Entanglement of Internal Degrees of Freedom, Finiteness and Holography

<!--HTML--><p>I will discuss notions of entanglement of internal degrees of freedom in theories of large N matrices. Along with usual base space entanglement, this kind of entanglement (which do not pertain to a subregion of base space) should play a key role in holography, particularly in the emergence of a smooth internal space in the bulk. One such notion is target space entanglement, which appears as entanglement of D branes in the bulk. For the c=1 Matrix Model we will show how the target space entanglement entropy is perturbatively divergent, but non-perturbatively finite. Yet another notion relates to geometriztion of internal symmetries, as in familiar examples of holography n AdS X Y.&nbsp; We provide evidence that the&nbsp;entropy of this kind of entanglement is measured by Ryu-Takayanagi surfaces which are smeared along the AdS directions, and anchored on a subregion of the internal space Y.</p>