Radial coordinates for defect CFTs

TL;DR

This paper introduces new radial coordinates for defect conformal field theories that simplify the computation of conformal blocks and improve the convergence of OPE expansions, aiding numerical bootstrap methods.

Contribution

The paper defines two new cross ratios with geometric interpretation, enabling efficient conformal block calculations and demonstrating their convergence properties in defect CFTs.

Findings

New cross ratios have simple geometric interpretation.

Conformal blocks can be efficiently computed in power expansion.

Two-point function expansion converges everywhere with these coordinates.

Abstract

We study the two-point function of local operators in the presence of a defect in a generic conformal field theory. We define two pairs of cross ratios, which are convenient in the analysis of the OPE in the bulk and defect channel respectively. The new coordinates have a simple geometric interpretation, which can be exploited to efficiently compute conformal blocks in a power expansion. We illustrate this fact in the case of scalar external operators. We also elucidate the convergence properties of the bulk and defect OPE decompositions of the two-point function. In particular, we remark that the expansion of the two-point function in powers of the new cross ratios converges everywhere, a property not shared by the cross ratios customarily used in defect CFT. We comment on the crucial relevance of this fact for the numerical bootstrap.