# A Lorentzian inversion formula for defect CFT

## TL;DR

This paper introduces a Lorentzian inversion formula for defect conformal field theories (CFTs) that enables the analytic extraction of bulk data from defect data, facilitating the analytic bootstrap approach for defect CFTs.

## Contribution

It provides a new inversion formula for the bulk channel in defect CFTs, complementing existing formulas for the defect channel, and applies it to compute large-spin and twist defect data.

## Key findings

- Derived the bulk channel inversion formula for defect CFTs.
- Calculated the large-spin limit of bulk data for defect identity.
- Computed bulk data of the 3d Ising model's twist defect to first order in epsilon-expansion.

## Abstract

We present a Lorentzian inversion formula valid for any defect CFT that extracts the bulk channel CFT data as an analytic function of the spin variable. This result complements the already obtained inversion formula for the corresponding defect channel, and makes it now possible to implement the analytic bootstrap program for defect CFT, by going back and forth between both channels. A crucial role in our derivation is played by the Calogero-Sutherland description of defect blocks which we review. As first applications we obtain the large-spin limit of bulk CFT data necessary to reproduce the defect identity, and also calculate the bulk data of the twist defect of the $3d$ Ising model to first order in the $\epsilon$-expansion.