Dispersion relation from Lorentzian inversion in 1d CFT

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.