Radial coordinates for defect CFTs

TL;DR

This work introduces radial coordinates $(r,\eta)$ and $(\hat r,\hat\eta)$ for defect CFTs to analyze the two-point function of bulk scalars in the presence of a defect. It develops two complementary OPE channels (bulk and defect) and constructs conformal blocks in these coordinates, using Casimir and Zamolodchikov recurrences to obtain efficient radial expansions. The authors prove that the radial expansions converge in the natural OPE domains and show the defect OPE converges exponentially fast via density arguments, while highlighting limitations of the traditional $\xi$-expansion due to Landau singularities. The framework provides both geometric intuition and practical tools for defect bootstrap, with explicit results for bulk and defect blocks and pathways to general spin-bearing operators. Overall, the radial-coordinate approach enhances analytic control and numerical bootstrap prospects for defect CFTs, including general defects beyond codimension-one.

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.

Figures (10)

• Figure 1: Both pictures represent the example of a $(d+2)=3$ dimensional embedding space, labelled by $P^M=(P^0,P^1,P^{2})$. We show the null cone, the defect plane and the Poincaré section (in red). In the left picture, the $P^-$ axis lies on the defect plane. The latter intersects the Poincaré section in a single point (in yellow), which in physical space corresponds to a flat defect. In the right picture, the $P^1$ direction is parallel to the defect plane, while $P^-$ does not lie on it. The defect plane intersects the Poincaré section along $P^1$ in two points. This is a spherical defect centred in the origin.
• Figure 2: The configuration corresponding to eq. \ref{['PiOnTheCylinder1']}. The defect is marked in red, and lies on one equator of the sphere at $\tau=0$. The local operators are placed at equal generic time, on the orthogonal equator, in opposite points.
• Figure 3: The configuration corresponding to eq. \ref{['B_Poin_Conf']}. The defect is spherical and orthogonal to the plane drawn in the figure, and crosses it at the position marked by the red dots. The operators $\mathcal{O}_1$ and $\mathcal{O}_2$ sit at the same radius $r$, and the position of $\mathcal{O}_1$ is parametrized by the complex coordinates $(\rho,\bar{\rho})$. The fundamental domain $\mathcal{D}$ is highlighted in gray.
• Figure 4: The configuration corresponding to eq. \ref{['D_Cyl_Conf']}. The red lines on the cylinder mark the position of the defect. Constant time slices are spheres, which in this three dimensional example are marked by the defect in two opposite poles. The operators $\mathcal{O}_1$ and $\mathcal{O}_2$ live on the equator of two spheres inserted at time $0$ and $\tau$ respectively.
• Figure 5: The configuration corresponding to eq. \ref{['D_Poin_Conf']}. The defect is flat and orthogonal to the plane drawn in the figure, and crosses it at the position marked by the red dot. The operator $\mathcal{O}_1$ sits at unit radius, while the position of $\mathcal{O}_2$ is parametrized by the complex coordinates $({\hat{\rho}},{\bar{\hat{\rho}}})$. The fundamental domain $\hat{\mathcal{D}}$ is highlighted in gray.