Holographic entanglement entropy and the internal space

We elaborate on the role of extremal surfaces probing the internal space in AdS/CFT dualities. Extremal surfaces in anti--de Sitter space quantify the ``geometric'' entanglement between different regions in physical space for the dual conformal field theory. This, however, is just one of many ways to split a given system into subsectors, and extremal surfaces in the internal space should similarly quantify entanglement between subsectors of the theory. For the case of AdS₅×S⁵, their area was interpreted as entanglement entropy between U(n) and U(m) subsectors of U(n+m) N=4 super Yang-Mills theory (SYM). Making this proposal precise is subtle for a number of reasons, the most obvious being that from the bulk one usually has access to gauge-invariant quantities only, while a split into subgroups is inherently gauge variant. We study N=4 SYM on the Coulomb branch, where some of the issues can be mitigated and the proposal can be sharpened. Continuing back to the original AdS₅×S⁵ geometry, we obtain a modified proposal, based on the relation of the internal space to the R-symmetry group.