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Entanglement Entropy Renormalization for the NC scalar field coupled to classical BTZ geometry

In this work, we consider a noncommutative (NC) massless scalar field coupled to the classical nonrotational BTZ geometry. In a manner of the theories where the gravity emerges from the underlying scalar field theory, we study the effective action and the entropy derived from this noncommutative model. In particular, the entropy is calculated by making use of the two different approaches, the brick wall method and the heat kernel method designed for spaces with conical singularity. We show that the UV divergent structures of the entropy, obtained through these two different methods, agree with each other. It is also shown that the same renormalization condition that removes the infinities from the effective action can also be used to renormalize the entanglement entropy for the same system. Besides, the interesting feature of the NC model considered here is that it allows an interpretation in terms of an equivalent system comprising of a commutative massive scalar field, but in a modified geometry; that of the rotational BTZ black hole, the result that hints at a duality between the commutative and noncommutative systems in the background of a BTZ black hole.