Abstract
We compute correlation functions, specifically 1-point and 2-point functions, in holographic boundary conformal field theory (BCFT) using geodesic approximation. The holographic model consists of a massive scalar field coupled to a Karch-Randall brane -- a rigid boundary in the bulk AdS space. Geodesic approximation requires the inclusion of paths reflecting off of this brane, which we show in detail. For the 1-point function, we find agreement between geodesic approximation and the harder $\Delta$-exact calculation, and we give a novel derivation of boundary entropy using the result. For the 2-point function, we find a factorization phase transition and a mysterious set of anomalous boundary-localized BCFT operators. We also discuss some puzzles concerning these operators.
Highlights
We examine geodesic approximation [13,14] for correlators of “heavy” (Δ ≫ d) CFT operators by summing exponentiated geodesic lengths between boundary insertions
The main result of this paper is to extend the geodesic approximation (2) to holographic models with boundaries such as KR braneworlds
We have studied d-dimensional boundary CFT (BCFT) using a bottomup17 holographic model involving a massive free scalar field with a Karch-Randall end-of-the-world brane
Summary
Recent work has highlighted the importance of KarchRandall (KR) braneworlds [1,2] to understanding gravitational phenomena. (I) a d-dimensional boundary CFT (BCFT) [6,7], (II) a CFT þ gravity on an asymptotically AdSd space, connected by transparent boundary conditions to a nongravitating d-dimensional BCFT bath, (III) Einstein gravity on an asymptotically AdSdþ space containing an end-of-the-world brane Much work in these systems has used duality between (II) and (III) to compute the fine-grained entanglement entropy of black hole information in (II) via the island rule [8–12] to get the semiclassical Page curve in (II) from the classical geometry of (III). This requires taking into account geodesics that reflect off of the boundary, as described originally in [20–22] in a nonholographic context We compute both the heavy 1-point and 2-point functions of a scalar BCFT operator at nonzero brane tension for the first time using geodesic approximation. Nonzero tension yields a mysterious extra set of “anomalous” operators [29]
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