Defects in conformal field theory

TL;DR

Defect CFTs are analyzed using embedding-space methods to fix correlation functions in the presence of flat or spherical defects. The authors classify tensor structures for bulk-to-defect and bulk-bulk correlators, derive Casimir equations and light-cone expansions for conformal blocks, and examine Ward identities that define the displacement operator and its couplings. For codimension-two defects, bulk blocks map to four-point blocks of a defect-free CFT, yielding closed-form results in many cases and enabling defect bootstrap approaches. In two dimensions, the displacement operator is related to the reflection coefficient with explicit unitarity bounds, linking boundary/defect data to scattering properties. Overall, the work provides a comprehensive toolkit for analyzing and bootstrapping defect CFTs with spinning operators and various defect geometries.

Abstract

We discuss consequences of the breaking of conformal symmetry by a flat or spherical extended operator. We adapt the embedding formalism to the study of correlation functions of symmetric traceless tensors in the presence of the defect. Two-point functions of a bulk and a defect primary are fixed by conformal invariance up to a set of OPE coefficients, and we identify the allowed tensor structures. A correlator of two bulk primaries depends on two cross-ratios, and we study its conformal block decomposition in the case of external scalars. The Casimir equation in the defect channel reduces to a hypergeometric equation, while the bulk channel blocks are recursively determined in the light-cone limit. In the special case of a defect of codimension two, we map the Casimir equation in the bulk channel to the one of a four-point function without defect. Finally, we analyze the contact terms of the stress-tensor with the extended operator, and we deduce constraints on the CFT data. In two dimensions, we relate the displacement operator, which appears among the contact terms, to the reflection coefficient of a conformal interface, and we find unitarity bounds for the latter.

Figures (5)

• Figure 1: The picture illustrates the choice of coordinates in this subsection: the spherical defect is drawn in brown and is placed on the plane spanned by the coordinates $x^{\tilde{a}}$.
• Figure 2: The angle $\phi$ is formed by the projections of the vectors $P_1$ and $P_2$ onto the $q$-dimensional space orthogonal to the defect. The left figure gives a perspective view of this angle, while the one on the right gives the top view. The defect is represented here by a brown line (or brown point in the top view).
• Figure 3: The light-cone limit relevant for the bulk Casimir equation: the defect is time-like and the operators are space-like separated from it.
• Figure 4: One way to do radial quantization is to quantize around a bulk point. We can always choose a sphere which does not intersect the defect and such that both insertions lie in its interior.
• Figure 5: The theory can also be quantized around a point on the defect. This is usually associated with the defect OPE.