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.
Highlights
The results of the analytic conformal bootstrap have been unified and extended through the so-called Lorentzian inversion formula [11].2 The formula exploits complex analyticity of the four-point function to extract its OPE decomposition from the double commutator, called double discontinuity and denoted dDisc[G(z, z)].3 More precisely, the formula computes the coefficient function I∆,J of the decomposition of the four-point function into a complete set of conformal partial waves labelled by their dimension ∆ and spin J
We prove that I∆ of a crossing-symmetric four-point function in a unitary theory can be expanded in the coefficient functions of crossing-symmetrized exchange Witten diagrams
It is analogous to the Lorentzian inversion formula of Caron-Huot [11], which applies to the principal series of the conformal group in more than one dimension
Summary
We will focus on the four-point function of identical operators φ(x) in a conformal field theory, denoted φ(x1)φ(x2)φ(x3)φ(x4). In order to avoid the branch cuts, we should take the limit in the upper half-plane as z = reiθ, r ∈ R and r → ∞ This limit of the four-point function is precisely the Regge limit of the u-channel, as explained in detail in section 2 of [36]. It will sometimes be useful to consider functions which are better-behaved than just bounded in the Regge limit. The four-point function G(z) is defined analogously to the bosonic case χ(x1)χ(x2)χ(x3)χ(x4) = χ(x1)χ(x2) χ(x3)χ(x4) G(z). Just like in the bosonic case, G(z) satisfies crossing symmetry G(z) = G(1−z) and boundedness in the Regge limit (the latter whenever the theory is unitary)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
