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

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.)

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

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 )

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

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

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.

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

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.

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.

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

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.

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　

Boundary State with boundary interaction

open string

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

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).

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

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

tachyon vev ? charge ? mass ?

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.

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

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

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

Using the boundary state, we obtain

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

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

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.

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.

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.

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

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

Three-parameter solution case 　　corresponds to the dilaton charge.

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

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

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

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

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

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

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

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

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

Oscillator picture boundary condition in the oscillator picture

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

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

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

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

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

integrate with finite finally, we obtain origin Gaussian brane origin

　　　　　tachyon origin 　　　　　tachyon origin

graviton, dilaton via Gaussian boundary state

graviton, dilaton via three-parameter solution

constant criterion : RR charge Q keeps its value

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

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)