Two Point Functions in Defect CFTs

Abstract: This paper is designed to be a practical tool for constructing and investigating two-point correlation functions in defect conformal field theory, directly in physical space, between any two bulk primaries or between a bulk primary and a defect primary, with arbitrary spin. Although geometrically elegant and ultimately a more powerful approach, the embedding space formalism gets rather cumbersome when dealing with mixed symmetry tensors, especially in the projection to physical space. The results in this paper provide an alternative method for studying two-point correlation functions for a generic $d$-dimensional conformal field theory with a flat $p$-dimensional defect and $d-p=q$ co-dimensions. We tabulate some examples of correlation functions involving a conserved current, an energy momentum tensor and a Maxwell field strength, while analysing the constraints arising from conservation and the equations of motion. A method for obtaining bulk-to-defect correlators is also explained. Some explicit examples are considered: free scalar theory on $\mathbb{R}^p \times ({\mathbb R}^q / {\mathbb Z}_2)$ and a free four dimensional Maxwell theory on a wedge.