Gauge invariant target space entanglement in D-brane holography

It has been suggested in arXiv:2004.00613 that in Dp-brane holography, entanglement in the target space of the D-brane Yang-Mills theory provides a precise notion of bulk entanglement in the gravity dual. We expand on this discussion by providing a gauge invariant characterization of operator sub-algebras corresponding to such entanglement. This is achieved by finding a projection operator which imposes a constraint characterizing the target space region of interest. By considering probe branes in the Coloumb branch we provide motivation for why the operator sub-algebras we consider are appropriate for describing a class of measurements carried out with low-energy probes in the corresponding bulk region of interest. We derive expressions for the corresponding Renyi entropies in terms of path integrals which can be directly used in numerical calculations.