Defect Conformal Field Theory and Locally Localized Gravity

TL;DR

This work connects locally localized gravity on an $AdS_d$ brane to a dual defect conformal field theory by mapping bulk gravity mode decomposition to the BOPE of ambient operators. The authors establish a precise criterion: localization occurs when a reduced operator in the BOPE acquires an anomalous dimension driving its dimension toward $\Delta \sim d-1$, with gravity calculations corroborating the field-theory picture at leading order in brane backreaction. They provide both a toy field-theory demonstration of defect operator anomalous dimensions and a gravity-based computation framework (Type I–III diagrams) to extract these dimensions from two-point functions, including how logarithmic terms encode anomalous dimensions. The results illuminate how backreaction and defect dynamics control localization, and they offer a concrete holographic approach for analyzing brane-induced localization in string-theoretic setups such as the D3/D5 system.

Abstract

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.

Figures (5)

• Figure 1: The two slicings of $AdS_5$. The horizontal axis is the direction $x$ transverse to the brane and the vertical axis is the radial direction of $AdS$ interpolating from the boundary (solid line) to the horizon (dashed line). The figure on the left shows lines of constant $\rho$ while the figure on the right shows lines of constant $r$.
• Figure 2: The leading diagram contributing to the $\langle j j \rangle$ current-current correlator.
• Figure 3: The leading diagram contributing to the $\langle {\cal J} j \rangle$ current-current correlator.
• Figure 4: The leading diagram contributing to the $\langle {\cal J} {\cal J} \rangle$ current-current correlator.
• Figure 5: Diagrams appearing in the computation of $\left\langle {\cal O}_4 {\cal O}_4 \right\rangle$. The circle represents the boundary of $AdS_5$, where we insert the operators, and the vertical straight line crossing it represents the $AdS_4$ brane. The other solid lines are bulk scalar propagators, and the curly line is a brane scalar propagator.