A Crossing-Symmetric OPE Inversion Formula

Understand

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.