Gaussian Brane and Open String Tachyon Condensation

1. 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

2. Motivation • Gravitational systems and string theory • Black holes = ? • Our universe = ? • Stringy effects • string length ? • non-perturbative effect ? → D-brane may be a clue to tackle such problems

3. X0 Xi Xμ D-brane • Open string endpoints stick to a D-brane • Properties • SO(1,p)×SO(9-p), RR-charged • (mass)  1/(coupling) → non-perturbative Dp-brane open string

4. low energy limit D-brane and Black p-brane String Field Theory Supergravity low energy limit α’ →0 D-brane classical solution ( Black p-brane )

5. More general D-branes • BPS D-brane • supersymmetric, static ~ BPS black hole • non-BPS D-brane • no SUSY • unstable (classical, quantum) ~ unstable BH,… • time-dependent, dynamical~ Cosmological model Tachyonic mode of open string on D-brane = Instability of the system

6. -brane system Tachyon Condensation Case 1 Unstable multiple branes Open string tachyon denotes the instability. D-branes and anti D-branes attracts together. Stable D-branes are left. case

7. Tachyon Condensation Kraus-Larsen (‘01) Case 2 The system extends to all directions. Gaussian in -direction localized at

8. Tachyon Condensation Case 3 Asakawa-SK-Matsuura, in preparation Haussian brane

9. How should we describe D-branes ? • Non-perturbative string theory • String Field Theory • Matrix Theory • Low energy effective theory • Metric around D-brane e.g.) Black p-brane solution, Three-parameter solution,… • D-brane action → Born-Infeld action, Kraus-Larsen action, …

10. Strings point particle open string closed string

11. spacetime world-sheet symmetry of world-sheet spacetime action

12. String in flat spacetime • Free motion of a one-dimensional object • Flat background spacetime cf.) action for the free relativistic point particle　→ δS=0 ⇔ eom of point-particle

13. σ = 0 σ =  τ = 2 τ = 2 τ = 1 τ = 1 τ = 0 τ = 0 τ = -1 τ = -1 world-line of point-particle world-sheet of string

14. Action for free string • In the flat spacetime • analogy to point-particle→ area of the world-sheet = action→ Nambu-Goto action → δS=0 ⇔ eom

15. Polyakov action cf.) Nambu-Goto action • Weyl invariance • δS = 0 ⇔ • mode expansion of → quantization → state of string

16. String in Curved Spacetime • String in curved background= non-linear sigma model → are couplings • Conformal inv. decides the behavior • This can be reproduced by SUGRA action

17. String with Boundary Interaction • Including the boundary interaction = Considering the D-brane string

18. Non-linear sigma model with boundary interaction eom EOM can be reproduced via the Born-Infeld action

19. String with tachyonic interaction Kraus-Larsen (‘01) • Unstable system has the tachyonic interaction EOM

20. Effective action for unstable D-brane Kraus-Larsen (‘01) : linear tachyon Gaussian brane

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

25. tachyon vev ? charge ? mass ?

26. New parametrization → During the tachyon condensation, the RR-charge does not change its value. → We need a new parametrization.

27. Asymptotic behavior of the solution

28. 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 ) coincident

29. source source Gravitational Field graviton

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

31. General Boundary State with ordinary boundary state

32. via the boundary state from the 3-parameter solution

33. Compared with each other, we find and the ADM mass and the RR-charge are

34. Case 1 c1does not correspond to the vev of tachyon ! (as opposed to the result of hep-th/0005242)

35. Case 2

36. Case 3

37. Summary • D-brane plays an important role in string theory • Black hole, Universe, non-perturbative, … • Symmetry of world-sheet → spacetime action • Tachyon condensation of unstable D-brane system→ Kraus-Larsen action • Metric around some unstable D-brane systems→ Three-parameter solution • New parametrization is needed. • DpDp system = the three-parameter solution with c_1 =0 • <T> ~ (mass) － (RR-charge) • c_1 corresponds to the full width at half-maximum.(hep-th/0409044, 0502XXX SK-Asakawa-Matsuura)

38. Future Works • Time-dependent solutions • feedback to SFT • Solving δB|B>=0 ( E-M conservation law in SFT ) (Asakawa, SK & Matsuura (‘03) ) • Application to a Cosmological Model (with K. Takahashi & Himemoto) • Stability analysis • Relation to open string tachyons( with K. Takahashi ) • Entropy counting via non-BPS boundary state • Massive modes analysis using the boundary state