Bulk vs. Boundary Dynamics in Anti-de Sitter Spacetime

TL;DR

This work develops a Lorentzian AdS bulk–boundary correspondence, arguing that both normalizable bulk modes (supporting bulk fluctuations and a Hilbert space) and non-normalizable modes (acting as non-fluctuating backgrounds that encode boundary operator insertions) are essential. It provides explicit constructions of scalar modes for arbitrary mass in AdS, in both Poincaré and global coordinates, and analyzes their organization into SL(2,R)×SL(2,R) representations in AdS3, relating normalizable modes to unitary representations and non-normalizable modes to non-unitary ones. The paper connects these bulk solutions to boundary CFT data on the cylinder and plane, clarifying how operator dimensions h_± arise and how primary and descendant states map across the bulk–boundary duality. It concludes with a discussion of the utility and limitations of the bulk–boundary description for bulk physics and black hole interiors, highlighting ambiguities in the effective action and suggesting directions for future work on horizon physics and background handling.

Abstract

We investigate the details of the bulk-boundary correspondence in Lorentzian signature anti-de Sitter space. Operators in the boundary theory couple to sources identified with the boundary values of non-normalizable bulk modes. Such modes do not fluctuate and provide classical backgrounds on which bulk excitations propagate. Normalizable modes in the bulk arise as a set of saddlepoints of the action for a fixed boundary condition. They fluctuate and describe the Hilbert space of physical states. We provide an explicit, complete set of both types of modes for free scalar fields in global and Poincaré coordinates. For $\ads{3}$, the normalizable and non-normalizable modes originate in the possible representations of the isometry group $\SL_L\times\SL_R$ for a field of given mass. We discuss the group properties of mode solutions in both global and Poincaré coordinates and their relation to different expansions of operators on the cylinder and on the plane. Finally, we discuss the extent to which the boundary theory is a useful description of the bulk spacetime.

Figures (2)

• Figure 1: Anti-de Sitter spacetime displayed as the interior of a cylinder. For the single cover of AdS the top and bottom boundaries should be identified, whereas for its universal covering space (CAdS) an infinite number of copies should be attached above and below the displayed region. The boundary of AdS is identified with the boundary of the cylinder. The coordinates indicated correspond to those in (89). Horizons in AdS are obtained by making two diagonal cuts through the cylinder, as shown. The cuts divide AdS into two regions, each of which is covered by a set of Poincare coordinates. The boundary divides into two diamond shaped regions, which are each conformal to copies of flat Minkowski space.
• Figure 2: Penrose diagrams for anti-de Sitter space. Displayed are vertical cross sections which cut through the center of the AdS cylinder. In each figure, regions demarcated by solid lines identify the portion of the spacetime covered by a single coordinate patch. a) Global coordinates. The boundary of the region is the surface of a cylinder. b) Poincare coordinates. Here AdS is divided into two patches, with the two boundaries at $r=0$ being conformal to flat Minkowski space. At the horizons, $r=\pm \infty$. c) BTZ coordinates. AdS is divided into twelve patches, eight of which appear in the two dimensional slice shown. The eight boundaries are each conformal to flat Minkowski space.