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a r X i v : 1 3 1 1
. 7 5 2 6 v 1 [ h e p - t h ] 2 9 N o v 2 0 1 3
Quantum Near Horizon Geometry of Black 0-Brane
Yoshifumi Hyakutake
College of Science, Ibaraki University Bunkyo 1-1, Mito, Ibaraki 310-0062, Japan
Abstract
We investigate a bunch of D0-branes to reveal its quantum nature from the gravityside. In the classical limit, it is well described by a non-extremal black 0-brane in typeIIA supergravity. The solution is uplifted to the eleven dimensions and expressed bya 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 quantumgravity. Thermodynamic property of the quantum near horizon geometry of the black0-brane is also studied by using Walds formula. Combining our result with that of theMonte Carlo simulation of the dual thermal gauge theory, we nd strong evidence forthe gauge/gravity duality in the D0-branes system at the level of quantum gravity.
http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v1http://arxiv.org/abs/1311.7526v18/13/2019 Hyakutake-Quantun Near Zero Geometry
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1 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 super-
string 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 su-
pergravity, 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 space-
time [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
denition 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 studiedby 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 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 gs correction,
it is possible to give a direct test for the gauge/gravity duality at the level of quantum
gravity [24]. ( = 2s is the string length squared and gs is the string coupling constant.)
The purpose of this paper is to derive quantum correction to the near horizon geometryof 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 and gs . For example, since four point amplitudes of gravitons at tree and one loop level
are nontrivial, there should exist terms like 3e 2 t8t8R4 and 3g2s t8t8R4 in the effective
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action, respectively [ 27][36]. These are called higher derivative terms and t8 represents
products of four Kroneckers deltas with eight indices. Especially we are interested in the
latter terms, which give nontrivial gs 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 localsupersymmetry in ten dimensions. So the supersymmetrization of 3g2s t8t8R4 is very im-
portant [ 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 eld and R-R 1-form eld only.
Notice that these elds are collected into the metric in eleven dimensional supergravity [39],
and the black 0-brane is expressed by M-wave solution. Then 3g2s t8t8R4 and other terms
which include the dilaton and R-R 1-form eld are simply collected into 6 pt8t8R4 terms in
eleven dimensions. Here p = s g1/ 3s 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 g2s . The M-wave geometry re-
ceives the quantum corrections and thermodynamic quantities for the M-wave are modied.
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 . Section 6 is devoted to conclusion and discussion.
Detailed calculations and discussions on the ambiguities of the higher derivative corrections
are collected in the appendices.
2 Classical Near Horizon Geometry of Black 0-Brane
In this section, we briey review the non-extremal solution of the black 0-brane which carries
mass and R-R charge. Especially we up lift 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 , the dilaton and R-R 1-form eld C , the relevant part of the
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type IIA supergravity action is given by
S (0)10 = 12210 d10x g e 2 R + 4 14G G , (1)
where 2210 = (2 )78s g2s and G is the eld strength of C . gs and s are the string coupling
constant and the string length, respectively. It is possible to solve the equations of motionby 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.)
ds210 = H 1
2 F dt 2 + H 12 F 1dr 2 + H
12 r 2d28, (2)
e = H 34 , C =
r +r
72
H 1dt,
H = 1 + r 7r 7
, F = 1 r 7+ r 7
r 7 .
The horizon is located at r H = ( r 7+
r 7 )17 . Parameters r are related to the mass M 0 and
the R-R charge Q0 of the black 0-brane by
M 0 = V S 82210
8r 7+ r 7 , Q0 = N s gs
= 7V S 82210
r + r 72 , (3)
where N is a number of D0-branes and V S 8 = 29 / 2
(9 / 2) = 2(2 )4
7 15 is the volume of S 8. Now the
parameters r are expressed as
r 7 = (1 + ) 1(2)215gs N7s , (4)
where is a non-negative parameter. The extremal limit r+ = r is saturated when = 0.
Let us rewrite the solution ( 2) in terms of U = r/ 2s and = gs N/ (2)23s , 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 s 0 while U , and/ 4s are xed. Then the near horizon limit of the solution ( 2) becomes [8]
ds210 = 2s H
12 F dt 2 + H
12 F 1dU 2 + H
12 U 2d28 , (5)
e = 3s H 34 , C = 4s H
1dt,
H = (2)415
U 7 , F = 1
U 70U 7
,
where U 70 = 24s (2)415 .
The type IIA supergravity is related to the eleven dimensional supergravity via circle
compactication. In fact, the eleven dimensional metric is related to the ten dimensional
one like ds211 = e 2/ 3ds210 + e4/ 3(dz C dx)2. The near horizon limit of the non-extremal
solution of the black 0-brane ( 5) can be up lifted to eleven dimensions as
ds211 = 4s H
1F dt 2 + F 1dU 2 + U 2d28 + ( 4s H
12 dz H
12 dt)2 . (6)
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This represents the near horizon limit of the non-extremal M-wave solution in eleven dimen-
sions. The solution is purely geometrical and the expressions become simple. Furthermore,
on the geometrical part, quantum corrections to the eleven dimensional supergravity are un-
der control. This is the reason why we execute analyses of the solution in eleven dimensions.
3 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 elds of the eleven dimensional supergravity consists of a vielbein ea , a Ma-
jorana gravitino and a 3-form eld A . 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.
2211S (0)11 = d11xeR, (7)
where 2211 = (2 )89 p = (2 )89s g3s . Notice that after the dimensional reduction this becomes
the action ( 1), which contains the dilation and the R-R 1-form eld as well as the graviton
in ten dimensions [ 39].Of course there are other terms which depend on and A , which are completely
determined by the local supersymmetry. For example, a variation of the vielbein under
the local supersymmetry is given by [e] = [ ]. Here we use a symbol [X ] to abbreviate
indices and gamma matrices in X , and represents a parameter of the local supersymmetry.
Then the variation of the scalar curvature is written by [eR] = [eR ]. In order to cancel
this, we see that a variation of the Majorana gravitino should include [] = [D ] + and simultaneously there should exist a term like [ e2] in the action. Here 2 represents
the eld strength of the Majorana gravitino. By continuing this process, it is possible todetermine 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
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like [eR 4]. This is directly uplifted to the eleven dimensions and we see that the effective
action of the M-theory should include terms like B1 = [eR 4]7. The subscript 7 indicates
that there are potentially 7 independent terms if we consider possible contractions of 16
indices out of 4 Riemann tensors. (To be more precise, we excluded terms which contain
Ricci tensor or scalar curvature, since these can be eliminated by redenition of the graviton.Discussions on these terms will be found in the appendix C.) As in the case of the eleven
dimensional supergravity, it is possible to determine other corrections by requiring the local
supersymmetry. For example, variations of B1 under the local supersymmetry contain terms
like V 1 = [eR 4 ]. In order to cancel these terms, B11 = [e 11AR 4]2 and F 1 = [eR 3 2]92should exist in the action. The structures of B1 , B11 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 B1 are completely determined up to over all factor [ 35, 36]. The result become
as follows.
2211 S (1)11 =
26 p3 284! d11x e t8t8R4
14! 11 11
R4
=26 p
3 284! d11x e 24 Rabcd Rabcd Refgh Refgh 64Rabcd Raefg Rbcdh Refgh+ 2 Rabcd Rabef Rcdgh Refgh + 16 Racbd Raebf Rcgdh Regfh
16Rabcd Raefg Rbefh Rcdgh 16Rabcd Raefg Rbfeh Rcdgh . (8)
Here t8 is products of four Kroneckers deltas with eight indices and 11 is an antisymmetric
tensor with eleven indices. Local Lorentz indices are labelled by a,b, = 0, 1, , 10.Although all indices are lowered, it is understood those are contracted by the at metricab . The Riemann tensor with local Lorentz indices is dened by Rabcd = e ce d( ab ab + a eeb a eeb), where ab is a spin connection and , 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 eld are irrelevant to the analyses for the M-wave. In summary the
effective action of the M-theory is described by
S 11 = S (0)11 + S
(1)11 =
12211 d11x e R + 12s t8t8R4 14! 11 11R4 , (9)
where = 2
3 28 4!g2s6s
= 6
27 32 2N 2 . Notice that the parameter remains nite 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.
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Now we derive equations of motion for the action ( 9). Although the derivation is straight-
forward, 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 elds with respect to the vielbein.
e = eei e i = eij eij ,cab = ecab = ( k[a b]icj +
k[a b] j ci +
kc i [a b] j )Dke
ij ,
R abcd = e cRabd + e dRabc + e ce dRab = 2eij Rabi [cd] j + 2D [cd]ab , (10)R ab = eij Rajib + eij Rai bj + Dbcac Dcbac,
where eij ei ej . Then variations of the higher derivative terms are evaluated as
e t8t8R4
14! 11 11R
4
= 24 e 4(Rabcd )Rabcd Refgh Refgh 64(Rabcd )Rabce Rdfgh Refgh+ 8( Rabcd )Rabef Rcdgh Refgh + 64( Rabcd )Raecg Rbfdh Refgh
64(Rabcd )Rabeg Rcfeh Rdfgh 64(Rabcd )Refag Refch Rgbhd+ 32( Rabcd )Rabef Rcegh Rdfgh
= e(Rabcd )X abcd
= 2 eeij Rabci X abc j 2eX abcd Ddcab= 2eeij Rabci X abc j
2e( k
abicj + k
abj ci + k
cia bj )DkDdX abcd eij
= 2 eR abci X abc j eij 2eD cD d(X cijd + X cjid + X ijcd )eij= e(3Rabci X abc j Rabcj X abc i)eij 2eD cD d(X cijd + X cjid )eij= e 3Rabci X abc j Rabcj X abc i 4D (a Db)X a ij b eij , (11)
where we dened
X abcd = 12
X [ab][cd] + X [cd][ab] , (12)
X abcd = 96 Rabcd Refgh Refgh 16Rabce Rdfgh Refgh + 2 Rabef Rcdgh Refgh+ 16 Raecg Rbfdh Refgh 16Rabeg Rcfeh Rdfgh 16Refag Refch Rgbhd+ 8 Rabef Rcegh Rdfgh .
Finally we obtain the equations of motion for the effective action ( 9).
E ij R ij 12
ij R + 12s 12
ij t8t8R4 14! 11 11
R4
+ 32
R abci X abc j 12
Rabcj X abc i 2D (a Db)X a ij b = 0 . (13)
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by
E 1 = 63x34f 1 9x35f 1 49x41h1 + 49 x34(1 x7)h2 + 23 x35(1 x7)h
2 + 2 x
36(1 x7)h2
+ 98 x41h3 + 7 x42h3 63402393600x14 + 70230343680x7 + 1062512640 = 0 , (15)
E 2 = 63x34f 1 + 9 x35f 1 + 7 x34(9 2x7)h1 + 9 x35(1 x7)h 1 112x34(1 x7)h216x35(1 x7)h
2 98x41h3 7x42h
3 2159861760x7 5730600960 = 0, (16)
E 3 = 133x34 f 1 + 35 x35f 1 + 2 x
36f 1 + 28x34(3 10x7)h1 + 7 x35(4 7x7)h
1 + 2 x
36(1 x7)h1
7x34(5 26x7)h2 21x35(1 2x7)h2 2x36(1 x7)h
2 + 98x
41h3 + 7 x42h3 (17)
+ 5669637120x7 8626383360 = 0,E 4 = 259x34 f 1 + 53 x35f
1 + 2 x
36f 1 + 147x34(1 3x7)h1 + x35(37 58x7)h
1
+ 2 x36(1
x7)h
1 + 147x41h2 + 21 x42h
2 + 294x41h3 + 21 x42h
3 (18)
63402393600x14 + 133632737280x7 71292856320 = 0,E 5 = 49x34h1 + 7 x35h
1 + 49 x
34h2 x35h2 x36h
2 98x34h3 22x35h
3 x36h
3
63402393600x7 + 70230343680 = 0 . (19)
Here we dened E 1 = 4U 80 4s x36 1E 00 , E 2 = 4U 80 4s x36
1E 11 , E 3 = 4U 80 4s x36 1E 22 ,
E 4 = 4U 80 4s x36 1E 1010 and E 5 = 4U 80 4s x
652 (1 + x7)
12 1E 010 . Note that the above
equations are derived up to the order of , and a part of 0 is zero since the ansatz ( 14) is
a uctuation around the classical solution ( 6).Now we solve these equations to obtain hi and f 1. We will see that hi 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.
19x28(x7 1)
(E 1 + E 2) = 7x6h1 x7h1 + 7 x
6h2 79
x7h 2 29
x8h 2
+ 518676480
x28 7044710400
x21
= x7h1 + x7h2 29x8h
2 +
352235520x20
19210240x27
= 0 . (20)
From this equation h1 is expressed in terms of h2 as
h1 = h2 29
xh 2 + c1x7
+ 352235520
x27 19210240
x34 , (21)
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where c1 is an integral constant. Next let us evaluate E 5.
1x28
E 5 = 49x6h1 + 7 x7h1 + 49 x
6h2 x7h2 x8h
2 98x6h3 22x7h
3 x8h
3
63402393600
x21 +
70230343680x28
= 7x7h1 + 7 x7h2 x8h2 14x7h3 x8h
3 + 3170119680x20 2601123840x27
= 14x7h2 239
x8h 2 14x7h3 x8h3 +
5635768320x20
2735595520x27
= 0 . (22)
In the last line, we removed h1 by using the eq. ( 21). Thus a linear combination of h3 is
expressed in terms of h2 as
14x7h3 + x8h3 = 14x
7h2 23
9 x8h 2 + c2 +
5635768320x20
2735595520x27
, (23)
where c2 is an integral constant. From the eqs. ( 21) and ( 23), it is possible to remove h1 and
h3 out of E 1, E 3 and E 4. After some calculations, we obtain three equations remaining to
be solved.
E 1 = 63x34f 1 9x35f 1 + 49 x
34h2 + x35(23 30x7)h2 + 2 x
36(1 x7)h2
49c1x34 + 7 c2x34 41211555840x14 + 52022476800x7 + 1062512640 = 0 , (24)E 3 = 133x34f 1 + 35 x35f
1 + 2 x
36f 1
+ 49 x34h2 79
x35(23 62x7)h2
29
x36(32 53x7)h2
49
x37(1 x7)h2 (25)
49c1x34
+ 7 c2x34
125748080640x14
+ 301493283840x7
37672266240 = 0,E 4 = 259x34f 1 + 53 x35f
1 + 2 x
36f 1
+ 147 x34h2 79
x35(5 26x7)h2
29
x36(32 53x7)h2
49
x37(1 x7)h2 (26)
147c1x34 + 21 c2x34 81366405120x14 + 324970168320x7 95670650880.
Notice, however, that three functions E 1 , E 3 and E 4 are not independent because of the
identity
E 4 = 2
7xE 1
9E 1 +
16
7 E 3. (27)
This corresponds to the energy conservation, Da E ab = 0. Thus we only need to solve
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following two equations.
12
E 1 + 14
(E 3 E 4) = 114
xE 1 + 74
E 1 928
E 3
= 49x34h2 x35(15 22x7)h2 x36(1 x7)h
2 + 7(7 c1 c2)x34
+ 9510359040x14 31880459520x7 + 13968339840 = 0 , (28)12
(E 3 E 4) = 17
xE 1 + 92
E 1 914
E 3
= 63x34f 1 9x35f 1 49x34h2 7x35(1 2x7)h
2 + 7(7 c1 c2)x34
22190837760x14 11738442240x7 + 28999192320 = 0 . (29)
By solving the eq. ( 28), nally we obtain h2 as
h2 = 19160960
x34 58528288
x27 +
221356813x20
122976013x13
+ c1 c27 + 2459520x6 + c43136x7 + 1054080 2 1x7 I (x), (30)I (x) =
c3944455680
+ log(x 1) + c4
6611189760 log(1 x
7)
n =1 ,3,5
cos n7 log x2 + 2 x cos n7 + 1
2n =1 ,3,5
sin n7 tan 1 x + cos
n7
sin n7, (31)
where c3 and c4 are integral constants. Although the form of I (x) seems to be complicated,
its derivative becomes
I (x) = 7
x7 11 +
c4 x 1
6611189760. (32)
So far there are four integral constants, but these will be xed by appropriate conditions.
In fact it is natural to require that hi (1) are nite and hi (x) O(x 8) when x goes to
the innity. In order to satisfy these conditions, it is necessary to choose c2 = 7c1, c3 =
944455680(sin 7 + sin 3
7 + sin 5
7 ) and c4 = 6611189760. Inserting these values into theeqs. (30), (31) and ( 32), we obtain
h2 = 19160960
x34 58528288
x27 + 2213568
13x20 1229760
13x13
2108160
x7 +
2459520x6
+ 1054080 2 1x7
I (x), (33)
I (x) = log x7(x 1)
x7 1 n =1 ,3,5cos n7 log x
2 + 2 x cos n7 + 1
2n =1 ,3,5
sin n7 tan 1 x + cos
n7
sin n7 2
, (34)
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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 in ten dimensions.
Let us summarize the quantum correction to the near horizon geometry of the non-
extremal M-wave and the black 0-brane. By solving the eqs. ( 15)(19), we obtained thequantum near horizon geometry of the non-extremal M-wave,
ds211 = 4s H
11 F 1dt
2 + F 11 U 20 dx
2 + U 20 x2d28 +
4s H
12
2 dz H 1
2
3 dt2 , (42)
H i = (2)415
U 70
1x7
+2
U 60h i , F 1 = 1
1x7
+2
U 60f 1.
In stead of , we introduced dimensionless parameter
= 2
= 6
2732N 2 0.835
N 2 , (43)
and the functions hi and f 1 are uniquely determined as
h1 = 1302501760
9x34 57462496
x27 +
1205164813x20
478240013x13
3747840
x7 +
4099200x6
1639680(x 1)(x7 1)
+ 117120 18 23x7
I (x),
h2 = 19160960
x34 58528288
x27 +
221356813x20
122976013x13
2108160
x7 +
2459520x6
+ 1054080 2 1x7
I (x),
h3 = 3611104009x34 59840032x27 2402131213x20 5807200013x13 (44)
2108160
x7 +
2459520x6
+ 117120 18 41x7
I (x),
f 1 = 1208170880
9x34 +
161405664x27
+5738880
13x20 +
956480x13
+819840
x7 I (x).
The function I (x) is dened by the eq. ( 34). In order to x the integral constants, we
required that hi (1) are nite and hi (x), f 1(x) O(x 8) when x goes to the innity. After
the dimensional reduction to ten dimensions, we obtain
ds210 = 2s H
11 H
12
2 F 1dt2 + H
12
2 F 11 U
20 dx
2 + H 12
2 U 20 x
2d28 , (45)
e = 3s H 34
2 , C = 4s H
12
2 H 1
2
3 dt.
This represents the quantum near horizon geometry of the non-extremal black 0-brane.
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5 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 Walds formula [ 41, 42]. These quantities are quite
important when we test the gauge/gravity duality.
In the following, quantities are calculated up to O( 2). First of all, let us examine thelocation of the horizon xH . This is dened by F 1(xH ) = 0 and becomes
xH = 1 f 1(1)
7U 60 , (46)
where U 0 U 0/13 is a dimensionless parameter. Temperature of the black 0-brane is derived
by the usual prescription. We consider the Euclidean geometry by changing time coordinateas t = i and require the smoothness of the geometry at the horizon. This xes theperiodicity of direction and its inverse gives the temperature of the non-extremal black
0-brane. Then the dimensionless temperature T = T/13 of the black 0-brane is evaluated
as
T = 14
U 10 H 1
2
1 F 1 xH
13 = a1 U
52
0 1 + a 2 U 60 , (47)
where a1 and a2 are numerical constants given by
a1 = 7
163 15 0.00206,a2 =
914
f 1(1) + 17
f 1(1) 12
h1(1) 937000. (48)
Inversely solving the eq. ( 47), the dimensionless parameter U 0 is written in terms of the
temperature T as
U 0 = a 2
5
1 T
25 1
25
a125
1 a2 T 12
5 , (49)
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-extremalblack 0-brane. In practice, we consider the quantum near horizon geometry of the non-
extremal M-wave because of its simple expression. Since the effective action ( 9) includes
higher derivative terms, we should employ Walds entropy formula which ensures the rst
law of the black hole thermodynamics. The Walds entropy formula is given by
S = 2 H d8dz h S 11R N N , (50)13
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The specic heat is evaluated as
1N 2
d E dT
= 95
a3 T 95
35
a3a4 T 3
5 . (55)
Notice that the specic heat becomes negative in the region where T < ( a 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 isalso veried from the Monte Carlo simulation of the dual gauge theory [ 24].
6 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 eld.
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 eld 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
S D0 = T 0 dte g dx
dtdx
dt + T 0 C
=
T 04
s dtH
12
2 H 1
1 F
1 F 1
1 U 2
0 x2 + T
04
s dtH
12
2 H
12
3 . (56)
The momentum conjugate to x is evaluated as
p = T 04s H 1
2
2F 11 U 20 x
H 11 F 1 F 11 U 20 x2, (57)
and the energy of the probe D0-brane is given by
E D0 = px + T 04s H 1
2
2 H 11 F 1 F 11 U 20 x2 T 04s H 12
2 H 1
2
3
= T 04s H
12
2
H 11 F 1
H 11 F 1 F 11 U 20 x2 T 04s H
12
2 H 1
2
3
= T 04s H 1
2
1 H 1
2
2 F 12
1 1 + pF 12
1 H 12
2T 04s U 0
2
T 04s H 1
2
2 H 1
2
3
12
H 1
2
1 H 12
2 F 32
1 p2
T 04s U 20+ T 04s H
12
1 H 1
2
2 F 12
1 H 1
2
2 H 1
2
3 . (58)
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In the nal line we took the non-relativistic limit. From this we see that the potential energy
for the probe D0-brane is expressed as
V D0 = T 04s H 1
2
1 H 1
2
2 F 12
1 H 1
2
2 H 1
2
3 . (59)
The rst term corresponds to the gravitational attractive force and the second one does tothe R-R repulsive force.
When we take N = , the potential energy becomes V D0 = T 04s H 1( F 1). The part
( F 1) shows that the gravitational attractive force overcomes the R-R repulsive force.Similarly, when N is nite, we regard F 1 as the gravitational attractive force to the probeD0-brane. The function of F 1 is plotted in g. 1. From this we see that the gravitationalforce becomes repulsive near the horizon xH .
x0.8 1.0 1.2 1.4 1.6 1.8 2.0
2
4
6
8
10
Figure 1: The function F 1(x) with F 1(x) = 1 1/x 7 + 0 .000001f 1(x).
7 Conclusion and Discussion
In this paper we studied quantum nature of the bunch of D0-branes in the type IIA super-string 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
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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 explicitform 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 Walds 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 eld rede-
nitions. 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 nd connections of
our results to the other approaches to the eld theory on the D0-branes is important as
well [43, 44]. 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 [ 45, 46].
Acknowledgement
The author would like to thank Masanori Hanada, Goro Ishiki and Jun Nishimura for dis-
cussions and collaborations. He would also like to thank Masafumi Fukuma, Hideki Ishihara,
Hikaru Kawai and Yukinori Yasui for discussions. This work was partially supported by the
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Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (B)
19740141, 2007 and 24740140, 2012.
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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 is calculated as
R00 = 4U 80 x24s98f 1 + 30 xf 1 + 2 x2f 1 + 49(2 7x7)h1 + 3 x(10 17x7)h 1 + 2 x2(1 x7)h 1
+ 147 x7h2 + 21 x8h2 + 196x
7h3 + 14 x8h3 ,
R11 =
4U 80 x24s 98f 1 30xf 1 2x2f
1 35(1 8x7)h1 21x(1 2x7)h
1 2x2(1 x7)h
1
7(9 + 12x7)h2 + 7 x(1 4x7)h2 + 2 x
2(1 x7)h2 196x7h3 14x8h
3 , (60)
R a a =
2U 80 x24s 14f 1 2xf 1 7(1 x7)h1 x(1 x7)h
1 + 7(1 x7)h2 + x(1 x7)h
2 ,
R =
4U 80 x24s98f 1 + 14 xf
1 + 49(1 3x7)h1 + 7 x(1 x7)h
1
+ 49(1 x7)h2 + 23 x(1 x7)h 2 + 2 x2(1 x7)h 2 + 196x7h3 + 14 x8h 3 ,R0 =
x3/ 2 x7 14U 80 4s
49h1 + 7 xh1 + 49 h2 xh
2 x2h
2 98h3 22xh
3 x2h
3 .
Here we used instead of 10 and a = 2 , , 9. Ricci scalar up to the linear order of becomes like
R =
2U 80 x24s 161f 1 39xf 1 2x2f
1 98(1 3x7)h1 3x(10 17x7)h
1 2x2(1 x7)h
1
+ 49(1 4x7)h2 + x(23 44x7)h2 + 2 x
2(1 x7)h2 98x7h3 7x8h
3 . (61)
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) with = 0, because
the equations of motion are solved up to the linear order of . First of all, each component
of Rabcd is calculated as
R0101 = 28
U 20 x24s, R0a 0a =
72U 20 x24s
,
R011 = 28 x7 1U 20 x
112 4s
, R0a a = 7 x7 1
2U 20 x112 4s
,
R1 1 = 28(x7 1)
U 20 x94s, R1a 1a =
72U 20 x94s
, (62)
R a a = 7(x7 1)
2U 20 x94s, R a ba b =
1U 20 x94s
.
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We used instead of 10 and a, 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
X 0101 = 20321280U 60 x2012s
, X 0a 0a = 1270080U 60 x2012s
,
X 011 = 20321280 x7 1U 60 x472 12s
, X 0a a = 1270080 x7 1U 60 x472 12s
,
X 1 1 = 20321280(x7 1)
U 60 x2712s, X 1a 1a =
1270080U 60 x2712s
, (63)
X a a = 1270080(x7 1)
U 60 x2712s, X a ba b =
1192320U 60 x2712s
.
By using these results we are ready to calculate higher derivative terms in the eq. ( 13). The
R4 terms are calculated as
t8t8R4
14! 11 11 R
4=
531256320U 80 x3616s . (64)
The RX terms become
Rabc0X abc 0 = 1066867200
U 80 x2916s, Rabc1X abc 1 =
1066867200U 80 x3616s
,
Rabc X abc = 1066867200(x7 1)
U 80 x3616s, Rabca X abc b =
1088640U 80 x3616s
a b, (65)
Rabc0X abc = Rabc X abc 0 = 1066867200 x7 1
U 80 x652 16s
,
and the DDX terms are evaluated as
D (a Db) X a 00 b = 198132480(47 + 40x7)
U 80 x2916s, D(a Db) X a 11 b =
2177280(513 + 124x7)U 80 x3616s
,
D (a Db) X a b = 198132480(4787x7 + 40 x14)
U 80 x3616s, D(a Db) X a a b
b = 236234880(43x7)
U 80 x3616s a b,
D (a Db) X a 0 b = D (a Db)X a 0b = 198132480(47 + 40x7) x7 1
U 80 x652 16s
. (66)
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.
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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 Rabcd =
Rbcad Rcabd , it is possible to show that there are only 7 independent terms.B1 = Rabcd Rabcd Refgh Refgh , B2 = Rabcd Raefg Rbcdh R efgh ,
B3 = Rabcd Rabef Rcdgh Refgh , B4 = Racbd Raebf Rcgdh R egfh ,
B5 = Rabcd Raefg Rbefh Rcdgh , B6 = Rabcd Raefg Rbfeh Rcdgh , (67)
B7 = Racbd Raefg Rbefh Rcgdh .
In the main part of this paper we considered the R4 terms t8t8R4 14! 11 11 R4 = 24( B1 64B2 + 2B3 + 16B4 16B5 16B6) which is explicitly written in the eq. ( 8). In order toderive equations of motion, we need to calculate variations of ( 67). These are evaluated as
B1 = 4( Rabcd )Rabcd Refgh R efgh , B2 = ( Rabcd )Rabce Rdfgh Refgh ,
B3 = 4( Rabcd )Rabef Rcdgh R efgh , B4 = 4( R abcd )Raecg Rbfdh Refgh ,
B5 = 2( Rabcd )Rabeg Rcfeh Rdfgh + 2( Rabcd )Refag Refch Rgbhd , (68)
B6 = 2( Rabcd )Rabeg Rcfeh Rdfgh + 2( Rabcd )Refag Refch Rgbhd 2(Rabcd )Rabef Rcegh Rdfgh ,B7 = 4( Rabcd )Raefg Rcefh Rgbhd .
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 thescalar curvature. Since the procedure for the classication is straightforward, we employ a
Mathematica code. As as result those are classied into 19 terms.
B8 = Rabcd Rabcd Ref Ref , B9 = Rabcd Rabcd R2, B10 = Rabcd Rbcdf Ref Rae ,
B11 = Rabcd Raefg RbcdgRef , B12 = Rabcd RbcdeRae R, B 13 = Racbd Rcedf Ref Rab ,
B14 = Rabcd Rabeg Rcdfg Ref , B15 = Racbd Raebg Rcfdg Ref , B16 = Rabcd Rabef Rcdef R,
B17 = Racbd Raebf Rcedf R, B 18 = Racbd RabRcdR, B 19 = Rabcd Rcdef Rae Rbf , (69)
B20 = Racbd Rcedf Rae Rbf , B21 = Racbd Rae RbeRcd , B22 = RabRabRcdRcd ,
B23 = RabRabR2, B24 = RabR cdRac Rbd, B25 = RabRac RbcR,
B26 = R4.
Then the effective action ( 9) is generalized into the form of
S 11 = 12211 d11x e R + 12s t8t8R4 14! 11 11R4 +
26
n =8bn Bn . (70)
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The coefficients bn (n = 8, , 26) cannot be determined from the results of scattering am-plitudes in the type IIA superstring theory, since we can remove or add these terms by
appropriate eld redenitions 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 conrm
this in the appendix D.Let us derive equations of motion for the effective action ( 70). The variations of 19 terms
in (69) are evaluated as
B8 = ( Rabcd ) 2Rabcd R ef R ef + 2Refgh Refgh Rac bd ,
B9 = ( Rabcd ) 2Rabcd R2 + 2 Refgh Refgh ac bdR ,
B10 = ( Rabcd ) RebcdRaf Ref + Rafgh Refgh Rcebd ,
B11 = ( Rabcd ) RebcdRafeg R fg 12 Raefg Rcefg Rbd 12 Reghi R fghi Reafc bd ,B12 = ( Rabcd ) RebcdRae R + 12 Raefg Rcefg bdR + 12 Reghi R fghi Ref ac bd ,B13 = ( Rabcd ) 2Rebfd Rac R ef + Ragch Regfh Ref bd ,
B14 = ( Rabcd ) Rabeg Rcdfg Ref + 2 Rabef Refgd R cg + Refgh Refai Rghci bd ,
B15 = ( Rabcd ) Raecg Rbfdg Ref + 2Raecf Regfd Rbg + Refgh Reagi R fchi bd ,
B16 = ( Rabcd ) 3Rabef Rcdef R + Refgh Refij Rghij ac bd ,
B17 = ( Rabcd ) 3Raecf Rbedf R + Refgh Reigj R fihj ac bd , (71)
B18 = ( Rabcd ) Rac RbdR + 2 Raecf Ref bdR + Refgh Reg R fh ac bd ,
B19 = ( Rabcd ) 2Rcdef Rae Rbf + 2Raegh Rcfgh Ref bd ,B20 = ( Rabcd ) 2Rebfd Rae Rcf + 2Rageh Rcgfh Ref bd ,
B21 = ( Rabcd ) Rae RceRbd + 2 Rafeg RceR fg bd + Rebfd RegR fg ac ,
B22 = 4( Rabcd )Rac Ref Ref bd,
B23 = ( Rabcd ) 2Rac bdR2 + 2 Ref Ref ac bdR ,
B24 = 4( Rabcd )Ref Rae Rcf bd,
B25 = ( Rabcd ) 3Rae RcebdR + R fg Ref Reg ac bd ,
B26 = 4( Rabcd )ac bdR3
.
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evaluated as
Y 0101 = 1
U 60 x2712s11907
2 b11 (1 + x7) 21609b14 (1 + x7)
30872
b15(1 + x7)
85176b16
10458b17 ,
Y 0a0a = 1
U 60 x2712s11907
8 (1 + 4 x7)b11 +
634
(5 1372x7 )b14
63
4 (17 + 98x7)b15 85176b16 10458b17 ,
Y 011 = x7 1U 60 x
472 12s
119072
b11 + 21609b14 + 3087
2 b15 ,
Y 0a a = x7 1U 60 x
472 12s
119072
b11 + 21609b14 + 3087
2 b15 ,
Y 0 0 = 1
U 60 x
27
12s
11907
2
b11
21609b14
3087
2
b15
85176b16
10458b17 (74)
Y 1 1 = 1
U 60 x2712s 11907
2 b11 (2 x7) + 21609(2 x7)b14
+ 3087
2 b15(2 x7) + 85176b16 + 10458b17 ,
Y 1a1a = 1
U 60 x2712s 35721
8 b11 +
861214
b14 + 7245
4 b15 + 85176b16 + 10458b17 ,
Y a a = 1
U 60 x2712s11907
8 b11 (3 + 4 x7) +
634
b14(1367 1372x7)+
634
b15(115 98x7) + 85176b16 + 10458b17 ,Y a ba b = 1U 60 x2712s
119074
b11 3152 b14 + 10712 b15 + 85176b16 + 10458b17 ,
where a, b = 2, , 9. By using these results it is possible to evaluate the higher derivativeterms which depend on the Y tensor in the eq. ( 73). The RY terms are calculated as
Rabc0Y abc 0 = 1
U 80 x2916s416745b11 1214514b14 71442b15 ,
Rabc1Y abc 1 = 1
U 80 x3616s 416745b11 + 1214514b14 + 71442b15 ,
Rabc Y abc
= x7
1
U 80 x3616s 416745b11 1214514b14 71442b15 , (75)Rabca Y abc b =
14U 80 x3616s
a b 416745b11 1214514b14 71442b15 ,
Rabc0Y abc = Rabc Y abc 0 = x7 1U 80 x
652 16s
416745b11 1214514b14 71442b15 ,
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and DDY terms become
D (a Db) Y a 00 b = 1701U 80 x3616s
72
(459 235x7 + 540 x14)b11+ (
6507
2397x7 + 6860x14)b14 +
12
(
999
282x7 + 980 x14)b15
+ ( 36504 + 31772x7)b16 + ( 4482 + 3901x7)b17 ,D (a Db) Y a 11 b =
1701U 80 x3616s 7(31 + 46x
7)b11 + 4(6 + 505 x7)b14
+ 12
(75 + 376x7)b15 + 676(9 + 16x7)b16 + 83(9 + 16x7)b17 ,D (a Db)Y a b =
1701U 80 x3616s
72
(1034 1455x7 + 540x14 )b11+ (13724 18897x7 + 6860x14 )b14 +
12
(2021 2742x7 + 980x14 )b15+ 676(47
33x7)b16 + 83(47
33x7)b17 , (76)
D (a Db)Y a a bb = 4 + 3 x7
U 80 x3616s a b
19170274
b11 6013035
2 b14
5596292
b15
16098264b16 1976562b17 ,D (a Db) Y a 0 b = D (a Db)Y a 0b
= x7 1U 80 x
652 16s
595352
(115 108x7)b11 11907(1031 980x7)b14
11907(73 70x7)b15 + 8049132b16 + 988281b17 .
As mentioned before, only b11 , b14 , b15 , b16 and b17 appeared in the calculations.By using the ansatz ( 14) and inserting values of X and Y tensors into the equations of
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nal form of the solution becomes
h1 = 440559
4 b11 +
7687752
b14 + 53333
2 b15 + 927472 b16 + 113876b17 +
13025017609
1x34
+ 23814b11 86436b14 6174b15 57462496 1x27
+ 12051648
13x20 4782400
13x13
3747840
x7 + 4099200
x6 1639680(x 1)(x7 1)
+ 117120 18 23x7
I (x),
h2 = 11907
4 b11 +
3152
b14 1071
2 b15 170352b16 20916b17 + 19160960
1x34
+ 23814b11 86436b14 6174b15 58528288 1x27
+ 2213568
13x20 1229760
13x13
2108160
x7 +
2459520x6
+ 1054080 2 1x7
I (x), (82)
h3 = 11907
4 b11 +
760272
b14 + 8225
2 b15 94640b16 11620 b17 +
3611104009
1x34
+ 23814b11 86436b14 6174b15 59840032 1x27
2402131213x20
58072000
13x13 2108160
x7 +
2459520x6
+ 117120 18 41x7
I (x),
f 1 =440559
4 b11
7309192
b14 48685
2 b15 889616b16 109228b17
12081708809
1x34
+ 130977b11 + 432810 b14 + 28728 b15 + 1022112 b16 + 125496 b17 + 161405664 1x27
+ 5738880
13x20 +
956480x13
+819840
x7 I (x).
The function I (x) is given by the eq. ( 34) and integral constants are determined so as to
satisfy that hi (1) are nite and hi (x), f 1(x) O(x 8) when x goes to the innity. Noticethat b11 , b14, b15 , b16 and b17 only appear in the coefficients of x 27 and x 34 . The solution
is reliable up to O( 2).
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 ( 82) by following the arguments in the section 5. Although the solution is
modied, the results obtained until the eq. ( 50) do not change. Since the effective action is
modied as in the eq. ( 70), the eq. ( 51) should be replaced with
S 11R
= 12211
g[g] + 12s (X + Y ) . (83)
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The entropy of the quantum near horizon geometry of the black 0-brane is evaluated as
S = 42211 H d8dz h 1 1212s (X + Y )N N
= 422
11 Hd8dz h 1 220s U 20 H
11 (X
txtx + Y txtx )
= 42211 H d8dz h 1 + U 60 40642560
23814b11 + 86436b14 + 6174b15 + 170352b16 + 20916b17=
449
a1N 2 U 92
0 1 + 914
f 1(1) + 12
h2(1) + 40642560
23814b11 + 86436b14 + 6174b15 + 170352b16 + 20916b17 U 60
= 449
a 4
5
1 N 2 T
95 1 + a
125
1 95
f 1(1) 935
f 1(1) + 910
h1(1) + 12
h2(1) + 40642560
23814b11 + 86436b14 + 6174b15 + 170352b16 + 20916b17 T
125
= a3N 2 T 95 1 + a 5 T
125 . (84)
Notice that f 1(1), f 1(1), h1(1) and h2(1) depend on b11 , b14 , b15 , b16 and b17 . The value of
a3 is given in the section 5, and a5 is given by
a5 = a125
1 95
f 1(1) 935
f 1(1) + 910
h1(1) + 12
h1(1) + 40642560
23814b11 + 86436b14 + 6174b15 + 170352b16 + 20916b17 . (85)It seems that a5 depends on b11 , b14, b15 , b16 and b17 . The explicit calculation, however,
shows that a5 = a4 and the result does not depend on the ambiguities of the effective action.
Thus the physical quantities of the black 0-brane are free from the ambiguities and uniquely
determined.
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