More on boundary holographic Witten diagrams

In this paper we discuss geodesic Witten diagrams in general holographic conformal field theories with boundary or defect. In boundary or defect conformal field theory, two-point functions are nontrivial and can be decomposed into conformal blocks in two distinct ways; ambient channel decomposition and boundary channel decomposition. In our previous work [A. Karch and Y. Sato, J. High Energy Phys. 09 (2017) 121.] we only consider two-point functions of same operators. We generalize our previous work to a situation where operators in two-point functions are different. We obtain two distinct decomposition for two-point functions of different operators.

Figure 1

A schematic figure of the Janus geometry (1.1). The dashed lines are at constant $r$ and represent sliced ${\mathrm{AdS}}_d$ spacetimes.

Figure 2

Schematic picture of the ambient channel decomposition. The dashed line is a geodesic. The point $r=r^{\prime }$ where the source is inserted is integrated.

Figure 3

Schematic picture of the boundary channel decomposition. The dashed lines are geodesics. We need to integrate about the point $r=r^{\prime }$ where the source is inserted. Anomalous dimension terms are omitted.

Figure 4

Two distinct decompositions in the D3/D5 brane system.

Figure 5

A leading correction of two-point functions in the Takayanagi’s proposal.