Gaussian Brane and Open String Tachyon Condensation

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


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)


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()(


1)(

64

)3)(1(

)ln(7

1)(


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

Gaussian Brane and Open String Tachyon Condensation

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