1d Conformal Field Theory and Dispersion Relations

TL;DR

This paper extends conformal dispersion relations to one-dimensional CFTs, deriving a new integral relation, and computes scalar Witten diagrams in AdS2 for holographic theories, enhancing understanding of 1D conformal dynamics.

Contribution

It introduces a dispersion relation for 1D CFT four-point functions, connecting it with the Lorentzian inversion formula and computes scalar Witten diagrams in AdS2.

Findings

Derived a dispersion relation for 1D CFT four-point functions.

Established an integral relation between dispersion and inversion kernels.

Computed scalar Witten diagrams in AdS2 at tree and loop levels.

Abstract

We study conformal field theory in $d=1$ space-time dimensions. We derive a dispersion relation for the 4-point correlation function of identical bosons and fermions, in terms of the double discontinuity. This extends the conformal dispersion relation of arXiv:1910.12123, which holds for CFTs in dimensions $d\geq 2$, to the case of $d=1$. The dispersion relation is obtained by combining the Lorentzian inversion formula with the operator product expansion of the 4-point correlator. We perform checks of the dispersion relation using correlators of generalised free fields and derive an integral relation between the kernel of the dispersion relation and that of the Lorentzian inversion formula. Finally, for $1$-$d$ holographic conformal theories, we analytically compute scalar Witten diagrams in $AdS_2$ at tree-level and $1$-loop.