Hyakutake-Quantun Near Zero Geometry

8/13/2019 Hyakutake-Quantun Near Zero Geometry

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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/1311.7526v1


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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.

12


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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

16


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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

17


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Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (B)

19740141, 2007 and 24740140, 2012.

18


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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.

20


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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

25


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28/32

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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Hyakutake-Quantun Near Zero Geometry

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