1. Open String Tachyon in Supergravity Solution
Shinpei Kobayashi
( Research Center for the Early Universe, The University of Tokyo )
Based on hep-th/
in collaboration with
Tsuguhiko Asakawa and So Matsuura ( RIKEN )
2005/01/18
at KEK

2. Motivation
We would like to apply the string theory to the analyses of the gravitational systems.
We have to know
how we should apply string theory to realistic gravitational systems,
or what stringy (non-perturbative) effects are,
or what stringy counterparts of the BHs or Universe in the general relativity are.
→ D-branes may be a clue to tackle such problems (BH entropy, D-brane inflation, etc.)

3. Contents D-branes and Classical Descriptions D/anti D-brane system
Three-parameter solution
Conclusions
Discussions and Future Works

4. 1. D-branes and Classical descriptions
String Field Theory
Supergravity
low energy limit
α’ →　0
classical description
( Black p-brane )
low energy limit
D-brane
( Boundary State )

5. D-brane ( BPS case ) Open string endpoints stick to a D-brane
Properties
SO(1,p)×SO(9-p) ( BPS case ), RR-charged
(mass)  1/(string coupling)
Dp-brane X0 Xμ Xi
open string

6. BPS black p-brane solution
Symmetry : SO(1,p)×SO(9-p), RR-charged
setup : SUGRA action
ansatz :

7. BPS black p-brane solution (D=10)
・ SO(1,p)×SO(9-p),
・ (mass)=(RR-charge), which are the same as D-branes
it must be large for
the validity of SUGRA
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.

8. coincide Relation between the D-brane ( the boundary state)
and the black p-brane solution
(Di Vecchia et al. (1997))
asymptotic behavior of the black p-brane = difference from the flat background = graviton, dilaton, RR-potential in SUGRA
massless modes of the closed strings from the boundary state ( D-brane in closed string channel ) = graviton, dilaton, RR-potential in string theory
( string field theory )
coincide

9. Boundary State ( = D-brane)
Boundary states are defined as sources of closed strings ( = D-branes in closed string channel ).
As closed strings include gravitons, the boundary state directly relates to a black p-brane solution.

11. e.g. ) asymptotic behavior of Φ of black p-brane
leading term at infinity
coincident
<B|
　|φ>
We can reproduce the leading term of a black p-brane
solution ( asymptotic behavior ) via the boundary state.

12. String Field Theory Supergravity eom eom D-brane ( Boundary State )
low energy limit
α’ →　0
eom
eom
D-brane
( Boundary State )
classical solution
( Black p-brane )
low energy limit
BPS case → OK (Di Vecchia et al. (1997))
non-BPS case → ?
We study non-BPS systems
( e.g. D/anti D-brane system ).
non-BPS cases
are more realistic
in GR sense

13. We verify their claim using the boundary state.
BPS case
Dp-brane 　　 black p-brane
Non-BPS case
D/anti D-brane system with a constant tachyon vev Three-parameter solution ? ( guessed by Brax-Mandal-Oz (2000))
( other non-BPS system corresponding classical solution ?)
We verify their claim using the boundary state.

14. 2. D/anti D-brane system tachyon condensation closed string emission
D-branes and
anti D-branes
attracts together
Unstable multiple branes
Open string tachyon
represents its instability
Stable
D-branes are left
　 case　

15. Boundary State with boundary interaction

16. open string

17. Boundary state for D/anti D-brane with a constant tachyon vev
RR-charge
mass

18. Change of the Mass during the tachyon condensation
D-branes, 　　anti D-branes coincide with each other. ( t = 0 ) 　
During the tachyon condensation ( t = t0 ) tachyon vev is included in the mass.
Final state ( t = ∞ ) The mass will decrease through the closed string emission, and the value of the mass will coincide with that of the RR-charge (BPS).

19. Boundary state for D/anti D-brane
constant tachyon
RR-charge
mass

20. 3. Three-parameter solution ( Zhou & Zhu (1999) )
SUGRA action
ansatz : SO(1, p)×SO(9-p) ( D=10 )
same symmetry as the D/anti D-brane system

21. tachyon vev ?
charge ?
mass ?

22. Property of the three-parameter solution
ADM mass
RR charge
We can extend it to an arbitrary dimensionality.
We re-examine the correspondence
between the D/anti D-brane system
and
the three-parameter solution
using the boundary state. ~ ? ~ ?
From the form of the boundary state,
Brax-Mandal-Oz claimed
that c_1 corresponds to the tachyon vev.

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

24. Asymptotic behavior of the three-parameter solution (= graviton, dilaton, RR-potential in SUGRA )

25. graviton, dilaton, RR-potential in string theory
<B|
　|physical field>

26. Using the boundary state, we obtain

27. Results and Comparison
asymptotic behavior of
the three-parameter solution
massless modes
via the boundary state

29. Results and Comparison
asymptotic behavior of
the three-parameter solution
massless modes
via the boundary state

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

31. Conclusions
Using the boundary state, we find that the three-parameter solution with c_1=0 corresponds to the D/anti D-brane system with a constant tachyon vev.
New parametrization is needed to keep the RR-charge constant during the tachyon condensation.
The vev of the open string tachyon is only seen as a part of the ADM mass.
c_1 does not corresponds to the tachyon vev as opposed to the proposal made so far.
We find that we can extend the correspondence between D-branes and classical solutions to non-BPS case.
First discovery of the correspondence in non-BPS case.
It may be a clue to describe “realistic” gravitational systems which are generally non-BPS.

32. Discussion : Why was c_1 thought to be the open string tachyon vev ?
Parametrization → during the t.c., the RR-charge does not change its value. →
The relation between the mass and the scalar charge → cf. Wyman solution in D=4 case c_1 corresponds to the dilaton charge.

33. Wyman solution in Schwarzschild gauge
Static, spherically symmetric, with a free scalar

34. Wyman solution in isotropic gauge
r → R
In this gauge,
we can compare it with the 3-para. sln.

35. Three-parameter solution case
　　corresponds to
the dilaton charge.

36. Discussion : Stringy counterpart of c_1 ?
has something to do with the brane.
We can not relate these parts
with an ordinary
boundary state
counterpart of
the D/anti D-brane system

37. We can not relate these parts with an ordinary boundary state
counterpart of
the D/anti D-brane system

38. Do we have such a deformation in string theory ?
Deformation of the boundary state
Do we have such a deformation
in string theory ?
→ 　　 with open string tachyon
We can reproduce the 3-para. sln with non-zero
by adjusting α・β

39. Construction of (Asakawa-Sugimoto-Terashima, JHEP 0302 (2003) 011)
boundary interaction

40. →ordinary boundary state
δ-function with ｔ → ∞
→ordinary boundary state

41. From Gaussian Boundary State to BPS Dp-brane
lower-dimensional
BPS D-brane
t → ∞
tachyon has some
configuration

42. -direction infinitely
extend to
-direction infinitely
Gaussian in
-direction
localized at

43. Consider that each or is made from
So far, we treat 　　　　　　　　
Consider that each　　 or is made from
boundary state is deformed as follows:
ordinary
Deformed
origin
Gaussian brane
origin

44. Gaussian boundary state
D9-tachyon
Mixture of Neumann b.c. and Dirichlet b.c.
→　smeared boundary condition

45. Oscillator picture
boundary condition in the oscillator picture

46. cf. ordinary boundary state
open string σ τ
closed
string
closed tree graph τ σ
boundary state
D-brane
open 1-loop graph

47. boundary conditions Longitudinal to the D-brane
Transverse to the D-brane

48. Gaussian boundary state case
・ Longitudinal to the Dp-brane
・Transverse to the Dp-brane

49. Oscillator part 0-mode part combine them to ordinary boundary
state with ｔ→∞
combine them

50. thus, in the limit （D9-tachyon vanishes）
tension part via SFT
（Kraus-Larsen, PRD63 (2001) ）
From a
to one
thus, in the limit （D9-tachyon vanishes）

51. integrate with finite
finally, we obtain
origin
Gaussian brane
origin

53. 　　　　　tachyon
origin
　　　　　tachyon
origin

54. graviton, dilaton via Gaussian boundary state

55. graviton, dilaton via three-parameter solution

56. constant
criterion :
RR charge Q keeps
its value

57. Thus, we compare them as
→ The effect of can be interpreted as
D9-tachyon t.

58. Future Work
c_1 and a Gaussian brane (SK, Asakawa & Matsuura, hep-th/0502XXX )
Entropy counting via non-BPS boundary state
Construction of a time-dependent solution
feedback to SFT
Solving δB|B>=0 ( E-M conservation law in SFT ) (Asakawa, SK & Matsuura, JHEP 0310 (2003) 023)
Application to cosmology (SK, K. Takahashi & Himemoto)
Stability analysis ( K. Takahashi & SK)