# A Crossing-Symmetric OPE Inversion Formula

## TL;DR

This paper derives a crossing-symmetric Lorentzian OPE inversion formula for $sl(2, ext{R})$, enabling a manifestly crossing-symmetric approach to conformal bootstrap and connecting to Polyakov bootstrap and analytic bootstrap functionals.

## Contribution

It introduces a new inversion formula applicable to crossing-symmetric four-point functions in $sl(2, ext{R})$, linking it to Polyakov bootstrap and analytic bootstrap functionals.

## Key findings

- Inverting a single conformal block yields the crossing-symmetric sum of Witten exchange diagrams.
- The inversion kernel has poles at double-trace dimensions, which cancel in solutions to crossing.
- The coefficient function of the principal series is meromorphic with poles only at expected locations.

## 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.