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A<sc>bstract</sc> Realizations of the holographic correspondence in String/M theory typically involve spacetimes of the formAdS×YwhereYis some internal space which geometrizes an internal symmetry of the dual field theory, hereafter referred to as an “Rsymmetry”. It has been speculated that areas of Ryu-Takayanagi surfaces anchored on the boundary of a subregion ofY, 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 definiteRcharge 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 pureAdS×Yand for flow geometries, whereAdS×Yarises 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 sufficiently larger than the AdS scale. For non-compactY, this provides a notion of approximate locality. WhenYis compact, clustering happens only when the size ofYis 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. 

A bstract We study minimum area surfaces associated with a region, R , of an internal space. For example, for a warped product involving an asymptotically AdS space and an internal space K , the region R lies in K and the surface ends on ∂R . We find that the result of Graham and Karch can be avoided in the presence of warping, and such surfaces can sometimes exist for a general region R . When such a warped product geometry arises in the IR from a higher dimensional asymptotic AdS, we argue that the area of the surface can be related to the entropy arising from entanglement of internal degrees of freedom of the boundary theory. We study several examples, including warped or direct products involving AdS 2 , or higher dimensional AdS spaces, with the internal space, K = R m , S m ; Dp brane geometries and their near horizon limits; and several geometries with a UV cut-off. We find that such RT surfaces often exist and can be useful probes of the system, revealing information about finite length correlations, thermodynamics and entanglement. We also make some preliminary observations about the role such surfaces can play in bulk reconstruction, and their relation to subalgebras of observables in the boundary theory.