  We explore higher-dimensional conformal field theories (CFTs) in the presence
of a conformal defect that itself hosts another sub-dimensional defect. We
refer to this new kind of conformal defect as the composite defect. We
elaborate on the various conformal properties of the composite defect CFTs,
including correlation functions, operator expansions, and conformal block
expansions. As an example, we present a free O(N) vector model in the presence
of a composite defect. Assuming the averaged null energy condition (ANEC) does
hold even for the defect systems, we conclude that some boundary conditions can
be excluded. Our investigations shed light on the rich phenomenology arising
from hierarchical defect structures, paving the way for a deeper understanding
of critical phenomena in nature.
