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Boundary States Boundary States and Black p-branesand Black p-branes
Shinpei Kobayashi Shinpei Kobayashi (( RESCEURESCEU ))
in collaboration with in collaboration with
Tsuguhiko Asakawa (RIKEN)Tsuguhiko Asakawa (RIKEN)
So Matsuura (RIKEN)So Matsuura (RIKEN)2004/05/19, 関東ゼミ
1. Introduction1. Introduction
How should we apply string theory How should we apply string theory to gravitational systems ? to gravitational systems ?What is ‘string cosmology’ ?What is ‘string cosmology’ ?
→ → D-brane is thought to be a key to D-brane is thought to be a key to describe well-known gravitational systems describe well-known gravitational systems
via string theory. via string theory.
D-braneD-brane
Open string endpoints can stick to Open string endpoints can stick to D-braneD-brane
D-branes carry D-branes carry RR chargesRR charges
)(|:)1,...,1(
)(0|:),...,1,0(
0
0
DirichletxXDpiX
NeumannXpXiii
X0
X Xi
σ=0 σ=π
τ
Dynamical non-BPS D-brane systems are Dynamical non-BPS D-brane systems are very important in string theory, very important in string theory,
(e.g.)(e.g.) D(2p+1)-brane in type IIA stringD(2p+1)-brane in type IIA string D(2p)-brane in type IIB stringD(2p)-brane in type IIB string D/anti D-brane systemD/anti D-brane system
(c.f.) BPS D-branes, (c.f.) BPS D-branes, stablestable non-BPS D-brane non-BPS D-brane
But no one has succeeded in describing But no one has succeeded in describing the dynamics of non-BPS D-brane.the dynamics of non-BPS D-brane.
Non-BPS D-brane system (1)Non-BPS D-brane system (1)D(2p+1)-braneD(2p+1)-brane
Closed string vacuum
Non-BPS D-brane system (2)Non-BPS D-brane system (2)D/anti D-brane systemD/anti D-brane system
closed string vacuum
lower-dimensional D-brane
Importance of dynamical D-brane Importance of dynamical D-brane systemssystems
String theoryString theory Searching for ‘real’ vacuum of string theorySearching for ‘real’ vacuum of string theory String interaction & dynamicsString interaction & dynamics
→ → non-perturbative string theorynon-perturbative string theory Gravitation & CosmologyGravitation & Cosmology
D-brane inflationD-brane inflation Black hole evaporation Black hole evaporation
→ → Application to physics at Planck scaleApplication to physics at Planck scale
Trials to dynamical D-brane Trials to dynamical D-brane systemssystems
Via ‘non-perturbative’ string theoryVia ‘non-perturbative’ string theory Open string field theory (A.Sen, …)Open string field theory (A.Sen, …) Closed string field theory Closed string field theory
(Asakawa, SK &Matsuura (’03), …) (Asakawa, SK &Matsuura (’03), …)
Via conformal field theoryVia conformal field theory Logarithmic CFT description Logarithmic CFT description
(Asakawa, Ishimoto, SK & Matsuura, (Asakawa, Ishimoto, SK & Matsuura, work in progress) work in progress)
Trials to dynamical D-brane Trials to dynamical D-brane systemssystems
Via low-energy effective theoryVia low-energy effective theory (Zhou & Zhu (‘99), Ohta & Yokono (‘02)(Zhou & Zhu (‘99), Ohta & Yokono (‘02) Brax, Mandal & Oz (‘01)) Brax, Mandal & Oz (‘01))
Time-dependent solutions have not found yet.Time-dependent solutions have not found yet. Stable BPS solution →Stable BPS solution → OKOK
black p-branes : black p-branes : Today’s themeToday’s theme Non-BPS solution → ?Non-BPS solution → ?
Dynamical systemHawking radiation,
Inflation, etc.
unknownobject
SUGRA String theory
Unknown non-BPSblack p-brane
(BPS) black p-brane
non-BPS D-brane
BPS D-brane
?
D-brane/black p-brane relationD-brane/black p-brane relation Stable BPS D-brane caseStable BPS D-brane case (Unstable non-BPS case)(Unstable non-BPS case)
Black p-brane Black p-brane from boundary state (= D-brane) from boundary state (= D-brane)
(Difference between D-brane (Difference between D-brane and black p-brane) and black p-brane)
2. Black p-brane 2. Black p-brane
Classical solution of SUGRAClassical solution of SUGRA It has same symmetry, charge and mass It has same symmetry, charge and mass
as a D-braneas a D-brane
→ → Low-energy description of a D-brane.Low-energy description of a D-brane. But no one has provedBut no one has proved..
(Non-BPS black p-branes have not been (Non-BPS black p-branes have not been found yet)found yet)
String Theory and SUGRAString Theory and SUGRA
String Field Theory action
Classical solution ofSting theory
Dp-brane
Classical solution ofSUGRA
Black p-brane
Supergravity actionmassless
massless
EO
M
EO
M
SUGRA action & ansatzSUGRA action & ansatz
11
21
2
)(
,||)!1(2
1
2
1
nn
naD
dAF
Fen
RgxdS
・ Φ : dilaton ・ A : n-form potential ・ F : (n+1)-form field strength
X0
XXi
σ=0 σ=π
τ
jiij dxdxrgdxdxrbds
)()( 222
Black p-brane solutionBlack p-brane solution
.42
)3)(1(2
,1
)3(
21)(
,)()(,)()(
,)()(
2
32
1...01
2
2
1
2
32
aD
pDp
rpD
TrH
where
rHrArHre
dxdxrHdxdxrHds
pDpD
p
p
a
jiij
D
p
D
pD
3. Boundary state3. Boundary state
D-brane in closed string channelD-brane in closed string channel Source of closed stringsSource of closed strings
← Such properties are guaranteed by ← Such properties are guaranteed by conformal symmetry of the world-sheet conformal symmetry of the world-sheet
conformal transformationconformal transformationζ→ ζ→ ff(ζ), where ζ=σ+iτ(ζ), where ζ=σ+iτ
Using the conformal transformation, we can Using the conformal transformation, we can change the boundary condition for open change the boundary condition for open strings into that for closed strings.strings into that for closed strings.
),(),(.
,
iii
)1,...,1(|
),...,1,0(0|
0
0
DpixX
pXii
)1,...,1(|
),...,1,0(0|
0
0
DpixBX
pBXi
Xi
X
Closed string
Boundary state
Closed string tree graph
Open string
D-brane
Open string 1-loop graph
We can rewrite the boundary condition with We can rewrite the boundary condition with using the oscillators.using the oscillators.
).exp(
,)(1
iz
zizXn
nn
0)ˆ(;0ˆ
),(,0)~(
,0,0)~(;0)~(
Xii
X
ijXnn
Xin
inXnn
BxqBp
SBS
nBB
000)ˆ(2
~
1
~1)1(
pexq
TB
n
SniipDp
X
nn
)1,...,1(|
),...,1,0(0|
0
0
DpixBX
pBXi
Xi
X
4. Black p-brane solution4. Black p-brane solution from boundary state from boundary state
pp
p
p
p
jiij
pp
rprGrGTrH
rHArHe
dxdxrHdxdxrHds
7)8(
1...01
4
3
8
1
8
72
)7(
1)(),(21)(
,)(,)(
,)()(
).(2
),(2
3
,)(8
1,)(
8
72
)1(...01
)1(
)1(
rGTA
rGTp
rGTp
rGTp
h
pp
p
ijpp
<B| |massless>
kmasslessge
BL
masslessmassless
BL
;0~.).
1
1
11
0
0
(e.g.) dilaton (10-dim.)(e.g.) dilaton (10-dim.)
)(21)7(
21)(
,)(
78
2
32ˆ222
rGTrp
TrH
rHee
ppp
p
p
)(22
3)(ˆ rGT
pr p
<B| |φ> +…
We can extract each mode which are incluWe can extract each mode which are included in Φ, for example, dilaton, graviton, anded in Φ, for example, dilaton, graviton, antisym.tensor and so on. tisym.tensor and so on.
Such modes corresponds to the leading teSuch modes corresponds to the leading term of the classical solution.rm of the classical solution.
SFT action and source termSFT action and source term
tensionbraneDT
stateboundarysourceB
fieldantisymgravitondilaton
fieldstring
BTg
QS
p
p
:
)(:
,....).,,(
:
,32
1
Calculation of fieldsCalculation of fields
BBgTBT
eXdX
BTg
QS
pp
S
p
32
,32
1
<B| |> +<B|
<B||> +…
Here, we do not know how strings interact, Here, we do not know how strings interact, so we use 3-point coupling of SUGRA.so we use 3-point coupling of SUGRA.
SUGRASFT
(e.g.) dilaton (10-dim.)(e.g.) dilaton (10-dim.)
)(21)7(
21)(
,)(
78
2
32ˆ222
rGTrp
TrH
rHee
ppp
p
p
22 )(22
3)(
22
3)(ˆ rGT
prGT
pr pp
<B| |φ> +<B|
<B| |φ> +…
59
9
2
2
97
2
21429
22
22)2(
||)2()8(2
32
9
)5)(7(
3
22
1
)7(28
29
)3(
)(22
3ˆ
pi
xik
pi
p
pp
p
pp
p
p
k
ekd
p
pp
pp
pT
rp
Tp
p
rGTp
i
AAh
hg
Fp
pghRgxdI p
&
,2
,|ˆ|ˆ2
3exp
)!2(8
1ˆˆ2
1),(
2
1 222
10
ΦΦ h A AΦ+
hμν
Φ
Φ
Φ
A
A
k1k1
k2k2
k3 k3
).()2(28
3)(
),()2(2
1)(
321)9(9
21...01...01
321)9(9
2121
kkkkkp
AA
kkkkkkkh
pppp
pp
,1
22ˆ
),,...,,,...,(1
2
,1
22
3ˆ
21...01
21
21
ippp
ipp
ipp
kVTAB
bbaadiagk
VThB
k
pVTB
kΦ
k
hμν
59
9
2
22
97
2
21429
22
22)2(
||)2()8(2
32
9
)5)(7(
3
)7(22
1
)7(28
29
)3(
)(22
3ˆ
pi
xik
pi
p
pp
p
pp
p
p
k
ekd
p
pp
pp
p
p
T
rp
Tp
p
rGTp
i
(c.f.) SUGRA(c.f.) SUGRA
JJTJTX pp32
Φhμν
Bμν
・・・
+ +…
5. Summary5. Summary
Black p-branes are the classical solutions Black p-branes are the classical solutions of SUGRA and they are thought to of SUGRA and they are thought to correspond to D-branes in low energy limit.correspond to D-branes in low energy limit.
Boundary states are another representation Boundary states are another representation of D-branes, which are written in closed of D-branes, which are written in closed string channel.string channel.
Using 3-point coupling of SUGRA, we can Using 3-point coupling of SUGRA, we can reproduce the asymptotic behavior of a reproduce the asymptotic behavior of a black p-brane from a boundary state .black p-brane from a boundary state .
6. Problem6. Problem
STF coupling ⇔ SUGRA coupling ?STF coupling ⇔ SUGRA coupling ?
Degrees of freedom of field-redefinitionDegrees of freedom of field-redefinitiongraviton of SFT ⇔ graviton of SUGRA ?graviton of SFT ⇔ graviton of SUGRA ?
Difference between D-brane Difference between D-brane and black p-brane and black p-brane → → massive mode effectmassive mode effect→ Hawking radiation, etc.→ Hawking radiation, etc.
7. Future Works7. Future Works We are now investigating...We are now investigating...
Classical solution for unstable non-BPS D-Classical solution for unstable non-BPS D-branebrane
D-brane deformation using String Field D-brane deformation using String Field Theory or CFTTheory or CFT
ApplicationApplication Hawking radiation in terms of D-braneHawking radiation in terms of D-brane D/anti-D brane inflationD/anti-D brane inflation