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Entanglement Entropy and Matter-Gravity Couplings for Fuzzy Geometry

In this talk I discuss some features of the entanglement entropy for fuzzy geometry, focusing on its dependence on the background fields and the spin connection of the emergent continuous manifold in a large $N$ limit. Using the Landau-Hall paradigm for fuzzy geometry, this is argued to be given by a generalized Chern-Simons form, making a point of connection with the thermodynamic view of gravity. Matter-gravity couplings are also considered in the same framework; they naturally lead to certain specific nonminimal couplings involving powers of the curvature.