Open String Tachyon in Supergravity Solution

1. Open String Tachyon in Supergravity Solution Shinpei Kobayashi ( Research Center for the Early Universe, The University of Tokyo ) Based on hep-th/0409044 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. Supergravity low energy limit α’ →0 classical description ( Black p-brane ) low energy limit 1. D-branes and Classical descriptions String Field Theory D-brane ( Boundary State )

5. X0 Xi Xμ 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 open string

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

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

12. low energy limit String Field Theory Supergravity low energy limit α’ →0 eom eom D-brane ( Boundary State ) classical solution ( Black p-brane ) 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. 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. Stable D-branes are left case 2. D/anti D-brane system 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

15. Boundary State with boundary interaction

16. open string

17. constant tachyon 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. constant tachyon Boundary state for D/anti D-brane 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. <B| |physical field> graviton, dilaton, RR-potential in string theory

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

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 ? → with open string tachyon • Deformation of the boundary state 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. 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. extend to -direction infinitely Gaussian in -direction localized at

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

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

45. Oscillator picture • boundary condition in the oscillator picture

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

47. Longitudinal to the D-brane • boundary conditions 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 to ordinary boundary state with ｔ→∞ combine them

50. tension part via SFT （Kraus-Larsen, PRD63 (2001) 106004） From a to one thus, in the limit （D9-tachyon vanishes）