1. Boundary States and Black p-branes Shinpei Kobayashi （ RESCEU ） in collaboration with Tsuguhiko Asakawa (RIKEN) Tsuguhiko Asakawa (RIKEN) So Matsuura (RIKEN) So Matsuura (RIKEN) 2004/05/19, 関東ゼミ

2. 1. Introduction How should we apply string theory to gravitational systems ? What is ‘string cosmology’ ? How should we apply string theory to gravitational systems ? What is ‘string cosmology’ ? → D-brane is thought to be a key to describe well-known gravitational systems via string theory. via string theory.

3. D-brane  Open string endpoints can stick to D-brane  D-branes carry RR charges X0X0 XX XiXi σ=0 σ=π τ

4.  Dynamical non-BPS D-brane systems are very important in string theory, (e.g.)  D(2p+1)-brane in type IIA string  D(2p)-brane in type IIB string  D/anti D-brane system (c.f.) BPS D-branes, stable non-BPS D-brane But no one has succeeded in describing the dynamics of non-BPS D-brane.

5. Non-BPS D-brane system (1) D(2p+1)-brane Closed string vacuum

6. Non-BPS D-brane system (2) D/anti D-brane system closed string vacuum lower-dimensional D-brane

7. Importance of dynamical D-brane systems  String theory  Searching for ‘real’ vacuum of string theory  String interaction & dynamics → non-perturbative string theory → non-perturbative string theory  Gravitation & Cosmology  D-brane inflation  Black hole evaporation → Application to physics at Planck scale → Application to physics at Planck scale

8. Trials to dynamical D-brane systems  Via ‘non-perturbative’ string theory  Open string field theory (A.Sen, …)  Closed string field theory (Asakawa, SK &Matsuura (’03), …)  Via conformal field theory  Logarithmic CFT description (Asakawa, Ishimoto, SK & Matsuura, work in progress)

9. Trials to dynamical D-brane systems  Via low-energy effective theory (Zhou & Zhu (‘99), Ohta & Yokono (‘02) Brax, Mandal & Oz (‘01))  Time-dependent solutions have not found yet.  Stable BPS solution → OK black p-branes : Today’s theme black p-branes : Today’s theme  Non-BPS solution → ?

10. Dynamical system Hawking radiation, Inflation, etc. unknown object SUGRAString theory Unknown non-BPS black p-brane (BPS) black p-brane non-BPS D-brane BPS D-brane ?

11.  D-brane/black p-brane relation  Stable BPS D-brane case  (Unstable non-BPS case)  Black p-brane from boundary state (= D-brane)  (Difference between D-brane and black p-brane)

12. 2. Black p-brane  Classical solution of SUGRA  It has same symmetry, charge and mass as a D-brane → Low-energy description of a D-brane. But no one has proved. → Low-energy description of a D-brane. But no one has proved.  (Non-BPS black p-branes have not been found yet)

13. String Theory and SUGRA String Field Theory action Classical solution of Sting theory Dp-brane Classical solution of SUGRA Black p-brane Supergravity action massless EOM

14. SUGRA action & ansatz ・ Φ ： dilaton ・ A ： n-form potential ・ F ： (n+1)-form field strength

15. X0X0 XX XiXi σ=0 σ=π τ

16. Black p-brane solution

17. 3. Boundary state  D-brane in closed string channel  Source of closed strings ← Such properties are guaranteed by conformal symmetry of the world-sheet conformal transformation ζ→ f(ζ), where ζ=σ+iτ conformal transformation ζ→ f(ζ), where ζ=σ+iτ

18. Using the conformal transformation, we can change the boundary condition for open strings into that for closed strings.

19.   Closed string Boundary state Closed string tree graph   Open string D-brane Open string 1-loop graph

20. We can rewrite the boundary condition with using the oscillators.

21. 4. Black p-brane solution from boundary state

22. <B||massless>

23. (e.g.) dilaton (10-dim.) <B| |φ> +…

24.  We can extract each mode which are included in Φ, for example, dilaton, graviton, antisym.tensor and so on.  Such modes corresponds to the leading term of the classical solution.

25. SFT action and source term

26. Calculation of fields <B||  > + <B| |  > +…

27. Here, we do not know how strings interact, so we use 3-point coupling of SUGRA. SUGRA SFT

28. (e.g.) dilaton (10-dim.) <B| |φ> + <B| |φ> +…

30. ΦΦ ｈ A AΦ + h μν Φ Φ Φ A A k1k1 k1k1 k2k2 k2k2 k3k3 k3k3

32. k Φ k

34. (c.f.) SUGRA Φ h μν B μν ・・・ ++…

35. 5. Summary  Black p-branes are the classical solutions of SUGRA and they are thought to correspond to D-branes in low energy limit.  Boundary states are another representation of D-branes, which are written in closed string channel.  Using 3-point coupling of SUGRA, we can reproduce the asymptotic behavior of a black p-brane from a boundary state.

36. 6. Problem  STF coupling ⇔ SUGRA coupling ?  Degrees of freedom of field-redefinition graviton of SFT ⇔ graviton of SUGRA ?  Difference between D-brane and black p-brane → massive mode effect → Hawking radiation, etc.

37. 7. Future Works  We are now investigating...  Classical solution for unstable non-BPS D- brane  D-brane deformation using String Field Theory or CFT  Application  Hawking radiation in terms of D-brane  D/anti-D brane inflation