Light-sheets and AdS/CFT

Abstract

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 $L={L}^{+}\ensuremath{\cap}{L}^{\ensuremath{-}}$, where ${L}^{+}$ (${L}^{\ensuremath{-}}$) 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.