Gravity may be "locally localized" over a wide range of length scales on a
d-dimensional anti-de Sitter (AdS) brane living inside AdS_{d+1}. In this paper
we examine this phenomenon from the point of view of the holographic dual
"defect conformal field theory". The mode expansion of bulk fields on the
gravity side is shown to be precisely dual to the "boundary operator product
expansion" of operators as they approach the defect. From the field theory
point of view, the condition for localization is that a "reduced operator"
appearing in this expansion acquires negative anomalous dimension. In
particular, a very light localized graviton exists when a mode arising from the
reduction of the ambient stress-energy tensor to the defect has conformal
dimension Delta ~ d-1. The part of the stress tensor containing the defect
dynamics has dimension Delta = d-1 in the free theory, but we argue that it
acquires an anomalous dimension in the interacting theory, and hence does not
participate in localization in the regime of small backreaction of the brane.
We demonstrate that such an anomalous dimension is consistent with the
conservation of the full stress-energy tensor. Finally, we analyze how to
compute the anomalous dimensions of reduced operators from gravity at leading
order in the interactions with the brane.Comment: 38 pages, LaTeX, 5 figures. v2: typos fixe
