Defects in conformal field theory

TL;DR

This paper explores how conformal symmetry breaking by extended operators affects correlation functions, analyzing tensor structures, conformal blocks, and constraints on CFT data, with special focus on defect codimension two and two-dimensional cases.

Contribution

It adapts the embedding formalism for defect CFTs, classifies tensor structures, solves Casimir equations, and relates defect data to reflection coefficients, providing new insights into defect conformal field theories.

Findings

Two-point functions are fixed up to OPE coefficients.

Casimir equations reduce to hypergeometric equations in certain cases.

Constraints on CFT data from stress-tensor contact terms and unitarity bounds.

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.