its relation to conventional crossing. These results deepen analytic control over 1D conformal dynamics and offer pathways for AdS2 perturbation theory and nonperturbative bootstrap in simplified settings.

Abstract

We derive a Lorentzian OPE inversion formula for the principal series of . Unlike the standard Lorentzian inversion formula in higher dimensions, the formula described here only applies to fully crossing-symmetric four-point functions and makes crossing symmetry manifest. In particular, inverting a single conformal block in the crossed channel returns the coefficient function of the crossing-symmetric sum of Witten exchange diagrams in AdS, including the direct-channel exchange. The inversion kernel exhibits poles at the double-trace scaling dimensions, whose contributions must cancel out in a generic solution to crossing. In this way the inversion formula leads to a derivation of the Polyakov bootstrap for . The residues of the inversion kernel at the double-trace dimensions give rise to analytic bootstrap functionals discussed in recent literature, thus providing an alternative explanation for their existence. We also use the formula to give a general proof that the coefficient function of the principal series is meromorphic in the entire complex plane with poles only at the expected locations.

Paper Structure

This paper contains 21 sections, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Kinematics and the Euclidean inversion formula
  3. The four-point function
  4. Review of the Euclidean inversion formula
  5. The Lorentzian inversion formula
  6. The general form
  7. Constraining the inversion kernels
  8. Explicit formulas for the inversion kernels
  9. The bosonic case
  10. The fermionic case
  11. Exchange Witten diagrams
  12. Inverting a single block in the crossed channel
  13. Fermionic Polyakov blocks
  14. Explicit results for the coefficient function
  15. Polyakov bootstrap and extremal functionals
  16. ...and 6 more sections

Figures (1)

  • Figure 1: The contour deformation bringing the "top-of-the-branch-cut" contributions of the Lorentzian inversion formula to the Euclidean region. The left figure depicts the second and third term on the RHS of equation \ref{['eq:lInvInterm']}. The two semi-infinite contours can be combined with a semi-circular contour at infinity and collapsed on top of the interval $z\in(0,1)$, giving the RHS of equation \ref{['eq:contourDef']}, as depicted in the figure on the right. The branch cuts of $\widetilde{\mathcal{G}}(z)$ are shown in red and the branch cut of $H_{\Delta}\!\left(\hbox{$\frac{1}{z}$}\right)$ in green.