Light-sheets and AdS/CFT

TL;DR

This paper proposes a covariant geometric construction of the holographic domain in AdS/CFT, relating boundary subsets to bulk regions using light-sheets and the covariant entropy bound, with implications for black holes and cosmology.

Contribution

It introduces a new geometric method to determine the bulk region dual to a boundary subset in AdS/CFT, based on light-sheets and the covariant entropy bound.

Findings

Proves that the causal set C equals the intersection of light-sheets L.

Shows the holographic domain is determined by covariant geometric principles.

Discusses implications for bulk reconstruction and black hole interiors.

Abstract

One may ask whether the CFT restricted to a subset b of the AdS boundary has a well-defined dual restricted to a subset H(b) of the bulk geometry. The Poincare patch is an example, but more general choices of b can be considered. We propose a geometric construction of H. We argue that H should contain the set C of causal curves with both endpoints on b. Yet H should not reach so far from the boundary that the CFT has insufficient degrees of freedom to describe it. This can be guaranteed by constructing a superset of H from light-sheets off boundary slices and invoking the covariant entropy bound in the bulk. The simplest covariant choice is L, the intersection of L^+ and L^-, where L^+ (L^-) is the union of all future-directed (past-directed) light-sheets. We prove that C=L, so the holographic domain is completely determined by our assumptions: H=C=L. In situations where local bulk operators can be constructed on b, H is closely related to the set of bulk points where this construction remains unambiguous under modifications of the CFT Hamiltonian outside of b. Our construction leads to a covariant geometric RG flow. We comment on the description of black hole interiors and cosmological regions via AdS/CFT.