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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/0409044
in 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.)
ContentsContents
1.1. D-branes and Classical D-branes and Classical Descriptions Descriptions
2.2. D/anti D-brane systemD/anti D-brane system
3.3. Three-parameter solution Three-parameter solution
4.4. ConclusionsConclusions
5.5. 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 Xiopen 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
2
3210
2||
)!2(2
1
2
1
2
1p
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)(
8
1
8
72
pp
pp
pr
p
pr
jiij
p
p
p
p
rp
NTrf
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 = graviton, dilaton, RR-potential in string theory 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
,)( 2
32ˆ222
p
p rfee
2
78
78 )7(22
3
)7(22
3)(ˆ
pp
p
pp
p
rp
Tp
rp
Tpr
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
22
3;0
ipp
NMMN k
VTp
BDk
<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 D/anti D-brane system with a constant
tachyon vevtachyon vev 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
DXXAXT
XTDXXAXM
)()(
)()()(
)() XAdXiScf b
braneD braneD
branesD
N N
N
branesD
open string
DXAT
TDXA
0
0
0
T
T
NN
N
N
N
N 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 = During the tachyon condensation ( t = tt00 ) )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 the mass will coincide with that of the RR-charge (BPS).RR-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
2
3210
2||
)!2(2
1
2
1
2
1p
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
7
0
2
2/122
)(
21
21
21
cpp
p
pk
r
rrf
rf
rfrh
rkhcrkh
rkhce
rkhcrkhp
rhcpp
r
rkhcrkhp
rhcpp
ffp
rB
rkhcrkhp
rhcpp
rA
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
pprNkcc
pM
)(),(,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
31 001
2 pp NQvck
pvNM
).0(1
1,2 2
22
070 v
vc
k
vr p
.12,22
3 70
2/122
7021
pp
pp rkNcQrNkcc
pM
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
,1
16
)7)(1(1
4
3)(
,1
4
31
8
11
,1
4
31
8
71
)7(270)(
)7(27012
)7(270
12)(2
)7(270
12)(2
ppr
pp
ijppijrB
pprA
rre
rrk
vcppv
pr
rrvc
k
pv
pe
rrvc
k
pv
pe
graviton, dilaton, RR-potential graviton, dilaton, RR-potential in string theory in string theory
.,,22
~;0)(
2
1||
2/12/1
2
ijMNMNi
pT
NSNSpNMMN
SSk
VetrNN
T
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
||
2
1||
)(
78
||
2
1
)(
)(
2
2
2
2
p
rp
TetrNNr
p
k
VTetrNN
kJk
pp
rp
TetrNNrh
pp
k
VTeNNN
kJkJkh
pp
pT
i
ppT
ABAB
ijpp
pTMN
iji
ppT
MNAB
ABAB
AB
MNMN
pr
p
ijpMN
re
rk
vcppv
pr
pp
rvc
k
pvrh
70)(
7012
70
12
16
)7)(1(1
4
3)(
,8
1,
8
7
4
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
rp
TNNe
p
rp
TNN
NN
etrr
pp
rp
TNN
NN
etrrh
asymptotic behavior of the three-parameter solution
massless modesvia the boundary state
Results and Comparison
k
vc
p
ppv
k
vcpv
NN
etr T1212
||
)3(4
)7)(1(1
4
31
21
2
01 c
pr
p
ijpMN
re
rk
vcppv
pr
pp
rvc
k
pvrh
70)(
7012
70
12
16
)7)(1(1
4
3)(
,8
1,
8
7
4
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
rp
TNNe
p
rp
TNN
NN
etrr
pp
rp
TNN
NN
etrrh
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,
,2
112||
2
NN
etrv
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
p
NNT
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 Using the boundary state, we find that the
three-parameter solution with c_1=0 three-parameter solution with c_1=0 corresponds to the D/anti D-brane system with corresponds to the D/anti D-brane system with a constant tachyon vev.a constant tachyon vev. New parametrization is needed to keep the RR-New parametrization is needed to keep the RR-
charge constant during the tachyon condensation.charge 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 between D-branes and classical solutions to non-BPS case. non-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
.
,2
1
2)2(
2)(22)(22)(22
24
dredredteds
RgxdS
rCrBrA
.,/2
1)(
),(ln)(
,)()()(
22
2
2)2(
21222
qm
m
r
mrF
rFm
qr
drrFdrrFdtrFds
Wyman solution in isotropic Wyman solution in isotropic gaugegauge
r → Rr → R
,/2
12
1
r
mr
mrR
22
2
2)2(
22222
2
2
,/2
1)(
,)(
)(ln2)(
),)(()()(
)(
qm
m
R
mRF
RF
RF
m
qr
dRdRRFRFdtRF
RFds
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)(
),(
ckr
rrf
f
fcr
drdrffdtf
fds kk
k
22
221
4
qm
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
,1
16
)7)(1(1
4
3)(
,1
4
31
8
11
,1
4
31
8
71
)7(270)(
)7(27012
)7(270
12)(2
)7(270
12)(2
ppr
pp
ijppijrB
pprA
rre
rrk
vcppv
pr
rrvc
k
pv
pe
rrvc
k
pv
pe
We can not relate these parts with an ordinary boundary statecounterpart of
the D/anti D-brane system
.1
,1
16
)7)(1(1
4
3)(
,1
4
31
8
11
,1
4
31
8
71
)7(270)(
)7(27012
)7(270
12)(2
)7(270
12)(2
ppr
pp
ijppijrB
pprA
rre
rrk
vcppv
pr
rrvc
k
pv
pe
rrvc
k
pv
pe
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
4
3~
21
8
1,
21
8
7~
2
1)()1(
2
1
)(
)()1(
ppp
k
VJ
pp
k
V
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
DXXAXT
XTDXXAXM
XMdPTre bS
:,
)()(
)()(
ˆexpˆ
boundary interaction
ppi
XtTA ii
9,,1
,0
99DD
zero
Mxt
oscr
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( ppixi
lower-dimensional BPS D-brane
braneDpBPS
),,1,0( px
)9,,1( ppi
xi
systemDD 99
extend to -direction infinitely
localized at
braneDpBPS
iixxte~
)9,,1( ppixi ),,1,0( px
)9,,1( ppi
xi
),,1,0( px
0t t
ix0ix
braneDp
Gaussian
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
tGtFteNNNT
ppT ;)]'([]2)[(
2~ 92/19
2
origin
DpNDpN Gaussian braneorigin
MNS
MNS
ordinary
Deformed
Gaussian boundary state Gaussian boundary state
XXdt
dXtG ip
)(ˆ
2exp; 2
tGXit
XetXidXtGP
pi
Xt
ipi
i
;ˆ
))((;ˆ2
2
0;ˆˆ tGXitP pii
..;
..;0
cbDirichlett
cbNeumannt
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
)~(1
2
'
~1
2
'
)()(ˆ),0(ˆ
in
nnn e
npP
00 )~(
1
'2
1),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
BxBXppiX
BXpX
0
0
|:)9,,1(
0|:),,1,0(
στ
open string
open 1-loop graph
boundary state
boundary conditions boundary conditions
)0(0)~(
0ˆ
nforB
Bp
nn
)0(0)~(
ˆ
nforB
BxBxin
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
tGnt
nt
tGn
t
n
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 nt
ntSwhere
)/'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
x
xxxF
x
,)'(~ 9
9p
p 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
tGtFteNNNT
ppT ;)]'([]2)[(
2~ 92/19
2
origin
DpNDpN Gaussian braneorigin
t
zero
Mxt
oscr
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
k
Ve
t
xkk
exdtTNNGk
i
i
92
1)(4
19
02)(
010
)'2(1
1,,,;|
2
20
))1(,)1(()'2()(2
))1(,()1()1()'2()(2
;~
0;
9)(
4
1
2
19
99)(
4
1
2
19
2/12/1
2
2
ijp
kt
i
p
ijpp
kt
i
p
pNM
i
i
ek
VNN
T
CBAek
VNN
T
tGDbbk
.)9(1)1()1(1
,)9(1)1()1(1
1,'
11,
)'2(1
)'2(1,
'21
'21,
,2
,1
1)(1
1
9
9
1
1)2/1(
2
CBApBA
BApBA
Ct
t
t
t
tS
eNN
NB
tOtF
tA
p
p
ijijt
ijijMN
T
tachyonorigin
tachyon origin
99DD
DpDp
graviton, dilaton via Gaussian graviton, dilaton via Gaussian boundary stateboundary state
2
4
19
2
1)1(
2
1)1(
)'2)((~
8
))(1(
4
1,
8
))(7(
4
72
2
~)(
)7()1()3(222
~)(
iktppp
ij
i
ppMN
i
pp
eNNTT
pppp
k
VTkh
pppk
VTk
graviton, dilaton via three-parameter graviton, dilaton via three-parameter solutionsolution
,)7(2
4
)3(
21
8
1)(
,)7(2
4
)3(
21
8
7)(
,)7(2
16
)1)(7(
21
4
3)(
2
87
01
2)1(
2
87
01
2)1(
2
87
01
2)1(
iji
ppp
ij
i
ppp
i
ppp
k
Vprc
k
pprh
k
Vprc
k
pprh
k
Vprc
k
pppk
12/1221
11
70
70 )()1(,, kcccrr
wherepp
pp
pp
pp
pp
rcpp
p
pc
pN
rkccp
NM
rNrkcNQ
70
21
21
7021
70
70
2/122
16
)1)(7(
7
)8(212
2
3
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 The effect of can be interpreted as as
D9-tachyon t. D9-tachyon t.
1c
t
pe
NN
N
t
ap
Cp
BAppp
T
'8
72)9(2
4
7)9(2
4
7
4
1
2
2
tC
ck
c
'2
1
2
1)(
2
1
),( 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)
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