Lorentzian inversion in non-relativistic and classical limits

TL;DR

This paper explores the use of the conformal bootstrap and Lorentzian inversion formula in non-relativistic and classical limits of AdS/CFT, deriving new results on anomalous dimensions and classical correlation functions.

Contribution

It introduces a new Lorentzian inversion formula tailored for non-relativistic AdS and extends the understanding of classical limits in conformal bootstrap analyses.

Findings

T-channel conformal blocks yield classical correlation functions to linear order.

Computed anomalous dimensions from stress-tensor exchange in general dimensions.

Developed a saddle-point evaluation of the Lorentzian inversion formula for large operator dimensions and spins.

Abstract

We study tools of the conformal bootstrap in simplifying limits, primarily a limit of large operator dimensions and small cross-ratios corresponding to non-relativistic physics in AdS. We show that T-channel conformal blocks give the classical limit of correlation functions to linear order in an interaction potential. We use the Lorentzian inversion formula to compute anomalous dimensions due to T-channel exchanges and compare with time-independent perturbation theory in AdS. These calculations include new results not restricted to any limit, including anomalous dimensions from stress-tensor exchange in general dimension. We also study a classical limit of large operator dimension and spin, in which the Lorentzian inversion formula is evaluated by saddle-point. Finally, we obtain a new `Lorentzian' inversion formula for non-relativistic AdS by taking a limit of the CFT inversion formula.