Gaussian Brane and Open String Tachyon Condensation Shinpei Kobayashi ( RESCEU, The University of Tokyo ) Tateyama, Chiba Yoshiaki Himemoto and Keitaro Takahashi ( The University of Tokyo ) Tsuguhiko Asakawa and So Matsuura ( RIKEN )

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

D-brane Open string endpoints stick to a D-brane Open string endpoints stick to a D-brane Properties Properties SO(1,p)×SO(9-p), RR-charged SO(1,p)×SO(9-p), RR-charged (mass)  1/(coupling) → non-perturbative (mass)  1/(coupling) → non-perturbative X0X0 XμXμ XiXi open string Dp-brane

Open string channel : D-brane X0X0 XαXα XiXi open string D-brane

σ τ closed string D-brane σ τ open string boundary state tree graph of the closed string 1-loop graph of the open string

closed string channel : boundary state σ τ closed string boundary state

Mode expansion

Boundary state ( = D-brane) mass RR-charge ← direction where the D-brane extends Coherent state of strings

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

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

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

Boundary state for DD-system mass RR-charge constant tachyon

During the tachyon condensation 1. D-branes, anti D-branes coincide with each other. ( t = 0 ) 1. D-branes, anti D-branes coincide with each other. ( t = 0 ) 2. During the tachyon condensation ( t = t 0 ) tachyon vev is included in the mass. 3. Final state ( t = ∞ ) The mass will decrease via the closed string emission, and the total mass of the system will become coincident with the RR-charge. 1.

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

Boundary State of the Gaussian D-brane

Ordinary Boundary State boundary state σ τ closed string source of closed strings

Mode expansion Boundary condition via oscillators Boundary condition via oscillators

Boundary condition by oscillators Boundary condition by oscillators Longitudinal to the boundary state Transverse to the boundary state We introduce to write both conditions

Longitudinal to the boundary state Transverse to the boundary state Gaussian boundary state case

Oscillator part which satisfies the boundary condition is 0-mode part is : Dp-brane : D9-brane Gaussian

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

Boundary State of the “ Haussian ” D-brane

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

point particle closed string open string Strings

spacetime world-sheet symmetry of world-sheet spacetime action

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

τ = -1 τ = 0 τ = 2 τ = 1 τ = 0 τ = 1 τ = 2 σ = 0 σ =  world-line of point-particleworld-sheet of string

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

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

mode expansion where

States of string first quantization first quantization quantization of center of mass quantization of oscillation mode

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

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

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

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

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

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

charge ? mass ? tachyon vev ?

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

Asymptotic behavior of the solution

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

source Gravitational Field graviton source

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

General Boundary State with ordinary boundary state

via the boundary state from the 3-parameter solution

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

Case 1 c 1 does not correspond to the vev of tachyon ! (as opposed to the result of hep-th/ )

Case 2

Case 3

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

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