Operators in the Internal Space and Locality

Realizations of the holographic correspondence in String/M theory typically involve spacetimes of the form $AdS \times Y$ where $Y$ is some internal space which geometrizes an internal symmetry of the dual field theory, hereafter referred to as an "$R$ symmetry". It has been speculated that areas of Ryu-Takayanagi surfaces anchored on the boundary of a subregion of $Y$, and smeared over the base space of the dual field theory, quantify entanglement of internal degrees of freedom. A natural candidate for the corresponding operators are linear combinations of operators with definite $R$ charge with coefficients given by the "spherical harmonics'' of the internal space: this is natural when the product spaces appear as IR geometries of higher dimensional AdS spaces. We study clustering properties of such operators both for pure $AdS \times Y$ and for flow geometries, where $AdS \times Y$ arises in the IR from a different spacetime in the UV, for example higher dimensional AdS or asymptotically flat spacetime. We show, in complete generality, that the two point functions of such operators separated along the internal space obey clustering properties at scales larger than the $AdS$ scale. For non-compact $Y$, this provides a notion of approximate locality. When $Y$ is compact, clustering happens only when the size of $Y$ is parametrically larger than the $AdS$ scale. This latter situation is realized in flow geometries where the product spaces arise in the IR from an asymptotically AdS geometry at UV, but not typically when they arise near black hole horizons in asymptotically flat spacetimes. We discuss the significance of this result for entanglement and comment on the role of color degrees of freedom.