Noncommutative correction to the entropy of charged BTZ black hole

Noncommutative geometry is an established potential candidate for including quantum phenomena in gravitation. We outline the formalism of Hopf algebras and its connection to the algebra of infinitesimal diffeomorphisms. Using a Drinfeld twist we deform spacetime symmetries, algebra of vector fields and differential forms leading to a formulation of noncommutative Einstein equations. We study a concrete example of charged BTZ spacetime and deformations steaming from the so called angular twist. The entropy of the noncommutative charged BTZ black hole is obtained using the brick-wall method. We used a charged scalar field as a probe and obtained its spectrum and density of states via WKB approximation. We provide the method to calculate corrections to the Bekenstein-Hawking entropy in higher orders in WKB, but we present the final result in the lowest WKB order. The result is that even in the lowest order in WKB, the entropy, in general, contains higher powers in $\hbar$, and it has logarithmic corrections, and polynomials of logarithms of the black hole area.


I. INTRODUCTION
The quest to understand the quantum aspects of gravity has a long history. Namely, since Einstein introduced the theory of general relativity (GR) [1] and the first black hole (BH) solution by Schwarzschild [2] there has been a lot of effort and research directions that have a goal to formulate some sort of quantum gravity theory. In the center of most of them is the study of quantum aspects of BHs, with entropy being the most interesting physical quantity.
The entropy of BHs was first understood by Bekenstein [3] and Hawking [4] in the framework of semi-classical gravity. Later it was argued that the origin of BH entropy is rooted in the statistical mechanics of in-falling particles [5] after a proper regularization of the UV divergence [6,7]. There are many approaches 1 to quantum gravity in which the entropy of BHs coincides with the Bekenstein-Hawking area law in the leading order.
In this paper, motivated by the possibility of co-existence of GR and Heisenberg's uncertainty principle [26,27], we use the noncommutative (NC) geometry framework as the underlying theory that captures quantum aspects of gravity [28][29][30][31][32]. It is natural to expect that at very small scales, i.e. Planck scale, the structure of smooth manifolds leading to the classical spacetime structure of GR will be modified in any theory of quantum gravity and therefore has to be replaced with some other structure. This new structure is the NC space, represented by some NC algebra and a corresponding NC differential geometry that will enable a formulation of a NC gravity theory [33,34]. In [26,27] it was explicitly shown that the principles of GR together with the Heisenberg's uncertainty principle lead to a NC structure of spacetime.
The charged BTZ (QBTZ) black hole is a solution of Einstein-Maxwell equations in 2+1 dimensions [62][63][64][65]. It is a 2+1 dimensional generalization/simplification of Reissner-Nordström-AdS black hole [66]. Since gravity in 2+1 dimesions has no propagating degrees of freedom and in principle can be quantized [67,68] it provides a perfect laboratory for analyzing quantum aspects of gravity and in particular entropy of BHs. This, together with the fact that angular twist renders NC corrections only to the coupling between the electromagnetic potential and charged scalar field while keeping the background classical [58][59][60][61] is our main motivation to study the entropy of a NC QBTZ.
The paper is organized as follows. In Section II we first outline the Hopf algebra approach to the symmetries of manifolds and then present its deformation via Drinfeld twist leading to a formulation of NC gravity theory. Using angular twist we show that QBTZ is a solution of NC Einstein equations and we derive NC corrections to the equation of motion for a charged scalar field. In Section III, the brick wall method for calculating entropy of BHs is outlined in general and then used to calculate the quantum and NC corrections to the entropy of QBTZ BH. In Section IV we study certain limiting cases and finally in Section V we conclude the paper with some final remarks.
Throughout the paper we are using units where k B = c = G = 1, but we explicitly write since we will be using WKB and expansions in .

II. NONCOMMUTATIVE QBTZ
In this section we will first motivate the use of Hopf algebra when dealing with infinitesimal diffemorphisms, then introduce the Drinfeld twist and show how it generates NC differential geometry ultimately leading to a formulation of NC gravity. Using this formalism we will analyze the deformation of the QBTZ-metric via the angular twist and derive the NC generalization of the equation of motion for the charged scalar field.

A. Infinitesimal diffeomorphisms and Hopf algebra
The symmetries of a manifold M are encoded in the algebra of infinitesimal diffeomorphisms. The infinitesimal diffeomorphism generated by some smooth vector field v acts on the algebra of smooth functions on M, i.e. C ∞ (M), as Lie derivatives where in the last equality we used a local basis {∂ µ } of smooth vector fields Ξ(M). The algebra of infinitesimal diffeomorphisms is described by the Lie algebra of vector fields where the Lie bracket [·, ·] : Ξ(M) × Ξ(M) −→ Ξ(M) is given by where in the local basis and v µ , w µ ∈ C ∞ (M). The Lie algebra of vector fields (Ξ(M), [·, ·]) together with its dual, i.e. one-forms Ω 1 (M) plays a central role in differential geometry and its action on any tensor field is represented via the Lie derivative due to the compatibility identity Note that for any vector field v ∈ Ξ(M) there exists v inv = −v that generates the inverse infinitesimal diffeomorphism. Also, the action of Lie algebra (Ξ(M), [·, ·]) on any tensor product between tensors T and T ′ , or better to say, product of representations T ⊗ T ′ is defined by the Leibniz rule If we now introduce a canonical map ǫ : Ξ(M) −→ C, such that it associates to each vector field the number zero, v → 0, the structure of a Hopf algebra emerges.
A Hopf algebra H is an algebra (also a co-algebra) together with three maps: coproduct ∆, counit ǫ and antipod S that satisfy where m(a ⊗ b) = ab, ∀a, b ∈ H is the algebra multiplication. In our example, the Lie algebra of vector fields (Ξ(M), [·, ·]), the Hopf algebra is given by H = (UΞ, m, ∆, ǫ, S) where UΞ is the universal enveloping algebra of the Lie algebra (Ξ(M), [·, ·]), m is the product in UΞ (namely composition of vector fields), ∆ : UΞ −→ UΞ ⊗ UΞ is a C-linear homomorphism that encodes the Leibniz rule (5) and is given by S : UΞ −→ UΞ is a C-linear antihomomorphism that encodes the algebra inverse and ǫ : UΞ −→ C is a C-linear map that encodes the normalization In conclusion one says that the symmetries of a manifold M are fully encoded in its corresponding Hopf algebra H = (UΞ, m, ∆, ǫ, S).

B. Deformed symmetries, twist and noncommutativity
Hopf algebra framework is suitable for investigating quantum or better to say deformed symmetries [69]. One of the most comprehensive ways of realizing this is by using the concept of Drinfeld twist [70][71][72]. A Drinfeld twist 2 F ∈ UΞ ⊗ UΞ is an invertible element that satisfies 3 (ǫ ⊗ 1)F = 1 = (1 ⊗ ǫ)F normalization condition (13) Provided that a Drinfeld twist exist, there is a well known theorem that enables us to construct a new Hopf algebra This new Hopf algebra H F describes the deformed infinitesimal diffeomorphisms, i.e. the symmetry of the deformed or noncommutative manifold. Notice that the usual geometry of a commutative manifold M is fully described by using its corresponding algebra of smooth functions (C ∞ (M), ·) =: A. This algebra A is covariant under the Hopf algebra H = (UΞ, m, ∆, ǫ, S). Namely we have 2 Here we need to extend the notion of universal enveloping algebra to formal power series in parameter λ. We replace the field C with the ring C[[λ]], so we promote U Ξ −→ U Ξ[[λ]], but for brevity we drop this notation (for more details see [34,69]). 3 We also demand F = 1 ⊗ 1 + O(λ).
However, the algebra A is commutative and fails to be covariant under the Hopf algebra H F , due to ∆ F which now encodes a deformed Leibniz rule. Luckily, the algebra A can be "fixed" by change the product, i.e. by promoting it into a noncommutative algebra A ⋆ = (C ∞ (M), ⋆) so that it becomes covariant under H F . The ⋆ is called the star-product and is fully determined by the Drinfeld twist F The algebra A ⋆ is in general noncommutative since f ⋆ g = g ⋆ f , but due to the cocycle condition for the twist (12) it is associative Also, the covariance is now given by Therefore it is often said that the Hopf algebra H F describes the symmetries of a noncommutative manifold underlying the algebra A ⋆ = (C ∞ (M), ⋆).
The twist F can be used to construct the ⋆-Lie algebra of vector fields Ξ ⋆ defined by deforming the Lie bracket (2) where F −1 =f A ⊗f A [33][34][35][36]. Explicitly for two vector fields we have where The deformed Lie derivative L ⋆ is given by and satisfies and a deformed Leibniz rule compatible with (22) where is the deformed tensor product. where Once again the twist F can be used to construct the NC one forms Ω 1 ⋆ as dual to vector fields 5 Ξ ⋆ via the ⋆-bilinear mapping ·, · ⋆ : and satisfies the A ⋆ -linearity properties D. Twisting the geometry and NC gravity The formalism of Hopf algebras and twists is suitable for further deforming the geometric objects like con- Here we immediately give the definitions of ⋆-objects (for more details see [33,34]).
There exists an associated covariant derivative ∇ ⋆ u (v) := u, ∇ ⋆ (v) ⋆ that due to (29) and (28) satisfies Notice that the ⋆-covariant derivative ∇ ⋆ u (f ) evaluated on a function is equal to the action of a ⋆-Lie derivative L ⋆ u (f ) as in the commutative case. The equations (30)-(32) are often considered as the "axioms" for defining ⋆-covariant derivative. The ⋆-curvature R ⋆ and ⋆-torsion T ⋆ associated to a connection ∇ ⋆ are defined as Notice that the definitions of R ⋆ and T ⋆ are analogous to the commutative ones up to the inclusion of R-matrix. Namely, since the noncommutativity of the ⋆-product is regulated by the R-matrix whenever we have to permute the order of the elements in the commutative definitions one needs to include the R-matrix contribution (see [36] for more details).
If we assume that there exist a local basis of vector fields {e i } and a dual basis of one-forms θ i such that we can define the coefficients R ⋆ l ijk and T ⋆ l ij as and ⋆-Ricci curvature tensor R ⋆ ij is given by 5 Notice that Ξ(M) and Ξ⋆ are isomorphic as vector spaces, as well as Ω 1 (M) and Ω 1 ⋆ At this point we are ready to write the NC version of the vacuum Einstein equation as Notice that all our algebraic (A ⋆ , Ξ ⋆ , Ω ⋆ , H F ) and geometric objects (∇ ⋆ , R ⋆ , T ⋆ ) reduce to the usual ones (C ∞ (M), Ξ(M), Ω(M), H, ∇, R, T ) in the commutative limit. The inclusion of full ⋆-Riemannian geometry will require the introduction of a metric tensor g and the notion of a torsion free and metric compatible connection Γ. Both of this can be defined in general (see [33,34]) which enables one to define the ⋆-Ricci scalar R and the full Einstein equation where T ⋆ ij is the noncommutative energy-momentum tensor and Λ is the cosmological constant. This defines our NC gravity. We will skip further discussion on the general ⋆-formalism and in the forthcoming subsections focus on concrete examples.

E. Angular twist
Angular twist is a special example of an Abelian Drinfeld twist defined by 6 The twist defined in (40) is Abelian since [∂ t , ∂ φ ] = 0 and Drinfeld because it satisfies (12). It is extensively studied in [58][59][60][61] where it was used to calculate NC corrections to the entropy of Reissner-Nordström BH [25] and field theory [57].
As described in previous subsections, the angular twist (40) defines the noncommutative algebra Therefore the algebra A ⋆ is a κ-deformed 7 spacetime where the commutation relations for coordinates are given by In the polar coordinates we have which is connected to the κ-cylinder algebra [73].
The angular twist (40) is also a Moyal twist since it can be written like 6 Here we consider that the twist, or better to say the vector fields ∂t and ∂ φ always act as a Lie derivatives. 7 It is called κ-deformed because the algebra (43) can be written in a general κ-deformed form where κ is the deformation parameter and uµ a constant unit vector. This Lie algebra type noncomutative space (42) can be equipped with a κ-igl Hopf algerba as its symmetry algebra [74,75]. Notice that the most famous example of (42) is the so called κ-Minkowski algebra (for which uµ = (1, 0)) together with its κ-Poincare Hopf algebra as its symmetry [52][53][54], and this explain the origin of the name "κ-deformed".
where A, B = {t, r, φ} and Θ tφ = −Θ φt = − ia 2 are the only non-zero elements of the constant deformation matrix Θ AB . In case of a Moyal twist we have a rather big simplification of the general formulation of the NC gravity. Namely, due to the trivial action of the twist on the basis {∂ µ } and {dx α } note that (46) and that the ⋆-covariant derivative is determined by which leads to and The metric tensor g is given by and it remains undeformed. One can show [33,34] that there exist a unique ⋆-torsion free and metric compatible ⋆ Levi-Civita connection that is explicitly given by where g ⋆ρσ is the unique ⋆-inverse satisfying and is explicitly given as where g αβ is the usual inverse Now it is easy to see that if the metric g has Killing vectors K α compatible with the Moyal twist, i.e. [K α , ∂ β ] = 0 that the ⋆-curvature and ⋆-Levi-Civita (53) are undeformed and in that case, any g which satisfies the commutative Einstein equation will also satisfy the ⋆-Einstein equation (39) provided that energy-momentum tensor inherits the symmetry, i.e. L Kα (T) = 0. However, the NC geodesic motion will change and so will the equation of motion for the metric perturbations (see [33,34]).

F. Noncommutative QBTZ
Using the formalism outlined so far we want to deform a particular manifold described by the QBTZ metric [62][63][64][65] with the angular twist (40). The QBTZ metric is given by and together with the electromagnetic potential A = A µ dx µ satisfies the coupled Einstein-Maxwell equations 8 . Here M and Q are the mass and charge of the black hole, while 1/l 2 is the cosmological constant rendering the QBTZ-metric asymptotically AdS. The QBTZ has time-translation and azimuthal symmetry since implying that ∂ t and ∂ φ are its Killing vector fields. Whenever we want to deform a manifold which has timetranslation and azimuthal symmetry (rotation in the x − y plane), that is, whenever the metric g has ∂ t and ∂ φ as Killing vectors, the angular twist (40) becomes an Abelian affine Killing twist [33]. One can easily see that deforming the QBTZ with the angular twist will lead to the conclusion that the whole NC differential geometry is undeformed and that QBTZ is also a solution of the ⋆-Einstein equation (39) with T ⋆ = T = T µν dx µ ⊗ dx µ , where T µν are the components of the Maxwell stress tensor. In order to see this more explicitly, it is enough to calculate where Σ ρ µν = Γ ρ µν = 1 2 g ρσ (∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν ) are the undeformed Levi-Civita coefficients leading ultimately to R ⋆ = R etc.
Therefore we can consider the QBTZ to be a fixed commutative and NC background at the same time, i.e. QBT Z = N CQBT Z. This is very convenient because we effectively have that the spacetime part, i.e. the geometry can be treated completely classically. However, when considering NC field theory, for example if we introduce a charged scalar probeΦ in the mix, things become interesting. As shown in [58][59][60], there will be a nontrivial NC correction to the coupling between the U (1) field and the charged scalar field. This we will look into more detail in the next subsection.

G. NC Klein-Gordon equation in (N C)QBT Z background
Since we shown that the NCQBTZ, that is the QBTZ deformed with an angular twist (40), can be treated as a classical curved background, the action for a NC charged scalar field is given by where D µ is the U (1) ⋆ -covariant derivative given by and where q and µ are the charge and mass of the scalar fieldΦ. Such defined action (61) is invariant under infinitesimal U (1) ⋆ gauge transformations whereΛ is the NC gauge parameter. In order to express the NC fieldΦ andÂ µ in terms of their commutative counterparts Φ and A µ one needs to use the so called Seiberg-Witten (SW) map [76,77]. The SW map for Abelian twist is known up to all orders in the deformation parameter a [78] and is explicitly given aŝ where F µν = ∂ µ A ν − ∂ ν A µ are the components of the electromagnetic field-strength 2-form. Using (64) and (40), the action (61) ,up to the first order in a, is given by where L(a) is the NC correction Now we vary the action (65) with respect to Φ + to obtain the equation of motion for Φ and after using the ansatz Φ = e − i Et R(r)e imφ we get the radial equation of motion where E and m are separation constant corresponding to the energy and angular momentum (magnetic quantum number) and we explicitly used the QBTZ-metric (57). Radial equation (67) is the central result of this paper and in the following sections we will use it to calculate the NC correction to the entropy of QBTZ via the brick-wall method.

III. BLACK HOLE ENTROPY IN THE BRICK WALL MODEL
In order to calculate the entropy of a BH in general it is crucial to regularize the UV divergence and isolate the relevant physical contribution [19,79]. One of the most simplest way to achieve that was presented long time ago by 't Hooft [6,7] using the so called brick wall method. Alongside the brick wall method, the BH entropy can be described using the entanglement of the degrees of freedom between the two sides of the horizon [80,81], and by using the Wald entropy formula within the effective field theory approach to quantum gravity [21,22,82]. It has been shown [19,83] that all three approaches are related due to the fact that all of them have almost identical UV divergence of the BH entropy.
The brick wall method is a semi-classical approach where the BH is considered a fixed background that is in a thermal equilibrium with a thermal bath of some surrounding quantum matter fields at the Hawking temperature. Therefore the canonical entropy of the matter field outside the BH horizon is related to the entropy of BH itself. When calculating the canonical entropy the crucial ingredient is the density of states. The density of states diverges at the horizon and this is the reason why one uses a cutoff h, i.e. the brick wall, imposing that at this point, r + + h, the matter fields outside the BH horizon vanish. The value of the cutoff is determined from the matching condition that the leading divergent part of the canonical entropy obeys the Bekenstein-Hawking area law. The canonical entropy is given by where F is the free energy of the matter field at the inverse temperature β and is given by [6,7] In order to evaluate (69), the key ingredients are the energy E (the spectrum of the field) and the density of states N (E), i.e. the number of eigenmodes of the matter field. Both of them are determined by solving the equation of motion for a surrounding field in the fixed BH background and imposing the brick wall boundary conditions, i.e. that the fields are vanishing close to horizon and in the spatial infinity. The first condition will regularize the UV infinities while the second regularize the IR infinities which will occur in the density of states due to the contribution coming from the vacuum surrounding the system at large distances, but this can be omitted [6]. For scalar fields, everything is governed by the radial part of the Klein-Gordon equation and one uses the WKB-method to solve it. Now we move on to our concrete example of NC scalar field on a NCQBTZ background.
A. WKB and the density of states N (E) The radial part of the equation of motion for the NC scalar field on NCQBTZ background is given in (67) and since we are unable to solve it analytically we use the WKB-method. In doing so, we first use an ansatz so that (67) can be put in the following form 9 where The V 2 function is related to the effective potential(for the µ = 0 case) together with the energy, A and C are the NC correction to ψ ′ and ψ term respectively, B is the correction to the ψ ′ term due to charge q and ∆ is the standard term related to the existence of the term R ′ in (67).
In order to solve (70) we use the WKB-ansatz to obtain a differential equation for P (r) We are looking for a solution that is a power series in , namely 9 Notice that the equation (71) slightly differs from the corresponding one in [84] due to appearance of the A, B and C term. They appear here because of two reasons. One is because we are looking at a Klein-Gordon equation for a charged scalar and the other is because of the NC corrections. When q and a go to zero, we recover the equation in [84].
Substituting the series (79) into (78) we get for the P n up to n = 2 From now on we will focus on the lowest order in the WKB-approximation 10 , i.e. on P 0 .
Using (80), (72) and choosing + sign we obtain Since we expect that the NC scale govern by the deformation parameter a is small compared the other scales in the problem, i.e. it is comparable with the Planck scale, we further expand P 0 up to 11 a 2 The density of states N (E) is defined by [6,7] N where r + is the outer horizon, h is the brick wall cutoff, L is the IR cutoff and m max is the maximal value of the magnetic quantum number m such that P remains real and is given by In the lowest order in WKB we have where and the linear term in a vanishes due to mmax −mmax mdm = 0. Now, we evaluate the Λ integration and using we obtain the density of states N (E) in the lowest WKB order and we postpone the r-integration for later. 10 Also notice that B = 0 for the QBTZ (57). 11 We expand up to a 2 because later we will see that the linear terms do not contribute to the entropy S.

B. Entropy of NCQBTZ
Once we have the density of states N 0 , using (69) and (68), we can obtain the free energy F 0 and canonical entropy 12 S 0 . After the E-integration we obtain where we used the ζ-function regularization and also subtracted the infinite contribution proportional to ζ(1) which originates from the electrostatic self-energy of the charge q of the scalar particle.
We are interested in the main contribution to the entropy S 0 coming from the horizon. In order to extract this contribution we split the integration over r into two parts where we introduced an intermediate scale R > 0 such that in the first term we have h ≪ r + ≪ R and in the second term r + ≪ R ≪ L. The second term is the contribution of the vacuum surrounding the system at large distances and can be omitted [6,7]. The first contribution can be evaluated in the near horizon limit, namely, we use the near horizon coordinate x = r − r + and h ≪ r + to obtain where the coefficients f n are given in Appendix up to n = 3 and we used that the inverse temperature β is related to the Hawking temperature via surface gravity κ The brick wall cutoff h is determined by imposing that the most divergent part of S 0 as h −→ 0 is equal to the Bekenstein-Hawking entropy S BH = A 4 [6,7] i.e.
leading to Now we can plug in the cutoff (97) into the full expression for the entropy (94) to obtain the dependance of the entropy on the area A, i.e. the quantum and NC corrections to the Bekenstein-Hawking area law where A = 2πr + is the area of the horizon and the function coefficients V k and W k depend on A and have an expansion in 12 Here we have also used the notation where the coefficients v kn and w kn are given in the Appendix up to n = 5 and k = 3. Notice that up to the linear order in , the expression for the entropy (98) is of the general form presented in [84], but due to the existence of terms ln k A l in higher orders in this is no longer true. This is due to the fact that the electromagnetic potential (58) in 2 + 1 dimensions has a logarithmic dependance, rather than r −1 , which then explains the appearance of higher powers of ln in the expression for the entropy (98).

IV. SOME LIMIT CASES
The equation for the entropy (98) looks very involved so we will analyze some interesting limiting cases of it.
A. Limit case q −→ 0 The limit q −→ 0 means that the charge of the surrounding scalar field Φ is negligible. In this case there is no coupling between the electromagnetic potential A µ and Φ and therefor there is no NC corrections to the entropy. However, we get quantum corrections to the entropy of QBTZ, i.e.
where we used This is in complete agreement with [84] and (100) represents the entropy of QBTZ in the lowest WKB-order.
The Q −→ 0 limit means that the charge of the BH is negligible. If so, there is no electromagnetic potential A µ , no coupling with Φ and no NC corrections. Also, the QBTZ reduces to the spinless BTZ metric, i.e. for the entropy we get where we used and This is also in complete agreement with [84] and (102) represents the entropy of BTZ in the lowest WKB-order.

C. The almost BTZ limit
The angular twist (40) does not deform either the BTZ metric or its coupling to the scalar probe. Hence taking limits of Q, q −→ 0 in (98) simply reproduces the commutative results. The NC corrections only appear if we have a charged BH (in our case QBTZ) and a charged scalar probe Φ. Therefore it is interesting to take a closer look at the QBTZ with a very small charge Q (but not negligible!) in order to compare the NC corrections with respect to the (almost) BTZ black hole. We will investigate (98) for small Q, that is, we expand everything to the lowest order in the BH charge Q. Since the NC correction is proportional to Q 2 it is enough to expand the S 0 up to quadratic terms in Q. In doing this we first need to find the expansion of r + . Since r + is given in terms of Lambert function W [85], in order to avoid the asymptotics and complex analysis, we solve the condition for the horizon of QBTZ metric f (r + ) = 0 perturbatively. First we write 13 and the expansion for r + is given by Now we expand the entropy (98) as a series in Q and obtain where s n and z n are calculated up to n = 2, i.e. and

V. FINAL REMARKS
The brick wall method is one of the most widely used methods for calculating corrections to the Bekenstein-Hawking entropy [84]. It is important to note that even the lowest order in WKB can provide corrections of the same structure as higher orders if one expands the metric beyond the linear order in the horizon [84]. Like in [25], we present a derivation of the NC correction to Bekenstein-Hawking entropy steaming from a Drinfeld twist which is compatible with the symmetries of the background metric, i.e. we are using a Killing twist [33,34]. Contrary to the general form for the entropy corrections found in [84] and [25], the NC corrections to the entropy of QBTZ exhibit extra terms that are polynomials in logarithms. This peculiarity is present due to the fact that both the electromagnetic potential (58) and QBTZ metric (57) have an explicit logarithmic dependance on r that later propagates in the WKB expansion. It is generally believed that the appearance of logarithmic dependance in the corrections to entropy of black holes is due to nonlocality of quantum nature of gravity. If so, we can conclude that in the special example of QBTZ deformed via the angular twist (40), the NC corrections to entropy seem to suggest that the quantum aspects of gravity in 2+1 dimensions have a higher degree of nonlocality than the corresponding theory in 3+1 dimensions [25].
It is important to note that the universal nature of the UV divergence of the BH entropy is related to the fact that the von Neumann algebra of observables in QFT in curved backgrounds is of Type-III [86][87][88]. As supported by the results in [25] and in this paper, it seems that from a perturbative standpoint these UV divergences persist even in the NC framework, suggesting that noncommutativity does not change the typology of the corresponding von Neumann algebra of observables.
Appendix B: Coefficients v kn and w kn v 00 =