Conformal Boundary Condition and Massive Gravitons in AdS/BCFT

Abstract

According to Witten [1], the conformal boundary condition of gravity, which specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and leads to well-defined perturbation theory of gravity about any classical solution. The conformal boundary condition was previously considered in [2, 3] in the context of AdS/BCFT, wherein the equation of motion of the end-of-the-world was derived and emphasized. In this paper, we investigate further other consequences of the conformal boundary condition in AdS/BCFT. We derive the boundary central charges of the holographic Weyl anomaly and show that they are exactly the same for conformal boundary condition and Dirichlet boundary condition. We analysis the metric perturbation with conformal boundary condition (CBC), Dirichlet boundary condition (DBC) and Neumann boundary condition (NBC) imposed on the end-of-the-world brane and show that they admit an interpretation as the fluctuation of the extrinsic curvature (case of CBC and DBC) and the induced metric (case of NBC) of Q respectively. In all cases, the fluctuation modes are massive, which are closely relevant to the massive island formation in the literature. Our results reveal that there are non-trivial gravitational dynamics from extrinsic curvatures on the conformal and Dirichlet branes, which may have interesting applications to the island. We also discuss, in passing, the localization of gravitons in brane world theory. We find that, contrary to NBC, the graviton for CBC/DBC is located on the brane with non-positive tension instead of non-negative tension.