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Gaussian Brane and Open String Tachyon Condensation. Shinpei Kobayashi ( RESCEU, The University of Tokyo ). Yoshiaki Himemoto and Keitaro Takahashi ( The University of Tokyo ) Tsuguhiko Asakawa and So Matsuura ( RIKEN ). 2005/02/17-19 @ Tateyama, Chiba. Motivation. - PowerPoint PPT Presentation
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Gaussian Brane Gaussian Brane and and
Open String Tachyon CondensationOpen String Tachyon Condensation
Shinpei KobayashiShinpei Kobayashi( RESCEU, The University of Tokyo )( RESCEU, The University of Tokyo )
2005/02/17-19@ Tateyama, Chiba
Yoshiaki Himemoto and Keitaro TakahashiYoshiaki Himemoto and Keitaro Takahashi
( The University of Tokyo ) ( The University of Tokyo )
Tsuguhiko Asakawa and So Matsuura Tsuguhiko Asakawa and So Matsuura
( RIKEN ) ( RIKEN )
MotivationMotivationGravitational systems and string theoryGravitational systems and string theory
Black holes = ?Black holes = ?Our universe = ?Our universe = ?
Stringy effects Stringy effects string length ?string length ?non-perturbative effect ?non-perturbative effect ?
→ → D-brane may be D-brane may be a clue to tackle such problems a clue to tackle such problems
D-braneD-braneOpen string endpoints stick to a D-braneOpen string endpoints stick to a D-branePropertiesProperties
SO(1,p)×SO(9-p), RR-chargedSO(1,p)×SO(9-p), RR-charged (mass) (mass) 1/(coupling) → non-perturbative 1/(coupling) → non-perturbative
X0
Xμ Xiopen string
Dp-brane
)10(9,,1:
.,,1,0:
DpiX
pXi
.1
)7(
21)(
,1)(),(
,)()(
7)8(
1)(4
3)(
8
1
8
72
pp
pp
pr
p
pr
jiij
p
p
p
p
rp
NTrfwhere
rferfe
dxdxrfdxdxrfds
String Field Theory
D-brane
Supergravitylow energy limit
α’ → 0
classical solution( Black p-brane )low energy limit
D-brane and Black p-braneD-brane and Black p-brane
x
ix
More general D-branesMore general D-branes
BPS D-braneBPS D-brane supersymmetric, static ~ BPS black holesupersymmetric, static ~ BPS black hole
non-BPS D-brane non-BPS D-brane no SUSYno SUSY unstable (classical, quantum) ~ unstable BH,…unstable (classical, quantum) ~ unstable BH,… time-dependent, dynamicaltime-dependent, dynamical ~ Cosmological model~ Cosmological model
Tachyonic modeTachyonic mode of open string on D-brane of open string on D-brane
= = InstabilityInstability of the system of the system
Tachyon CondensationTachyon Condensation
Case 1Case 1 DpDpDp
NN D-branes and anti D-branes
attracts together.
Unstable multiple branes Open string tachyon
denotes the instability.
Stable D-branes are left.
case
)( NN
NN
-brane systemDD
Tachyon CondensationTachyon Condensation
Case 2Case 2 DpDpGaussianDD 99
systemDD 99
The system extends to all directions.
localized at
braneDp
)9,,1( ppixi ),,1,0( px
)9,,1( ppixi
),,1,0( px
0ix
braneDpGaussian
Gaussian in -directionix
Kraus-Larsen (‘01)
Tachyon CondensationTachyon Condensation
Case 3Case 3 DpDD )1()1(
1
1
2)1(
2)1(
m
m
D
DmDpHaussian brane
Asakawa-SK-Matsuura,in preparation
How should we describe D-branes ?How should we describe D-branes ?
Non-perturbative string theoryNon-perturbative string theoryString Field TheoryString Field TheoryMatrix TheoryMatrix Theory
Low energy effective theoryLow energy effective theoryMetric around D-braneMetric around D-brane
e.g.) Black p-brane solution, e.g.) Black p-brane solution, Three-parameter solution,… Three-parameter solution,…
D-brane action → Born-Infeld action,D-brane action → Born-Infeld action, Kraus-Larsen action, … Kraus-Larsen action, …
point particle closed string open string
sl'
StringsStrings
X
),( X
X
X
,,, BG
,,TA
spacetime
world-sheet
symmetry of world-sheet
spacetime action
aaab
XXh
abh
Free motion of a one-dimensional objectFree motion of a one-dimensional object Flat background spacetime Flat background spacetime
cf.) action for the free relativistic point particlecf.) action for the free relativistic point particle
→ → δS=0 ⇔ eom of point-particle δS=0 ⇔ eom of point-particle
,dsmS
String in flat spacetimeString in flat spacetime
τ = -1τ = -1
τ = 0
τ = 2
τ = 1
τ = 0
τ = 1
τ = 2
σ = 0 σ =
world-line of point-particle world-sheet of string
X
XX
X
Action for free stringAction for free string
In the flat spacetimeIn the flat spacetimeanalogy to point-particleanalogy to point-particle
→ area of the world-sheet = action→ area of the world-sheet = action→→ Nambu-Goto action Nambu-Goto action
→ → δS=0 ⇔ eomδS=0 ⇔ eom
,,,
,det'2
12
1
baXXh
hddS
baab
abNG
Polyakov action Polyakov action cf.) Nambu-Goto actioncf.) Nambu-Goto action
Weyl invarianceWeyl invarianceδS = 0 ⇔δS = 0 ⇔mode expansion of mode expansion of
→ quantization → state of string → quantization → state of string
XXdS ba
ab 2
'2
1
02 XX
0)(;, ˆ xeknnstate xiklili
String in Curved SpacetimeString in Curved Spacetime
String in curved backgroundString in curved background= non-linear sigma model = non-linear sigma model → are couplings→ are couplings
Conformal inv. decides the behavior Conformal inv. decides the behavior
This can be reproduced by SUGRA actionThis can be reproduced by SUGRA action
,, BG
XXBiGdS ba
abab2
'2
1
,12
14 2 dBRG
String with Boundary InteractionString with Boundary Interaction Including the boundary interaction
= Considering the D-branestring
X
X
A
Non-linear sigma model with Non-linear sigma model with boundary interactionboundary interaction
AdXiXXGzdS 2
'2
1
0A
)1()2(
1 )'2det(
dAF
FGedS pBI
eom
EOM can be reproduced via the Born-Infeld action
String with tachyonic interactionString with tachyonic interaction
Unstable Unstable system has the tachsystem has the tachyonic interactionyonic interaction
AdXiS exp)exp( int
,)(expˆ)exp( int XMPTrS
DXXAXT
XTDXXAXM
)()(
)()()(
A
AT
T
D D
Kraus-Larsen (‘01)
EOM
99DD
Effective action for unstable D-braneEffective action for unstable D-braneKraus-Larsen (‘01)
k
I
IID TFTxdTS
1
222109 )('||'2exp2
)()(
)0()()2(ln21~
)2(2
)(4)(
2/1
2
xxOx
xxOx
x
xxxF
x
IIG
II XuxT 2/1')( ))(2exp()||'2exp( 222 I
I XuT
Gaussian brane
: linear tachyon
)9,,1( ppixi ),,1,0( px
)9,,1( ppixi
),,1,0( px
T
)(TV
T
)(TV
T
)(TV
actionDBPSnonactionDD 899
)( 99 uT
xdT
uFu
xdT
uFxuxdTS
Du
D
D
99
99
99
9299109
'22
'2
2
')(2
exp2
9
89)'2(2 DD TT
actionDBPSactionDD 799
),(, 9898 uuTT
xdT
uFu
uFu
xdT
uFuFxuxuxdTS
Duu
D
D
89
22
,
99
88
89
98299288109
'4
'2
'2
2
'')()(2
exp2
98
9u
792)'2( DD TT
Three-parameter solutionThree-parameter solution ( Zhou & Zhu (1999) )( Zhou & 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 system DD
.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 ?
New parametrizationNew parametrization
→ → During the tachyon condensation, During the tachyon condensation, the RR-charge does not change its value. the RR-charge does not change its value. → We need a new parametrization. → We need a new parametrization.
.,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 solution Asymptotic behavior of the solution
.1
)(
,1
16
)7)(1(1
4
3)(
,1
4
31
8
11
,1
4
31
8
71
)7(270
)7(27012
)7(27012)(2
)7(27012)(2
pp
pp
pprB
pprA
rrrC
rrv
k
cppv
pr
rrv
k
cpv
pe
rrv
k
cpv
pe
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 )
coincident
Relation between the D-brane ( the boundary state) and the black p-brane solution
(Di. Vecchia et al. (1997))
hg 1~
source
Gravitational Field graviton
)()( )(2 rCr d )()( )(2
rk
Cr d
i
source
,)( 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>
NSi
r
Nr
mMN
Mr
m
Nm
mMN
MmNSp
xp
bSbSC
B
0,0
~~exp2 2/1
)(
1
)(0
ij
MN B
AS
)2/1(
General Boundary StateGeneral Boundary State
with
1CBA ordinary boundary state
pp
pijpMN
rBpApCNr
rppCNBArh
7)1(
7)1(
1)7()1(
4
1)(
1)1(,)7()(
8
1)(
p
pijMN
rv
k
cppv
pr
rppv
k
cpvrh
7120)1(
7120)1(
1
16
)7)(1(1
4
3
4)(
1)1(,)7(
4
31
8)(
via the boundary statevia the boundary state
from the 3-parameter solutionfrom the 3-parameter solution
Bp
ApC
v
BAC
vk
c
4
7
4
11
)(
0
2
0
1
00 ,)( QCBAM
Compared with each other, we find Compared with each other, we find
and the ADM mass and the RR-charge are and the ADM mass and the RR-charge are
0,0~~exp
21
2
2/1
)(
1
)(
||02
ir
Nr
mMN
Mr
m
Nm
mMN
Mm
Tp
xpbSbS
eNN
NB
ij
MNS
2||21,1,1 Te
NN
NCBA
2||2
1
211
0
TeNN
Nv
c
Case 1Case 1 DpDpDp c1 does not correspond to the vev of tachyon !
(as opposed to the result of hep-th/0005242)
ir
Nr
mMN
Mr
m
Nm
mMN
Mm
Gp
ppbSbS
xFuG
,0~~exp
)(2
;
2/1
)(
1
)(
0
ij
rMNij
mMN
xrxr
S
xmxm
S
1
1,,
1
1, )()(
.))('4( 2 constux G
pxFCx
xBA
9)(,12
12,1
Case 2Case 2 DpDpGaussianDD 99
)12(2
)12(4)()38(
)()7(4)12)(1)(4(
)(2
7
)8(2
222
2221
x
xxFpxv
xxFpxxF
xF
p
pc
ir
Nr
mMN
Mr
m
Nm
mMN
Mm
Hp
ppbSbS
yGuH
,0~~exp
)(2
;
2/1
)(
1
)(
0
ij
rMNij
mMN
yryr
S
ymym
S ,1
1,,
1
1)()(
.)('4 2 constuy H
)(,1,12
12xGCB
y
yA
Case 3Case 3 DpDD )1()1(
)12(2
)12(4)()38(
)()1(4)12)(1)(4(
)(2
7
)8(2
222
2221
y
yyGpyv
yyGpyyG
yG
p
pc
SummarySummary D-brane plays an important role in string theoryD-brane plays an important role in string theory
Black hole, Universe, non-perturbative, …Black hole, Universe, non-perturbative, … Symmetry of world-sheet → spacetime actionSymmetry of world-sheet → spacetime action
Tachyon condensation of unstable D-brane systemTachyon condensation of unstable D-brane system→ Kraus-Larsen action→ Kraus-Larsen action
Metric around some unstable D-brane systemsMetric around some unstable D-brane systems→ Three-parameter solution→ Three-parameter solution New parametrization is needed.New parametrization is needed. DpDp system DpDp system
= the three-parameter solution with c_1 =0 = the three-parameter solution with c_1 =0 <T> ~ (mass) <T> ~ (mass) - - (RR-charge) (RR-charge) c_1 corresponds to the full width at half-maximum.c_1 corresponds to the full width at half-maximum.
(hep-th/0409044, 0502XXX SK-Asakawa-Matsuura)(hep-th/0409044, 0502XXX SK-Asakawa-Matsuura)
Future WorksFuture Works Time-dependent solutions Time-dependent solutions
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 (Asakawa, SK & Matsuura
(‘03) )(‘03) ) Application to a Cosmological Model Application to a Cosmological Model
(with K. Takahashi & Himemot(with K. Takahashi & Himemoto)o)
Stability analysis Stability analysis Relation to open string tachyonsRelation to open string tachyons
( with K. Takahashi )( with K. Takahashi ) Entropy counting via non-BPS boundary stateEntropy counting via non-BPS boundary state Massive modes analysis using the boundary stateMassive modes analysis using the boundary state