Light-sheets and AdS/CFT

One may ask whether the conformal field theory (CFT) restricted to a subset b of the anti-de Sitter (AdS) boundary has a well-defined dual restricted to a subset $H(b)$ of the bulk geometry. The Poincar\'e 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 L of H from light-sheets off boundary slices and invoking the covariant entropy bound in the bulk. The simplest covariant choice is L=L⁺∩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 renormalization-group flow. We comment on the description of black hole interiors and cosmological regions via AdS/CFT.