Open String Tachyon Open String Tachyon

in Supergravity in Supergravity SolutionSolutionShinpei KobayashiShinpei Kobayashi

( Research Center for the Early ( Research Center for the Early Universe, The University of Universe, The University of

Tokyo )Tokyo )

2005/01/18at KEK

Based on hep-th/0409044Based on hep-th/0409044in collaboration with in collaboration with

Tsuguhiko Asakawa and So Matsuura Tsuguhiko Asakawa and So Matsuura ( RIKEN ) ( RIKEN )

MotivationMotivation We would like to apply the string theory to tWe would like to apply the string theory to t

he analyses of the gravitational systems.he analyses of the gravitational systems. We have to know We have to know

how we should apply string theory to realistic ghow we should apply string theory to realistic gravitational systems, ravitational systems,

or what stringy (non-perturbative) effects are, or what stringy (non-perturbative) effects are, or what stringy counterparts of the BHs or Univor what stringy counterparts of the BHs or Univ

erse in the general relativity are.erse in the general relativity are. → → D-branes may be a clue to tackle D-branes may be a clue to tackle

such problems such problems (BH entropy, D-brane inflation, etc.) (BH entropy, D-brane inflation, etc.)

ContentsContents1.1. D-branes and Classical D-branes and Classical

Descriptions Descriptions 2.2. D/anti D-brane systemD/anti D-brane system3.3. Three-parameter solution Three-parameter solution 4.4. ConclusionsConclusions5.5. Discussions and Future Works Discussions and Future Works

String Field Theory

D-brane( Boundary State )

Supergravitylow energy limit

α’ → 　 0

classical description( Black p-brane )low energy limit

1. D-branes and Classical 1. D-branes and Classical descriptionsdescriptions

D-brane ( BPS case )D-brane ( BPS case ) Open string endpoints stick to a D-braneOpen string endpoints stick to a D-brane PropertiesProperties

SO(1,p)×SO(9-p) ( BPS case ), RR-chargedSO(1,p)×SO(9-p) ( BPS case ), RR-charged (mass) (mass) 1/(string coupling) 1/(string coupling)

X0

Xμ Xi open string

Dp-brane

)10(9,,1:

.,,1,0:

DpiX

pXi

BPS black p-brane solutionBPS black p-brane solution Symmetry : SO(1,p)×SO(9-p), RR-Symmetry : SO(1,p)×SO(9-p), RR-

charged charged setup : SUGRA actionsetup : SUGRA action

ansatz : ansatz :

22

23

2102 ||

)!2(21

21

21

p

p

Fep

RgxdS

)1()2(10)(

)1(

)(2)(22

,

),(

)(,

pppr

p

iiji

ijrBrA

dFdxdxdxe

r

xxrdxdxedxdxeds

BPS black p-brane solution BPS black p-brane solution (D=10)(D=10)

.1)7(

21)(

,1)(),(

,)()(

7)8(

1)(4

3)(

81

87

2

pp

pp

pr

p

pr

jiij

p

p

p

p

rpNT

rf

where

rferfe

dxdxrfdxdxrfds

Di Vecchia et al. suggested more direct method to check the correspondence between a Dp-brane

and a black p-brane solution using the boundary state.

it must be large for the validity of SUGRA

・ SO(1,p)×SO(9-p), ・ (mass)=(RR-charge), which are the same as D-branes

asymptotic behavior of the black p-brane asymptotic behavior of the black p-brane = difference from the flat background = difference from the flat background = graviton, dilaton, RR-potential in SUGRA= graviton, dilaton, RR-potential in SUGRA

massless modes of the closed strings from the massless modes of the closed strings from the boundary state ( D-brane in closed string boundary state ( D-brane in closed string channel ) channel ) = graviton, dilaton, RR-potential in string theory = graviton, dilaton, RR-potential in string theory

( string field theory )( string field theory )

coincide

Relation between the D-brane ( the boundary state) and the black p-brane solution

(Di Vecchia et al. (1997))

hg 1~

Boundary State ( = D-Boundary State ( = D-brane)brane) Boundary states are defined as Boundary states are defined as

sources of closed strings ( = D-sources of closed strings ( = D-branes in closed string channel ).branes in closed string channel ).

As closed strings include gravitons, As closed strings include gravitons, the boundary state directly relates to the boundary state directly relates to a black p-brane solution.a black p-brane solution.

)9,,1(,|

,,1,0,0|

0

0

piBxBX

pBXii

).,(

,)(2

)(2

)9()9(

ijMN

ippipp

S

RRxTN

NSNSxTN

B

iXX

0X

)(),(

,000~cos

~exp)(

,000~

sin

~exp)(

,)(2

)(2

~)1(

2/1

0

)9(

~2/1

0

)9(

NNforS

pdSd

SxRR

ghostpbSb

SxNSNS

RRNNT

NSNSNNT

B

ijMN

Rr

NrMN

Mr

n

NnMN

Mn

ip

r

NrMN

Mr

n

NnMN

Mn

ip

ppp

,)( 232ˆ222

p

p rfee

2

78

78 )7(22

3)7(22

3)(ˆ pp

pp

p

p

rpTp

rpTpr

sourcepropagatorfieldmassless

We can reproduce the leading term of a black p-brane solution ( asymptotic behavior ) via the boundary state.

leading term at infinity

e.g. ) asymptotic behavior of Φ of black p-brane

coincident

2111)( 1

223;0

ipp

NMMN k

VTpBDk

<B| 　 |φ>

String Field Theory Supergravity

classical solution( Black p-brane )

D-brane( Boundary State )

low energy limitα’ → 　 0

low energy limit

eom eom

BPS case → OK (Di Vecchia et al. (1997))

We study non-BPS systems ( e.g. D/anti D-brane system ).

non-BPS case → ?non-BPS cases

are more realistic in GR sense

BPS caseBPS case Dp-brane Dp-brane 　　 　　 black p-brane black p-brane

Non-BPS case Non-BPS case D/anti D-brane system with a constant tachyon D/anti D-brane system with a constant tachyon

vevvev Three-parameter solution ? Three-parameter solution ? ( ( guessedguessed by Brax-Mandal-Oz by Brax-Mandal-Oz (2000))(2000))

( other non-BPS system( other non-BPS system corresponding classical solution ?) corresponding classical solution ?)

We verify their claim using the boundary state.

2. D/anti D-brane system2. D/anti D-brane system 　　　　　　　　

NN D-branes and anti D-branes

attracts together

Unstable multiple branes Open string tachyon

represents its instability

Stable D-branes are left　 case 　

)( NN

NN

D/anti D-brane system

tachyon condensationclosed string emission

Boundary State with boundary Boundary State with boundary interaction interaction

pS BedXDpDp b ][

)(expˆ XMdTrPe bS

DXXAXTXTDXXA

XM)()(

)()()(

)() XAdXiScf b

braneD braneD

branesD

N N

NbranesD

open string

DXATTDXA

00

0

TT

NN

N

N

NN NN

NN

N

N

Boundary state for D/anti Boundary state for D/anti D-brane with a constant D-brane with a constant

tachyon vev tachyon vev 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　

)(),(

,000~cos

~exp)(

,000~

sin

~exp)(

,)(2

]2)[(2

,,;

~)1(

2/1

0

)9(

~2/1

0

)9(

|| 2

NNforS

pdSd

SxRR

ghostpbSb

SxNSNS

RRNNT

NSNSetrNNT

TNNB

ijMN

Rr

NrMN

Mr

n

NnMN

Mn

ip

r

NrMN

Mr

n

NnMN

Mn

ip

pTpp

massRR-charge

constant tachyon

Change of the Mass Change of the Mass during the tachyon during the tachyon

condensationcondensation1.1. D-branes, D-branes, 　　　　 anti D-branes coincide anti D-branes coincide

with each other. ( t = 0 )with each other. ( t = 0 )　　2.2. During the tachyon condensation ( t = tDuring the tachyon condensation ( t = t00 ) )

tachyon vev is included in the mass.tachyon vev is included in the mass.

3.3. Final state ( t = ∞ )Final state ( t = ∞ )The mass will decrease through the The mass will decrease through the closed string emission, and the value of closed string emission, and the value of the mass will coincide with that of the RR-the mass will coincide with that of the RR-charge (BPS).charge (BPS).

)(~),(~ braneDaofmassTNNTM pp

]2)[(~2||T

p etrNNTM

)(~ NNTM p

NN

Boundary state for D/anti Boundary state for D/anti D-brane D-brane

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　

)(),(

,000~cos

~exp)(

,000~

sin

~exp)(

,)(2

]2)[(2

,,;

~)1(

2/1

0

)9(

~2/1

0

)9(

|| 2

NNforS

pdSd

SxRR

ghostpbSb

SxNSNS

RRNNT

NSNSetrNNT

TNNB

ijMN

Rr

NrMN

Mr

n

NnMN

Mn

ip

r

NrMN

Mr

n

NnMN

Mn

ip

pTpp

massRR-charge

constant tachyon

3. Three-parameter solution3. Three-parameter solution ( Zhou & ( Zhou &

Zhu (1999) )Zhu (1999) ) SUGRA actionSUGRA action

ansatz : SO(1, p)×SO(9-p) ( D=10 ) ansatz : SO(1, p)×SO(9-p) ( D=10 )

22

23

2102 ||

)!2(21

21

21

p

p

Fep

RgxdS

.,

,

,

10)(112

)(

)(2)(22

prppp

r

jiij

rBrA

dxdxdxedF

ee

dxdxedxdxeds

same symmetry as the D/anti D-brane system

.16

)7)(1(7

)8(2

,1)(,)()(ln)(

,))(sinh())(cosh(

))(sinh()1(

,))(sinh())(cosh(ln4

3)(16

)1)(7()(

,))(sinh())(cosh(ln16

1)(64

)3)(1(

)ln(7

1)(

,))(sinh())(cosh(ln16

7)(64

)3)(7()(

21

70

2

2/122

)(

21

21

21

cppppk

rrrf

rfrfrh

rkhcrkhrkhce

rkhcrkhprhcppr

rkhcrkhprhcpp

ffp

rB

rkhcrkhprhcpprA

p

r

charge ?

mass ?

tachyon vev ?

Property of the three-parameter Property of the three-parameter solutionsolution

ADM massADM mass

RR chargeRR charge

We can extend it to an arbitrary We can extend it to an arbitrary dimensionality. dimensionality.

,)1(2 70

2/122

pprkNcQ

,22

3 7021

pprNkccpM

)(),(,16

)7)(8(2

8 pp

dd

ppp TvolVSvol

VppN

where

NNTp ~?

From the form of the boundary state, Brax-Mandal-Oz claimed

that c_1 corresponds to the tachyon vev.

]2[2T

p eNNNT ~?

We re-examine the correspondence between the D/anti D-brane system

and the three-parameter solution

using the boundary state.

New parametrizationNew parametrization

　 　 → → During the tachyon condensation, the RR-chDuring the tachyon condensation, the RR-charge arge does not change its value. does not change its value. → We need a new parametrization suitable for t.→ We need a new parametrization suitable for t.c. c.

.,4

31 0012 pp NQvc

kpvNM

).0(11,2 2

22

070 v

vc

kvr p

.12,22

3 70

2/122

7021

pp

pp rkNcQrNkccpM

Asymptotic behavior of the three-Asymptotic behavior of the three-parameter solution parameter solution

(= graviton, dilaton, RR-potential in (= graviton, dilaton, RR-potential in SUGRA )SUGRA )

.1

,116

)7)(1(14

3)(

,14

318

11

,14

318

71

)7(270)(

)7(27012

)7(270

12)(2

)7(270

12)(2

ppr

pp

ijppijrB

pprA

rre

rrkvcppvpr

rrvc

kpvpe

rrvc

kpvpe

graviton, dilaton, RR-potential graviton, dilaton, RR-potential in string theory in string theory

.,,22

~;0)(

21||

2/12/1

2

ijMNMNi

pT

NSNSpNMMN

SSkV

etrNNT

BDbbkkJ

)()( fieldMN

MN kJ <B| 　 |physical field>

.0,1,22

1,0,

22)(

)()()()(

lklklklk

k

MNNMMNMN

MhMN

MNhMN

hNM

hMN

Using the boundary state, we obtainUsing the boundary state, we obtain

.4

31)7(

22)(ˆ

,4

322

)()(ˆ

.8

1,8

71)7(

22)(ˆ

,8

1,8

722

)()()(ˆ

78

||

21||

)(

78

||

21

)(

)(

2

2

2

2

prp

TetrNNr

pkVT

etrNN

kJk

pprp

TetrNNrh

ppkVT

eNNN

kJkJkh

pp

pT

i

ppT

ABAB

ijpp

pTMN

iji

ppT

MNAB

ABAB

AB

MNMN

pr

p

ijpMN

re

rkvcppvpr

ppr

vckpvrh

70)(

7012

70

12

16)7)(1(1

43)(

,8

1,8

74

31)(

.1)7(

2

,4

31)7(

221)(ˆ

,8

1,8

71)7(

221)(ˆ

78

)(

78

||

78

||

2

2

pp

pr

pp

pT

ijpp

pT

MN

rpT

NNe

prp

TNN

NNetrr

pprp

TNN

NNetrrh

asymptotic behavior of the three-parameter solution

massless modesvia the boundary state

Results and Comparison

kvc

pppv

kvcpv

NNetr T

1212||

)3(4)7)(1(1

43121

2

01 c

pr

p

ijpMN

re

rkvcppvpr

ppr

vckpvrh

70)(

7012

70

12

16)7)(1(1

43)(

,8

1,8

74

31)(

.1)7(

2

,4

31)7(

221)(ˆ

,8

1,8

71)7(

221)(ˆ

78

)(

78

||

78

||

2

2

pp

pr

pp

pT

ijpp

pT

MN

rpT

NNe

prp

TNN

NNetrr

pprp

TNN

NNetrrh

asymptotic behavior of the three-parameter solution

massless modesvia the boundary state

Results and Comparison

We find that they coincide with each We find that they coincide with each other under the following identification, other under the following identification,

,2112||

2

NNetrv

T

RR-charge, constant during the tachyon condensation

v ^2 ~ M^2 – Q^2: non-extremality

→ tachyon vev can be seen as a part of the ADM mass01 c

,

)7(2

80

p

p

pNNT

c_1 does not corresponds to the vev of the open string tachyon.

The three-parameter solution with c_1=0 does correspond to the D/anti D-brane system.

ConclusionsConclusions Using the boundary state, we find that the three-Using the boundary state, we find that the three-

parameter solution with c_1=0 corresponds to parameter solution with c_1=0 corresponds to the D/anti D-brane system with a constant the D/anti D-brane system with a constant tachyon vev.tachyon vev. New parametrization is needed to keep the RR-charge New parametrization is needed to keep the RR-charge

constant during the tachyon condensation.constant during the tachyon condensation. The vev of the open string tachyon is only seen as a The vev of the open string tachyon is only seen as a

part of the ADM mass.part of the ADM mass. c_1 does not corresponds to the tachyon vev as c_1 does not corresponds to the tachyon vev as

opposed to the proposal made so far.opposed to the proposal made so far. We find that we can extend the correspondence We find that we can extend the correspondence

between D-branes and classical solutions to non-between D-branes and classical solutions to non-BPS case. BPS case. First discovery of the correspondence in non-BPS First discovery of the correspondence in non-BPS

case.case. It may be a clue to describe “realistic” gravitational It may be a clue to describe “realistic” gravitational

systems which are generally non-BPS. systems which are generally non-BPS.

1.1. Parametrization Parametrization → during the t.c., the RR-charge does not → during the t.c., the RR-charge does not change its value. change its value. → →

2.2. The relation between the mass and the scaThe relation between the mass and the scalar chargelar charge→ cf. Wyman solution in D=4 case→ cf. Wyman solution in D=4 case c_1 corresponds to the dilaton charge. c_1 corresponds to the dilaton charge.

),(),( 020 vcr

Discussion : Discussion : Why was c_1 thought to be Why was c_1 thought to be

the open string tachyon vev ?the open string tachyon vev ?

Wyman solution in Wyman solution in Schwarzschild gaugeSchwarzschild gauge

Static, spherically symmetric, with a Static, spherically symmetric, with a free scalarfree scalar

.

,21

2)2(

2)(22)(22)(22

24

dredredteds

RgxdS

rCrBrA

.,/21)(

),(ln)(

,)()()(

22

2

2)2(

21222

qmm

rmrF

rFmqr

drrFdrrFdtrFds

Wyman solution in isotropic Wyman solution in isotropic gaugegauge

r → Rr → R

,/2121

rmrmrR

22

2

2)2(

222222

2

,/21)(

,)()(ln2)(

),)(()()()(

qmm

RmRF

RFRF

mqr

dRdRRFRFdtRFRFds

In this gauge, we can compare it wit

h the 3-para. sln.

Three-parameter solution Three-parameter solution 　 　 casecase 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1,0,4 2 cpD

21

0

1

2)2(

22~2~22

~

2

4~,1)(

,ln)(

),(

ckrrrf

ffcr

drdrffdtffds kk

k

22

221

4qm

qc

　　 corresponds to the dilaton charge.

1c

Discussion : Stringy Discussion : Stringy counterpart of c_1 ?counterpart of c_1 ?

has something to do with the has something to do with the -brane. -brane.

99DD1c

.1

,116

)7)(1(14

3)(

,14

318

11

,14

318

71

)7(270)(

)7(27012

)7(270

12)(2

)7(270

12)(2

ppr

pp

ijppijrB

pprA

rre

rrkvcppvpr

rrvc

kpvpe

rrvc

kpvpe

We can not relate these parts with an ordinary boundary statecounterpart of

the D/anti D-brane system

.1

,116

)7)(1(14

3)(

,14

318

11

,14

318

71

)7(270)(

)7(27012

)7(270

12)(2

)7(270

12)(2

ppr

pp

ijppijrB

pprA

rre

rrkvcppvpr

rrvc

kpvpe

rrvc

kpvpe

We can not relate these parts with an ordinary boundary state

counterpart of the D/anti D-brane system

Deformation of the boundary stateDeformation of the boundary state

ijMNS )1(,)1(' ijMNS ,

8)1()7(1

43~

21

81,

21

87~

21)()1(

21

)(

)()1(

pppkV

J

ppkV

JJh

i

pMN

MN

iji

p

MNKL

LKKL

LK

MNMN

We can reproduce the 3-para. sln with non-zero by adjusting α ・ β

1c

Do we have such a deformation in string theory ?→ 　　 with open string tachyon99DD

Construction of Construction of (Asakawa-Sugimoto-Terashima, JHEP 0302 (2003) (Asakawa-Sugimoto-Terashima, JHEP 0302 (2003) 011)011)

XdXB ][9

matrixNNTAwhere

DXXAXTXTDXXA

XM

XMdPTre bS

:,

)()()()(

ˆexpˆ

boundary interaction

ppiXtTA ii

9,,1,0

99DD

zero

Mxtosc

r

Nr

rMN

Mr

Tp

p

xexdbSbi

eNNNatFT

tTNNG

i0

)(0

10)(

99

2

2

0~exp

]2)[()(2

,,,;

δ-function with ｔ → ∞→ordinary boundary state

From Gaussian Boundary From Gaussian Boundary State State

to BPS Dp-braneto BPS Dp-brane

systemDD 99

tachyon has some configuration

t → ∞

2

~)(

,~t

xxt

etV

ei

i

),,1,0( px

)9,,1( ppix i

lower-dimensional BPS D-brane

braneDpBPS

),,1,0( px

)9,,1( ppix i

systemDD 99

extend to -direction infinitely

localized at

braneDpBPS

iixxte~

)9,,1( ppix i ),,1,0( px

)9,,1( ppix i

),,1,0( px

0t t

ix0ix

braneDpGaussian

Gaussian in -directionix

So far, we treat So far, we treat 　　　　　　　　　　　　　　　　

Consider that eachConsider that each 　　 　　 or is or is made from made from

boundary state is deformed as boundary state is deformed as follows:follows:

99DD

DpNDpN

DpDp

tGtFteNNNTp

pT ;)]'([]2)[(2

~ 92/19 2

originDpNDpN Gaussian

braneorigin

MNS

MNS

ordinary

Deformed

Gaussian boundary state Gaussian boundary state XXdtdXtG ip

)(ˆ

2exp; 2

tGXit

XetXidXtGP

pi

Xtipi

i

;ˆ

))((;ˆ2

2

0;ˆˆ tGXitP pii

..;

..;0cbDirichlettcbNeumannt

D9-tachyon

Mixture of Neumann b.c. and Dirichlet b.c. → 　 smeared boundary condition

Oscillator pictureOscillator picture boundary condition in the oscillator boundary condition in the oscillator

picture picture

in

nnn

nnn

nn

i

en

ix

wwnix

ewwXX

00

00

)~(12

'

~12

'

)()(ˆ),0(ˆ

in

nnn e

npP

00 )~(1

'21),0(ˆ2

cf. ordinary boundary statecf. ordinary boundary state

iii xXppiX

XpX

0

0

|:)9,,1(0|:),,1,0(

D-brane

στ

closed string

closed tree graph

pi

pii

p

BxBXppiXBXpX

0

0

|:)9,,1(0|:),,1,0(

στ

open string

open 1-loop graph

boundary state

boundary conditions boundary conditions

)0(0)~(0ˆ

nforBBp

nn

)0(0)~(ˆ

nforBBxBx

in

in

ii

000~1exp)ˆ(2

~ ~1

)9(

pSn

xxT

B NnMN

Mn

n

iipp

0)~(),( BSS Nn

MN

MnijMN

Longitudinal to the D-brane

Transverse to the D-brane

0;ˆ tGP p 0;)~( tGpnn

0;~)/'2(1)/'2(1

0;~'21'21

tGntnt

tGn

tn

t

pin

in

pin

in

0;)ˆˆ( tGXtiP pii

・ Longitudinal to the Dp-brane

・ Transverse to the Dp-brane

Gaussian boundary state case

0~1exp;1

)(

N

nn

nMN

Mnoscp S

ntG

ijn

MN ntntSwhere

)/'2(1

)/'2(1,)(

ixtiip

ti

zerop xedxpedptGi

i

0)(

00

)(4

1

0

20

20

;

zeroposcppp tGtGTtG ;;~;

Oscillator part

0-mode part

combine them

to ordinary boundary state with ｔ→∞

)2(2)(4)(

2

xxxxF

x

,)'(~ 99

pp tFTT

xxOx

xxOxxF

;)(

0;)()2log2(1)(2/1

2

'2'2

0 )( tedxixti

o

pp

p TTT 99 )'2(~

99DD DpFrom a to one

tension part via SFT

thus, in the limit （ D9-tachyon vanishes ）t

（ Kraus-Larsen, PRD63 (2001) 106004 ）

)'()'( 2/1)( 20 tFttFedxixti

o integrate with finite

finally, we obtain

tGtFteNNNTp

pT ;)]'([]2)[(2

~ 92/19 2

originDpNDpN Gaussian

braneorigin

t

zero

Mxtosc

r

Nr

rMN

Mr

Tp

p

xexdbSbi

eNNNatFT

tTNNG

i0

)(0

10)(

99

2

2

0~exp

]2)[()(2

,,,;

p

i

pkt

p

zero

Mizero

i

xtpizero

kV

et

xkk

exdtTNNGk

i

i

92

1)(419

02)(

010

)'2(1

1,,,;|

2

20

))1(,)1(()'2()(2

))1(,()1()1()'2()(2

;~0;

9)(41

219

99)(41

219

2/12/1

2

2

ijpk

t

i

p

ijppk

t

i

p

pNM

i

i

ekV

NNT

CBAekV

NNT

tGDbbk

.)9(1)1()1(1

,)9(1)1()1(1

1,'

11,

)'2(1)'2(1,

'21'21,

,2,11)(11

9

9

1

1)2/1(

2

CBApBA

BApBA

Ct

tt

ttS

eNN

NBt

OtFt

A

p

p

ijijt

ijijMN

T

　　　　　 tachyonorigin

　　　　　 tachyon origin

99DD

DpDp

graviton, dilaton via Gaussian graviton, dilaton via Gaussian boundary stateboundary state

2

41

9

21)1(

21)1(

)'2)((~

8))(1(

41,

8))(7(

472

2

~)(

)7()1()3(222

~)(

iktppp

ij

i

ppMN

i

pp

eNNTT

pppp

kVT

kh

pppkVT

k

graviton, dilaton via three-parameter graviton, dilaton via three-parameter solutionsolution

,)7(2

4)3(

21

81)(

,)7(2

4)3(

21

87)(

,)7(2

16)1)(7(

21

43)(

28

70

1

2)1(

28

70

1

2)1(

28

70

1

2)1(

iji

ppp

ij

i

ppp

i

ppp

kVpr

ckpprh

kVpr

ckpprh

kVpr

ckpppk

12/1221

11

70

70 )()1(,, kcccrr

wherepp

pp

pp

pp

pp

rcppppcpN

rkccpNM

rNrkcNQ

70

21

21

7021

70

70

2/122

16)1)(7(

7)8(212

23

22

3

2)1(2

12/1221

11

70

70 )()1(,, kcccrr pp

constant

criterion : RR charge Q keeps

its valueDpDpGaussian

DpNNDpNDpN )(

Thus, we compare them as Thus, we compare them as

→ → The effect of can be interpreted as The effect of can be interpreted as D9-tachyon t. D9-tachyon t. 1

c

tpe

NNN

tap

CpBAppp

T

'872)9(2

47)9(2

47

41

2

2

tC

ckc

'21

21)(

21

),( 1

1

Future WorkFuture Work c_1 and a Gaussian brane c_1 and a Gaussian brane

(SK, Asakawa & Matsuura, hep-th/0502XXX )(SK, Asakawa & Matsuura, hep-th/0502XXX ) Entropy counting via non-BPS boundary stateEntropy counting via non-BPS boundary state Construction of a time-dependent solution Construction of a time-dependent solution

feedback to SFT feedback to SFT Solving δSolving δBB|B>=0 ( E-M conservation law in SFT ) |B>=0 ( E-M conservation law in SFT )

(Asakawa, SK & Matsuura, JHEP 0310 (2003) 023)(Asakawa, SK & Matsuura, JHEP 0310 (2003) 023) Application to cosmologyApplication to cosmology

(SK, K. Takahashi & Himemoto)(SK, K. Takahashi & Himemoto) Stability analysis Stability analysis

( K. Takahashi & SK)( K. Takahashi & SK)