Quantum Near Horizon Geometry of Black 0-Brane

We investigate a bunch of D0-branes to reveal its quantum nature from the gravity side. In the classical limit, it is well described by a non-extremal black 0-brane in type IIA supergravity. The solution is uplifted to the eleven dimensions and expressed by a non-extremal M-wave solution. After reviewing the effective action for the M-theory, we explicitly solve the equations of motion for the near horizon geometry of the M-wave. As a result we derive an unique solution which includes the effect of the quantum gravity. Thermodynamic property of the quantum near horizon geometry of the black 0-brane is also studied by using Wald's formula. Combining our result with that of the Monte Carlo simulation of the dual thermal gauge theory, we find strong evidence for the gauge/gravity duality in the D0-branes system at the level of quantum gravity.


Introduction
Superstring theory is a promising candidate for the theory of quantum gravity, and it plays important roles to reveal quantum nature of black holes. Fundamental objects in the superstring theory are D-branes as well as strings [1], and in the low energy limit their dynamics are governed by supergravity. The D-branes are described by classical solutions in the supergravity, which are called black branes [2,3]. A special class of them has event horizon like the black holes and its entropy can be evaluated by the area law. Interestingly the entropy can be statistically explained by counting number of microstates in the gauge theory on the D-branes [4]. This motivates us to study the black hole thermodynamics from the gauge theory. Furthermore it is conjectured that the near horizon geometry of the black brane corresponds to the gauge theory on the D-branes [5]. If this gauge/gravity duality is correct, the strong coupling limit of the gauge theory can be analyzed by the supergravity [6,7].
In this paper we consider a bunch of D0-branes in type IIA superstring theory. In the low energy limit, a bunch of D0-branes with additional internal energy are well described by non-extremal black 0-brane solution in type IIA supergravity [2,3]. After taking near horizon limit, the metric becomes AdS black hole like geometry in ten dimensional spacetime [8]. From the gauge/gravity duality, this geometry corresponds to the strong coupling limit of the gauge theory on the D0-branes [8], which is described by (1+0)-dimensional U (N ) super Yang-Mills theory [9]. This gauge theory is paid much attention as nonperturbative definition of M-theory [10,11], which is the strong coupling description of the type IIA superstring theory [12,13]. Recently nonperturbative aspects of the gauge theory are studied by the computer simulation [14]- [24]. (See refs. [25], [26] for reviews including other topics.) Especially in ref. [19], physical quantities of the thermal gauge theory, such as the internal energy, are evaluated numerically, and a direct test of the gauge/gravity duality is performed including α ′ correction to the type IIA supergravity. Furthermore, if the internal energy of the black 0-brane can be evaluated precisely from the gravity side including g s correction, it is possible to give a direct test for the gauge/gravity duality at the level of quantum gravity [24]. (α ′ = ℓ 2 s is the string length squared and g s is the string coupling constant.) The purpose of this paper is to derive quantum correction to the near horizon geometry of the non-extremal black 0-brane directly from the gravity side. In order to do this, we need to know an effective action which include quantum correction to the type IIA supergravity.
In principle the effective action can be constructed so as to be consistent with the scattering amplitudes in the type IIA superstring theory [27], and it is expressed by double expansion of α ′ and g s . For example, since four point amplitudes of gravitons at tree and one loop level are nontrivial, there should exist terms like α ′3 e −2φ t 8 t 8 R 4 and α ′3 g 2 s t 8 t 8 R 4 in the effective action, respectively [27]- [36]. These are called higher derivative terms and t 8 represents products of four Kronecker's deltas with eight indices. Especially we are interested in the latter terms, which give nontrivial g s corrections to the geometry. These higher derivative terms often play important roles to count the entropy of extremal black holes [37,38].
It is necessary that the effective action of the type IIA superstring should possess local supersymmetry in ten dimensions. So the supersymmetrization of α ′3 g 2 s t 8 t 8 R 4 is very important [28,29,30,33,35,36] to understand the structure of effective action. Although the task is not completed yet, since our interest is on the geometry of the black 0-brane, it is enough to know terms which contain the metric, dilaton field and R-R 1-form field only.
Notice that these fields are collected into the metric in eleven dimensional supergravity [39], and the black 0-brane is expressed by M-wave solution. Then α ′3 g 2 s t 8 t 8 R 4 and other terms which include the dilaton and R-R 1-form field are simply collected into ℓ 6 p t 8 t 8 R 4 terms in eleven dimensions. Here ℓ p = ℓ s g 1/3 s is the Planck length in eleven dimensions. Thus we consider the effective action for the M-theory and investigate quantum corrections to the near horizon geometry of the non-extremal M-wave. We show equations of motion for the effective action and explicitly solve them up to the order of g 2 s . The M-wave geometry receives the quantum corrections and thermodynamic quantities for the M-wave are modified.
Especially the internal energy of the M-wave is obtained quantitatively including quantum effect of the gravity.
Organization of the paper is as follows. In section 2, we review the classical near horizon geometry of the black 0-brane in ten dimensions, and uplift it to that of the M-wave in eleven dimensions. In section 3, we discuss the higher derivative corrections in the type IIA superstring theory and the M-theory, and solve the equations of motion for the near horizon geometry of the non-extremal M-wave in section 4. In section 5, we evaluate the entropy and the energy of the M-wave up to 1/N 2 . We probe the quantum near horizon geometry by D0-brane in section 6 and clarify the validity of our analyses in section 7. Section 8 is devoted to conclusion and discussion. Detailed calculations and discussions on the ambiguities of the higher derivative corrections are collected in the appendices.

Classical Near Horizon Geometry of Black 0-Brane
In this section, we briefly review the non-extremal solution of the black 0-brane which carries mass and R-R charge. Especially we uplift the solution to eleven dimensions and show that the black 0-brane is described by the M-wave solution.
In the low energy limit, the dynamics of massless modes in type IIA superstring theory are governed by type IIA supergravity. Since we are interested in the black 0-brane which couples to the graviton g µν , the dilaton φ and R-R 1-form field C µ , the relevant part of the type IIA supergravity action is given by where 2κ 2 10 = (2π) 7 ℓ 8 s g 2 s and G µν is the field strength of C µ . g s and ℓ s are the string coupling constant and the string length, respectively. It is possible to solve the equations of motion by making the ansatz that the metric is static and has SO(9) rotation symmetry. Then we obtain non-extremal solution of the black 0-brane. (See ref. [40] for example.) The horizon is located at r H = (r 7 + − r 7 − ) 1 7 . Parameters r ± are related to the mass M 0 and the R-R charge Q 0 of the black 0-brane by where N is a number of D0-branes and V S 8 = 2π 9/2 Γ(9/2) = 2(2π) 4 7·15 is the volume of S 8 . Now the parameters r ± are expressed as where δ is a non-negative parameter. The extremal limit r + = r − is saturated when δ = 0.
Let us rewrite the solution (2) in terms of U = r/ℓ 2 s and λ = g s N/(2π) 2 ℓ 3 s , which correspond to typical energy scale and 't Hooft coupling in the dual gauge theory, respectively.
The near horizon limit of the non-extremal black 0-brane is taken by ℓ s → 0 while U , λ and δ/ℓ 4 s are fixed. Then the near horizon limit of the solution (2) becomes [8] The type IIA supergravity is related to the eleven dimensional supergravity via circle compactification. In fact, the eleven dimensional metric is related to the ten dimensional one like ds 2 11 = e −2φ/3 ds 2 10 + e 4φ/3 (dz − C µ dx µ ) 2 . The near horizon limit of the non-extremal solution of the black 0-brane (5) can be uplifted to eleven dimensions as This represents the near horizon limit of the non-extremal M-wave solution in eleven dimensions. The solution is purely geometrical and the expressions become simple. Furthermore, on the geometrical part, quantum corrections to the eleven dimensional supergravity are under control. This is the reason why we execute analyses of the solution in eleven dimensions.

Quantum Correction to Eleven Dimensional Supergravity
The eleven dimensional supergravity is realized as the low energy limit of the M-theory.
A fundamental object in the M-theory is a membrane and if we could take account of interaction of membranes, the effective action of the M-theory would become the eleven dimensional supergravity with some higher derivative terms. Unfortunately quantization of the membrane has not been completed so far. It is, however, possible to derive the relevant part of the quantum corrections in the M-theory by requiring local supersymmetry. In this section we review the quantum corrections to the eleven dimensional supergravity.
Massless fields of the eleven dimensional supergravity consists of a vielbein e a µ , a Majorana gravitino ψ µ and a 3-form field A µνρ . Since we are only interested in the M-wave solution, we only need to take account of the action which only depends on the graviton.
where 2κ 2 11 = (2π) 8 ℓ 9 p = (2π) 8 ℓ 9 s g 3 s . Notice that after the dimensional reduction this becomes the action (1), which contains the dilation and the R-R 1-form field as well as the graviton in ten dimensions [39].
Of course there are other terms which depend on ψ µ and A µνρ , which are completely determined by the local supersymmetry. and simultaneously there should exist a term like [eψψ 2 ] in the action. Here ψ 2 represents the field strength of the Majorana gravitino. By continuing this process, it is possible to determine the structure of the 11 dimensional supergravity completely [39]. Now let us discuss quantum corrections to the eleven dimensional supergravity. Since the M-theory is related to the type IIA superstring theory by the dimensional reduction, the effective action of the M-theory should contain that of the type IIA superstring theory.
The latter can be obtained so as to be consistent with scattering amplitudes of strings, and it is well-known that leading corrections to the type IIA supergravity include terms should exist in the action. The structures of B 1 , B 11 and F 1 are severely restricted by the local supersymmetry. By continuing this process, it is possible to show that a combination of terms in B 1 are completely determined up to over all factor [35,36]. The result become as follows.
Although all indices are lowered, it is understood those are contracted by the flat metric η ab . The Riemann tensor with local Lorentz indices is defined by R abcd = e µ c e ν d (∂ µ ω νab − ∂ ν ω µab +ω µa e ω νeb −ω νa e ω µeb ), where ω µab is a spin connection and µ, ν are space-time indices.
The over all factor in eq. (8) is determined by employing the result of 1-loop four graviton amplitude in the type IIA superstring theory.
Since the near horizon limit of the M-wave solution (6) is purely geometrical, it is possible to examine the leading quantum corrections to it from the action (8). Other terms which depend on the 3-form field are irrelevant to the analyses for the M-wave. In summary the effective action of the M-theory is described by where γ = π 2 Notice that the parameter γ remains finite after the decoupling limit is taken. After the dimensional reduction, the action (9) becomes the effective action of the type IIA superstring theory, which includes the 1-loop effect of the gravity. Now we derive equations of motion for the action (9). Although the derivation is straightforward, we need to labor at many calculations because of the higher derivative terms in the action. Therefore in practice we use the Mathematica code for the calculations. Below we show the points of the calculations to build the code.
First of all we list variations of the fields with respect to the vielbein.
where δe ij ≡ e i µ δe µj . Then variations of the higher derivative terms are evaluated as where we defined Finally we obtain the equations of motion for the effective action (9).
As mentioned before the action (9) is not unique due to the ambiguity of field redefinitions, such as g µν → g ′ µν = g µν + γℓ 12 s R 2 R µν . Therefore the equations of motion are not unique as well. We will discuss, however, that physical quantities of the M-wave do not depend on these ambiguities. (See appendix D.)

Quantum Near Horizon Geometry of Black 0-Brane
In the previous section, we have explained the effective action of the M-theory (9), and derived the equations of motion (13). In this section we solve them up to the linear order of γ and obtain the non-extremal solution of the M-wave with quantum gravity correction.
In order to obtain the solution of (13), we relax the ansatz for the M-wave as where i = 1, 2, 3, and h i and f 1 are functions of a dimensionless variable x = U U 0 . This ansatz is static and possesses SO(9) rotation symmetry, and if we take N = ∞, the metric just becomes the classical solution (6). By solving the equations of motion (13), we determine functions h i (x) and f 1 (x).
Calculations are straightforward but complicated, so we use the Mathematica code to explicitly write down the equations of motion. Some of the results are listed in the appendices A and B. From the output we find that there are five nontrivial equations which are given by Here we defined Note that the above equations are derived up to the order of γ, and a part of γ 0 is zero since the ansatz (14) is a fluctuation around the classical solution (6). Now we solve these equations to obtain h i and f 1 . We will see that h i and f 1 are uniquely determined as functions of x by imposing reasonable boundary conditions. Because calculations below are a bit tedious, the results are summarized in the end of this section.
First let us evaluate the sum of E 1 and E 2 .
From this equation h 1 is expressed in terms of h 2 as where c 1 is an integral constant. Next let us evaluate E 5 . 1 In the last line, we removed h 1 by using the eq. (21). Thus a linear combination of h 3 is expressed in terms of h 2 as where c 2 is an integral constant. From the eqs. (21) and (23), it is possible to remove h 1 and After some calculations, we obtain three equations remaining to be solved.
Notice, however, that three functions E 1 , E 3 and E 4 are not independent because of the identity This corresponds to the energy conservation, D a E ab = 0. Thus we only need to solve following two equations.
By solving the eq. (28), finally we obtain h 2 as where c 3 and c 4 are integral constants. Although the form of I(x) seems to be complicated, its derivative becomes So far there are four integral constants, but these will be fixed by appropriate conditions.
In fact it is natural to require that h i (1) are finite and h i (x) ∼ O(x −8 ) when x goes to the infinity. In order to satisfy these conditions, it is necessary to choose c 2 = 7c 1 , c 3 = 944455680π(sin π 7 + sin 3π 7 + sin 5π 7 ) and c 4 = −6611189760. Inserting these values into the eqs. (30), (31) and (32), we obtain and Note that the function I(x) behaves as when x goes to the infinity. Now we remove h 2 out of the eq. (29), and obtain the differential equation only for f 1 .
Then f 1 is solved as Here the integral constant is set to be zero, because we imposed the boundary condition that The integral constant c 1 is chosen to be zero so as to satisfy h 1 (x) ∼ O(x −8 ) when x goes to the infinity. Finally from the eq. (23), we derive Thus h 3 is expressed as The integral constant is set to be zero, since this term can be removed by the general coordinate transformation on z direction. It corresponds to the gauge transformation on C µ in ten dimensions.
Let us summarize the quantum correction to the near horizon geometry of the nonextremal M-wave and the black 0-brane. By solving the eqs. (15)- (19), we obtained the quantum near horizon geometry of the non-extremal M-wave, In stead of γ, we introduced dimensionless parameter and the functions h i and f 1 are uniquely determined as This represents the quantum near horizon geometry of the non-extremal black 0-brane.

Thermodynamics of Quantum Near Horizon Geometry of Black 0-Brane
Since the quantum near horizon geometry of the non-extremal black 0-brane is derived in the previous section, it is interesting to evaluate its thermodynamics. In this section, we estimate the entropy and the internal energy of the quantum near horizon geometry of the non-extremal black 0-brane by using Wald's formula [41,42]. These quantities are quite important when we test the gauge/gravity duality.
In the following, quantities are calculated up to O(ǫ 2 ). First of all, let us examine the location of the horizon x H . This is defined by F 1 (x H ) = 0 and becomes is a dimensionless parameter. Temperature of the black 0-brane is derived by the usual prescription. We consider the Euclidean geometry by changing time coordinate as t = −iτ and require the smoothness of the geometry at the horizon. This fixes the periodicity of τ direction and its inverse gives the temperature of the non-extremal black 0-brane. Then the dimensionless temperatureT = T /λ 1 3 of the black 0-brane is evaluated asT where a 1 and a 2 are numerical constants given by Inversely solving the eq. (47), the dimensionless parameterŨ 0 is written in terms of the temperatureT asŨ By using this replacement, it is always possible to express physical quantities as functions of T .
Next we derive the entropy of the quantum near horizon geometry of the non-extremal black 0-brane. In practice, we consider the quantum near horizon geometry of the nonextremal M-wave because of its simple expression. Since the effective action (9) includes higher derivative terms, we should employ Wald's entropy formula which ensures the first law of the black hole thermodynamics. The Wald's entropy formula is given by where is the volume factor at the horizon and N µν is an antisymmetric tensor binormal to the horizon. The binormal tensor satisfies N µν N µν = −2 and nonzero component is only The effective action is given by the eq. (9), and in the formula the variation of the action is evaluated as if the Riemann tensor is an independent variable, that is, Now we are ready to evaluate the entropy of the quantum near horizon geometry of the non-extremal M-wave. Some useful results are collected in the appendix B. By using these, the entropy is evaluated as where numerical constants a 3 and a 4 are defined as This result includes the quantum gravity effect, and it gives quite nontrivial test of the gauge/gravity duality if we can evaluate the internal energy from the dual gauge theory. In fact it is possible by employing the Monte Carlo simulation and the result strongly concludes that the duality holds at this order [24].
The specific heat is evaluated as Notice that the specific heat becomes negative in the region whereT < (ǫa 4 /3) 5/12 ∼ 0.4N −5/6 . In this region the non-extremal black 0-brane behaves like Schwarzschild black hole and will be unstable. When N = ∞ the instability will be suppressed. This result is also verified from the Monte Carlo simulation of the dual gauge theory [24].

D0-brane Probe
In this section, we probe the quantum near horizon geometry of the non-extremal black 0-brane (45) via a D0-brane. Form the analysis it is possible to study how the test D0-brane is affected by the background field.
The bosonic part of the D0-brane action consists of the Born-Infeld action and the Chern-Simons one. Here we neglect an excitation of the gauge field on the D0-brane, so the Born-Infeld action is simply given by the pull-back of the metric. We also assume that the D0-brane moves only along the radial direction. Then the probe D0-brane action in the background of (45) is written as The momentum conjugate to x is evaluated as and the energy of the probe D0-brane is given by In the final line we took the non-relativistic limit. From this we see that the potential energy for the probe D0-brane is expressed as (59) The first term corresponds to the gravitational attractive force and the second one does to the R-R repulsive force.
When we take N = ∞, the potential energy becomes shows that the gravitational attractive force overcomes the R-R repulsive force.
Similarly, when N is finite, we regard √ F 1 as the gravitational attractive force to the probe D0-brane. The function of √ F 1 is plotted in fig. 1. From this we see that the gravitational force becomes repulsive near the horizon x H . (60) Here we used the relation eq. (49) by setting ǫ = 0. Then the supergravity approximation is valid when the string length √ α ′ is quite small compared to the curvature radius ρ and the effective string coupling g s e φ is also quite small, i.e.,T ∼ 0 and N ∼ ∞.
Now we consider the validity of our 1-loop analyses. From the effective action (9), we derived the internal energy (54) of the black 0-brane. However, if we include other higher derivative terms in the effective action, the Lagrangian is expected to be where R is the abbreviation of the Riemann tensor. The existence of these terms can be found in ref. [43]. By following the calculation of eq. (52) and using the dimensional analyses (60), the internal energy will be modified as E N 2 ∼ 7.41T 14 5 1 + T 9 5 +T 3 + · · · + 1 N 2 − 0.779 (62) T 9 5 andT 3 come from α ′3 and α ′5 terms at tree level, and 1 N 2n correspond to n-loop amplitudes. Numerical constants are assumed to be O(1) and at least this is true for the 1-loop result. The coefficient c 2 at 2-loop will be discussed later. From the above estimation, the 1-loop contribution of −0.779/(N 2T 12 5 ) becomes subleading when following conditions are satisfied,T 9 5 1/N 2T 12/5 = N 2T 21 5 ≤ s,T 9 5 ≤ s, where s < 1. Note that terms at n-loop (n > 2) are well suppressed both from the second and third inequalities. The above inequalities are equivalent to Then if c 2 is negative, the internal energy takes large negative value becuase of the negative power in c 2T −13/5 . Thus we expect c 2 is positive, and again the specific heat becomes negative. As a reference we mention that the numerical simulation suggests that the 2-loop coeficient becomes c 2 = +0.00459 [24].

Conclusion and Discussion
In this paper we studied quantum nature of the bunch of D0-branes in the type IIA superstring theory. In the classical limit, it is well described by the non-extremal black 0-brane in the type IIA supergravity. The quantum correction to the non-extremal black 0-brane is investigated after taking the near horizon limit.
In order to manage the quantum effect of the gravity, we uplifted the near horizon geometry of the non-extremal black 0-brane into that of the M-wave solution in the eleven dimensional supergravity. These two are equivalent via the duality between the type IIA superstring theory and the M-theory, but the latter is purely geometrical and calculations become rather simple. The geometrical part of the effective action for the M-theory (9) is derived so as to be consistent with the 1-loop amplitudes in the type IIA superstring theory.
And the quantum correction to the M-wave solution is taken into account by explicitly solving the equations of motion (13). The solution is uniquely determined and its explicit form is given by the eq. (45). It is interesting to note that a probe D0-brane moving in this background would feel repulsive force near the horizon. It means that the solution includes the back-reaction of the Hawking radiation.
We also investigated the thermodynamic property of the quantum near horizon geometry of the non-extremal black 0-brane. Since the effective action contains higher derivative terms, we examined the thermodynamic property of the black 0-brane by employing Wald's formula.
The entropy and the internal energy of the black 0-brane are evaluated up to 1/N 2 . The quantum correction to the internal energy becomes important when N is small. In ref. [24], the internal energy is also calculated from the dual thermal gauge theory by using the Monte Carlo simulation, and it agrees with the eq. (54) very well. This gives a strong evidence for the gauge/gravity duality at the level of quantum gravity.
Finally we give an important remark on the effective action for the M-theory. It contains higher derivative terms, but these cannot be determined uniquely because of the field redefinitions. In the appendices we have considered all possible higher derivative terms and shown that the ambiguities of the effective action have nothing to do with the thermodynamic properties of near horizon geometry of the non-extremal black 0-brane.
As a future work, it is important to derive quantum geometry of the non-extremal black 0-brane and obtain the solution (45) by taking the near horizon limit. The result will be reported elsewhere, but it is really possible. It is also interesting to examine quantum correction to the black 6-brane, which is also described by purely geometrical object, called Kaluza-Klein monopole, in the eleven dimensional supergravity. To find connections of our results to the other approaches to the field theory on the D0-branes is important as well [44,45]. Other approaches to probe curvature corrections by the black brane will also be related to our results in section 6 [46,47]. Since now we capture the quantum nature of the near horizon geometry of the black 0-brane, it is interesting to consider a recent proposal to resolve the information paradox on the black hole [48,49,50].

A Calculations of Ricci Tensor and Scalar Curvature
By using the ansatz (14) for the metric, each component of the Ricci tensor up to the linear order of γ is calculated as Here we used ♮ instead of 10 andā = 2, · · · , 9. Ricci scalar up to the linear order of γ becomes like

B Calculations of Higher Derivative Terms
In this appendix we summarize the values of higher derivative terms appeared in the eq. (13).
Note that we only need to evaluate these terms by using the ansatz (14) We used ♮ instead of 10 andā,b = 2, · · · , 9. The scalar curvature and each component of the Ricci tensor become zero, and each component of X abcd in the eq. (12) is evaluated as By using these results we are ready to calculate higher derivative terms in the eq. (13). The R 4 terms are calculated as The RX terms become By inserting these results into the eq. (13), we obtain the eqs. (15)- (19).

C Generic R 4 Terms, Equations of Motion and Solution
In this appendix, we classify independent R4 terms which consist of four products of the Riemann tensor, the Ricci tensor or the scalar curvature. The R4 terms which include the Ricci tensor or the scalar curvature cannot be determined from the scattering amplitudes in the type IIA superstring theory. So in general the effective action and equations of motion are affected by these ambiguities.
First let us review the R4 terms which only consist of the Riemann tensor. Since there are 16 indices, we have 8 pairs to be contracted. Naively it seems that there are so many possible patterns. However, carefully using properties of the Riemann tensor, such as R abcd = −R bcad − R cabd , it is possible to show that there are only 7 independent terms.
In the main part of this paper we considered the R4 terms which is explicitly written in the eq. (8). In order to derive equations of motion, we need to calculate variations of (72). These are evaluated as By using these results, we evaluated the eq. (11) and derived the equations of motion (13).
Next let us consider the R4 terms which necessarily depend on the Ricci tensor or the scalar curvature. Since the procedure for the classification is straightforward, we employ a Mathematica code. As as result those are classified into 19 terms.
Then the effective action (9) is generalized into the form of The coefficients b n (n = 8, · · · , 26) cannot be determined from the results of scattering amplitudes in the type IIA superstring theory, since we can remove or add these terms by appropriate field redefinitions of the metric. Therefore it is expected that these terms do not affect physical quantities such as the internal energy of the black 0-brane. We will confirm this in the appendix D.
Let us derive equations of motion for the effective action (75). The variations of 19 terms in (74) are evaluated as δB 8 = (δR abcd ) 2R abcd R ef R ef + 2R ef gh R ef gh R ac η bd , δB 9 = (δR abcd ) 2R abcd R 2 + 2R ef gh R ef gh η ac η bd R , δB 22 = 4(δR abcd )R ac R ef R ef η bd , δB 23 = (δR abcd ) 2R ac η bd R 2 + 2R ef R ef η ac η bd R , And as like the eq. (12), we define Y tensor as Following the similar calculations in the eq. (11), finally we obtain generic equations of In order to evaluate these equations, we need to insert the values of the Riemann tensor (67) into the above. Since the Ricci tensor and the scalar curvature become zero we obtain Below we repeat the similar calculations in the appendix B. Each component of Y abcd is evaluated as

D Thermodynamics of Black 0-Brane with Generic R 4 Terms
In this appendix, we examine thermodynamics of the quantum near horizon geometry of the black 0-brane (87) by following the arguments in the section 5. Although the solution is modified, the results obtained until the eq. (50) do not change. Since the effective action is modified as in the eq. (75), the eq. (51) should be replaced with ∂S 11 ∂R µνρσ = 1 2κ 2 11 g µ[ρ g σ]ν + γℓ 12 s (X µνρσ + Y µνρσ ) .
The entropy of the quantum near horizon geometry of the black 0-brane is evaluated as