Quantum M-wave and Black 0-brane

The effective action of superstring theory or M-theory is approximated by supergravity in the low energy limit, and quantum corrections to the supergravity are taken into account by including higher derivative terms. In this paper, we consider equations of motion with those higher derivative terms in M-theory and solve them to derive quantum M-wave solution. A quantum black 0-brane solution is also obtained by Kaluza-Klein dimensional reduction of the M-wave solution. The quantum black 0-brane is asymptotically flat and uniquely determined by imposing appropriate conditions. The mass and the R-R charge of the quantum black 0-brane are derived by using the ADM mass and the charge formulae, and we see that only the mass is affected by the quantum correction. Various limits of the quantum black 0-brane are also considered, and especially we show that an internal energy in the near horizon limit is correctly reproduced.


Introduction
Superstring theory is a promising candidate for the theory of quantum gravity. UV divergence of the gravity is well controlled and it unifies gauge theory and gravity consistently. Therefore much effort has been devoted to reveal the quantum nature of the gravity. In this paper, we proceed those discussions and describe the quantum geometry in the superstring theory.
The superstring theory contains not only fundamental strings but also Dp-branes which extend p spacial directions [1]. In the low energy limit, the superstring theory is well approximated by the supergravity, and Dp-branes are described by a classical solution which is called a black p-brane [2,3]. In the extremal case where the mass and Ramond-Ramond charge of the black p-brane are balanced, quantum corrections to the classical solution are suppressed.
Then for a special class of black brane solutions, a statistical derivation of its entropy is possible from the dual gauge theory on the corresponding D-branes [4]. Furthermore, by taking a near horizon limit of extremal black branes the geometry becomes anti-de-Sitter space-time, and it is conjectured that the superstring theory in the AdS background is dual to the gauge theory on the D-branes [5]. Correlation functions of the gauge theory can be calculated from the dual gravity theory [6,7]. On the other hand, the entropy counting or test of the gauge gravity duality of non-extremal black branes is quite difficult because quantum corrections become important which are not well understood so far. In this paper, we discuss quantum corrections to the non-extremal black 0-brane by explicitly solving equations of motion 1 .
In order to investigate quantum nature of the non-extremal black p-brane, we need to know quantum corrections to the supergravity. As the superstring theory is defined perturbatively, it is possible to derive corrections to the supergravity so as to be consistent with the scattering amplitude [10,11] or σ-model calculations [12,13]. Among these corrections, the structure of higher curvature R 4 terms is well studied [14,15] and they come from 4 gravitons amplitudes at tree and 1-loop levels [10,11,16]. R 4 terms are also derived by imposing local supersymmetry [17]- [22]. There are several attempts to solve the modified equations of motion [23]- [27]. For the black p-brane, however, it is difficult to solve the equations of motion consistently since the full form of the effective action including R-R gauge fields is not completely determined so far 2 . In this paper we concentrate on the black 0-brane in type IIA supergravity, thus at least the knowledge of quantum corrections to a metric, a dilaton field and a R-R 1-form filed are necessary. This problem is resolved by noting that these fields are gathered into the metric in 11 dimensions [29]. Fortunately supersymmetric 1 Higher derivative corrections are also important for the extremal case [8,9]. 2 In three dimensions the higher derivative corrections are well controlled. Although physical quantities are affected by the quantum corrections, the solution is the same as the classical one [28].
where 2κ 2 11 = (2π) 8 ℓ 9 p . The equations of motion become R M N − 1 2 g M N R = 0, and the following geometry becomes a solution.
This is the non-extremal M-wave solution, which contains two parameters r ± . The extremal case is saturated when r + = r − , and Schwarzschild black hole smeared along z direction is obtained when r − → 0.
The type IIA supergravity consists of a graviton g µν , a dilaton φ, a R-R 1-form field C µ , a NS-NS 2-form field B µν , a R-R 3-form field C µνρ , a Majorana gravitino ψ µ and a Majorana dilatino ψ. Here µ, ν, ρ are space-time indices in 10 dimensions, and both bosonic and fermionic fields have the same physical degrees of freedom as those in 11 dimensional supergravity. The type IIA supergravity is derived by dimensional reduction of the 11 dimensional supergravity if we express the metric in 11 dimensions as where z direction is the circle with the radius R 11 [41]. By inserting this metric into the action (1), we obtain the 10 dimensional action of the form where 2κ 2 10 = (2π) 7 ℓ 8 s g 2 s and G µν is the field strength of C µ . Notice that the coordinate transformation z ′ = z + χ(x) in 11 dimensions corresponds to the gauge transformation The non-extremal black 0-brane solution is obtained by the dimensional reduction of the M-wave solution (2). By applying eq. (3), we obtain [42] Thus the purely geometrical object in 11 dimensions becomes the charged black hole in 10 dimensions. The event horizon is located at where α is a dimensionless parameter. And the mass M and the R-R charge Q of the black 0-brane are evaluated as V S 8 = 2π 9/2 Γ(9/2) = 2(2π) 4 7·15 is the volume of S 8 . Since the charge of N D0-branes is quantized as Q = N ℓsgs in the type IIA superstring theory, the parameters r ± can be expressed as by introducing a non-negative parameter δ. There are three limits of the solution (5) which are important in later sections. (See table. 1.) First one is the extremal limit r + → r − which is equivalent to α → 0. Second one is the Schwarzschild limit r − → 0, where the charge Q goes to zero. Final one is the near horizon limit which is realized by [43]. This is equivalent to α → 0 by fixing r − α/r, ℓ 2 s /(r − α) and g 2 s N 2 /(r − α) 3 , so the near horizon limit corresponds to the near extremal limit.
Extremal limit α → 0 Schwarzschild limit r − → 0 Near horizon limit α → 0 by fixing r − α r , ℓ 2 s r − α and g 2 s N 2 (r − α) 3 Table 1: Extremal, Schwarzschild and near horizon limits 3 Quantum M-wave and Black 0-brane In this section, we consider leading quantum correction to the M-wave solution (2). Since the M-wave solution is purely geometrical in 11 dimensions, the 3-form A M N P is irrelevant to our analyses. Thus it is enough to investigate the effective action of the M-theory which depends only on higher curvature terms. The leading structure of those is known to be R 4 terms [14,15]. The explicit form of the effective action is given by Although we neglected fermionic terms, a part of them is also obtained in refs. [20,21,22].
The expansion parameter in the action is given by and a, b, c, · · · = 0, 1, · · · , 10 are local Lorentz indices. All indices are lowered for simplicity but should be contracted by the flat metric. Note that γ ∼ g 2 s ℓ 6 s , so when the effective action (9) is reduced to ten dimensions, it becomes one-loop leading corrections to the type IIA supergravity. By varying the effective action (9), equations of motion are obtained as Here D a is a covariant derivative for local Lorentz indices and The details of the derivation can be found in ref. [29].
Let us solve the eq. (11) up to the linear order of γ. The leading part of the metric (2) itself is not a solution of the eq. (11), we should relax the ansatz for the M-wave. Most general static ansatz with SO(9) rotation symmetry is given by Here α is given by α = (r 7 + /r 7 − − 1) 1 7 , and h i (i = 1, 2, 3) and f 1 are functions of r r − α . Note that, up to the linear order of γ, the coordinate transformation dz → dz + c g dt is interpreted It is clear that this corresponds to the gauge transformation of C µ in 10 dimensions.
Now we insert the ansatz (13) into the eq. (11). In order to make the equations of motion simple, we introduce following dimensionless coordinates, Then all components of the metric are expressed as a function of x, and we obtain following 5 differential equations for h i (x) and f 1 (x) 3 .
The above equations come from the linear order of γ, and equations of O(γ 0 ) are automatically satisfied. In following subsections, we will solve 5 equations of motion step by step and determine h i (x) and f 1 (x). Although there appear several integral constants, these are uniquely fixed by imposing appropriate conditions. For example, we require that h i (x) 3 In order to derive the equations of motion, 3 independent Mathematica codes are used. and f 1 (x) do not diverge around the event horizon, and h 3 (x) should be determined up to the gauge transformation (14). Consistency conditions with the extremal limit and the Schwarzschild one in table 1 are also taken into account. A complete form of the solution is summarized in section 4.

Solve
From this equation, ( There remain 4 equations to be solved.

Solve
Note that the eq. (22) is employed to eliminate f 1 (x) out of the equation. It is possible to integrate the above equation once, and the result becomes can be derived by integrating the eq. (24).
In order to execute the integral, we define the following function, It is useful to note following relations.
By using these relations it is possible to integrate the eq. (24), and the result becomes where c 2 is an integral constant and c 1 is redefined as Now we are ready to derive an explicit form of f 1 (x). By inserting the eq. (24) into the eq. (22), we obtain differential equation for f 1 (x), By using the relations (26), f 1 (x) can be solved as where c 3 is an integral constant. So far we solved There remain 3 equations to be solved.

Solve
Here we employed the eq. (24). This is a differential equation only on h 2 (x), and it is possible to integrate it once by employing the relations (26). The result is calculated as where c 4 is an integral constant. Now it is easy to integrate the above differential equation, and h 2 (x) is derived as where c 5 is an integral constant. Inserting this result into the eq. (28), we obtain We have already solved , h 2 (x) and f 1 (x). There remain 2 equations to be solved.

Solve E 5 = 0
The equation E 5 = 0 should be solved to determine h 3 (x).
This can be easily integrated once, and by using the eqs. (33) and (34) we obtain a differential equation for h 2 (x) + h 3 (x).
where c 6 is an integral constant. In order to obtain the last equality we used relations (26). This is a differential equation only on h 2 (x) + h 3 (x) and we obtain where c g is an integral constant. As explained in eq. (14), c g corresponds to the parameter of the coordinate transformation dz → dz + c g dt. Since the explicit form of h 2 (x) is given by the eq. (33), h 3 (x) is expressed as Now we have solved 4 equations and derived h i (x) and f 1 (x). There remain only one equation to be solved.

Solve E 1 = 0
The final independent equation which should be solved is E 1 = 0. By inserting solutions obtained so far, the equation E 1 = 0 gives a relation among integral constants.
From this, c 3 is expressed as There remain 4 integral constants c 2 , c 4 , c 5 and c 6 in h i (x), which should be fixed by imposing boundary conditions and requiring consistencies with various limits in table 1.

Determination of integral constants
Our remaining task is to determine integral constants c 2 , c 4 , c 5 and c 6 in h i (x). First of all, the solution should be asymptotically flat. This means that when x goes to the infinity, h 1 (x), h 2 (x) and f 1 (x) should vanish and h 2 (x) + h 3 (x) should do up to the coordinate transformation (14). Since the function I(x) is expanded around x ∼ ∞ as Due to the asymptotic flatness, constants in h 1 (x) and h 2 (x) should vanish in the above equations. Therefore we obtain and asymptotic behaviors of Note that f 1 (x) automatically goes to zero when x → ∞.
Below we consider consistency conditions with the extremal limit and the Schwarzschild one in the table 1. In the extremal limit α → 0, since the mass and the charge are fixed to be the same, 1 α 6 f 1 ( r r − α ) should be zero and 1 α 13 (h 2 ( r r − α )+h 3 ( r r − α )) should vanish up to the gauge transformation. These conditions are satisfied if c 3 ∼ α n (n ≥ 0) and c 4 ∼ c 6 ∼ α n (n ≥ 7).
Combining these restrictions with the eq. (39), we obtain and finally asymptotic behaviors of h 1 (x), h 2 (x), h 2 (x) + h 3 (x) and f 1 (x) become Therefore all integral constants except c g are uniquely determined by imposing asymptotic flatness and requiring consistencies with the extremal limit and the Schwarzschild one.

3 Limits of the Quantum Black 0-brane 4.1 The quantum black 0-brane solution
Let us summarize the quantum black 0-brane solution. The dimensional reduction of the quantum M-wave solution is identified with the quantum black 0-brane in 10 dimensions, and the geometry is described by where And the functions h i (x) and f 1 (x) are uniquely determined as

The extremal limit
The extremal limit is realized by α → 0. Because of the asymptotic behaviors (45), all quantum corrections are irrelevant and the solution is given by the eq. (46) with

Mass, R-R Charge and Internal Energy of the Quantum Black 0-brane
In this section we evaluate the mass, the R-R charge and the internal energy of the quantum black 0-brane. Formulae for the mass and the R-R charge including higher derivative corrections are discussed in the appendix. As a result, we can employ usual ADM mass and charge formulae.

The Mass
The mass of the quantum black 0-brane is calculated by using ADM mass formula. In order to use the formula the metric should be written in Einstein frame. The metric in the Einstein frame g E µν is written as g E µν = e − φ 2 g µν , so the line element in the Einstein frame is given by where r 2 = (x 1 ) 2 + · · · + (x 9 ) 2 and i, j = 1, · · · , 9. Then, by using the eq. (45), the deviation from the flat space-time is given by Now we are ready to apply the ADM mass formula. The mass of the quantum black 0-brane is evaluated as Thus the mass receives the nontrivial quantum correction at 1-loop level.

The R-R charge
The black 0-brane couples to the R-R 1-form field and carries the R-R charge. From the eq. (46), the R-R 1-form field C and 2-form field strength G 2 = dC are given by Then the R-R charge of the quantum black 0-brane is evaluated as Quantum corrections do not contribute to the R-R charge because h 2 ( r when r → ∞. Therefore the R-R charge remains the same as the classical one.

Near horizon limit of the internal energy
The internal energy E of the quantum black 0-brane is defined by E = M − Q. From the eqs. (56) and (58), the internal energy is written as Below we consider the near horizon limit of the internal energy, which was first derived in ref. [29] by using the black hole thermodynamics.
Since we will express E in terms of the temperature in the near horizon limit, first we need to derive the relation between parameters in the solution and the temperature. The location of the event horizon r = r H is obtained by solving F 1 (r H ) = 0 and modified up to the linear order of γ as The Hawking temperature of the black 0-brane is given by up to the linear order of γ. Now let us take the near horizon limit, Then the temperature (61) approaches tõ where Here dimensional quantitiesT ≡ T /λ 1 3 andŨ 0 ≡ U 0 /λ 1 3 are introduced to make expressions simple. From the eq. (63), it is possible to expressŨ 0 in terms ofT as By using this relation, the near horizon limit of the internal energy is expressed as a function of the temperature.
By inserting numerical values of (64), the dimensionless internal energy of the quantum black 0-braneẼ = E/λ 1 3 is written asẼ This is exactly the same as the result derived in ref. [29]. The above relation is also reproduced by the numerical simulation from the dual gauge theory [36], which strongly supports the gauge gravity duality at the quantum level.

Conclusion and Discussion
In give some astrophysical predictions to be observed in the future. For other black p-branes, first we need to know the effective action including R-R gauge fields.

A Higher Derivative Corrections to the ADM Mass and the R-R Charge
Since the effective action of the M-theory (9) contains higher curvature terms, the ADM mass formula and the charge of the non-extremal M-wave or the black 0-brane would be affected by these terms. In this section we derive higher curvature corrections to the ADM mass and the charge via Noether and Wald's method [44,45]. We use the vielbein formalism which is important for the supergravity [46].
First, the variation of the Lagrangian (9) is evaluated as where δe ij = e i M δe M j and Θ M (δ) is defined by E ij = 0 is the equations of motion given by the eq. (11). The variation of the Lagrangian becomes total derivative term up to E ij = 0. Since these variations are not covariant under the local Lorentz transformation, we consider following field dependent local Lorentz transformation simultaneously.
Then the variations of fields with (70) becomes δe M a = − 1 2 g M N δg N P e P a , δω N ab = 1 2 e aP e bQ (∇ Q δg P N − ∇ P δg QN ), where ∇ P represents a covariant derivative with respect to space-time indices. These variations are covariant under local Lorentz transformation. In the following we writeδ as δ for simplicity. Now we consider the variation for the general coordinate transformation x ′M = x M − ξ M . In this case the variation (71) becomes Inserting the eq. (72) into the eq. (69), Θ M (ξ) = Θ M (δ ξ ) is evaluated as Therefore the variation of the Lagrangian is given by Next, let us consider the variation of the Lagrangian by using the general covariance.
Since e −1 L transforms as a scalar field, the Lagrangian does like From the eqs. (74) and (75), the Noether current is constructed as HereQ M N (ξ) is an antisymmetric tensor and represents the ambiguity of the current. In order to fixQ M N (ξ), let us consider the variation of the current.
Since each non zero component of the Riemann tensor behaves like R abcd ∼ O( 1 r 9 ) asymptotically, in the above we dropped terms which depend on X abcd . The last equation is equivalent to the variation of the ADM mass [45]. Therefore the mass is simply given by the ADM mass formula.
When ξ is chosen as an asymptotic translation along z direction ξ Z , it satisfies ξ t Z Θ r (δ) − ξ r Z Θ t (δ) = 0, and the charge is given by As explained in the mass formula, terms which depend on X abcd are dropped, and the dimensional reduction (3) is used in the last step. Thus the R-R charge is given by the integral of the R-R flux as usual.
Finally we choose the Killing vector as ξ = ξ T + Ωξ Z , which becomes zero at the bifurcate horizon Σ. Then δ ξ e M a = δ ξ ω N ab = 0, and the symplectic current and the variation of the current also vanish. Furthermore, if the variation of the fields, such as δe M a , satisfy linearized equations of motion, the eq. (80) is simplified as By integrating the above equation over 10 dimensional asymptotically flat space with the horizon, we derive the first law of the black hole, Here κ is the surface gravity which is given by ∇ M ξ N = κN M N at the bifurcate horizon Σ, and N M N is an antisymmetric tensor which is binormal to the bifurcate horizon Σ. S corresponds to the entropy and is given by √ h is a volume factor of Σ.