Dispersion relation from Lorentzian inversion in 1d CFT

TL;DR

This paper derives a dispersion relation for 1d CFT four-point functions from the Lorentzian inversion formula, providing an explicit crossing symmetric kernel and applying it to holographic correlators in super Yang-Mills theory.

Contribution

It introduces a dispersion relation based on the Lorentzian inversion formula for 1d CFTs and explicitly constructs the crossing symmetric kernel for identical operators.

Findings

Successfully reproduces holographic correlators up to fourth order in large t'Hooft coupling expansion.

Provides an explicit integral representation for four-point functions in 1d CFTs.

Complements existing analytic functional approaches with a new dispersion relation method.

Abstract

Starting from the Lorentzian inversion formula, we derive a dispersion relation which computes a four-point function in 1d CFTs as an integral over its double discontinuity. The crossing symmetric kernel of the integral is given explicitly for the case of identical operators with integer or half-integer scaling dimension. This derivation complements the one that uses analytic functionals. We use the dispersion relation to evaluate holographic correlators defined on the half-BPS Wilson line of planar $\mathcal{N}=4$ super Yang-Mills, reproducing results up to fourth order in an expansion at large t'Hooft coupling.