Abstract
We derive a Lorentzian OPE inversion formula for the principal series of sl(2, 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, 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.
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Mazáč, D. A crossing-symmetric OPE inversion formula. J. High Energ. Phys. 2019, 82 (2019). https://doi.org/10.1007/JHEP06(2019)082
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DOI: https://doi.org/10.1007/JHEP06(2019)082