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In collaboration with Y. Baba(Tsukuba) and K. Murakami(KEK)

JHEP 05(2006):029JHEP 05(2007):020JHEP 10(2007):008

D-branes and Closed String Field Theory

N. Ishibashi (University of Tsukuba)

Progress of String Theory and Quantum Field Theory　　　 Friday 7th December - Monday 10th December 2007 　　　 @ Media Center, Osaka City University

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• D-branes can be realized as soliton solutions in

　　 open string field theory

• D-branes in closed string field theory?

Hashimoto and Hata

♦HIKKO♦A gauge invariant source term♦tension of the brane cannot be fixed

§1 Introduction

D-branes in string field theory

A.Sen,………

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For noncritical strings, the situation is better

c=0 noncritical strings

+ +double scalinglimit

String Field l

Describing the matrix model in terms of this field, we obtain the string field theory for c=0 noncritical strings

Kawai, N.I.

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joining-splitting interactions

Stochastic quantization of the matrix model Jevicki and Rodrigues

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correlation functions

Virasoro constraintsFKN, DVV

Virasoro constraints Schwinger-Dyson equationsfor the correlation functions

various vacua~

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Solitonic operators Fukuma and Yahikozawa

From one solution to the Virasoro constraint, one can generate another by the solitonic operator.

Hanada, Hayakawa, Kawai, Kuroki, Matsuo, Tada and N.I.

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These solitonic operators correspond to D-branes

state with D(-1)-brane ghost D-brane (Okuda and Takayanagi)

=

amplitudes with ZZ-branes are reproduced

the tension of the brane is fixed correctly

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critical strings?

noncritical case is simple

idempotency equation Kishimoto, Matsuo, Watanabe

For boundary states, things may not be so complicated

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We will show that

・ for OSp invariant SFT for the critical bosonic strings

・ we can construct BRS invariant observables using the boundary states for D-branes

imitating the construction of the solitonic operators for the noncritical string case.

Plan of the talk

§2 OSp Invariant String Field Theory

§3 D-brane States

§4 Conclusion and Discussion

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§2 OSp Invariant String Field Theory

light-cone gauge SFT

SO(25,1) symmetry

OSp invariant SFT =light-cone SFT withGrassmann

OSp(27,1|2) symmetrySiegel, Uehara, Neveu, West,Zwiebach, Kugo, Kawano,….

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variables

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action

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OSp theory 〜 covariant string theory with extra time and length

HIKKO

・ no

・ extra variables

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BRS symmetry

the string field Hamiltonian is BRS exact

The “action” cannot be considered as the usual action

looks like the action for stochastic quantization

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・ string length ・ joining-splitting interaction ・ extra time variable etc.

similar to noncriticalSFT

Green’s functions of BRS invariant observables = Green’s functions in 26D

S-matrix elements in 26D

∥

S-matrix elements derived from the light-cone gauge SFT

different from the usual formulation, but

One can show

Wick rotation, LSZ

We may be able to construct something like

Baba, Murakami, N.I.

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light-cone quantization

§3 D-brane States

Let us construct second quantized BRS invariant states, imitating the construction of the solitonic operator in the noncritical case.

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states with one soliton

will be fixed by requiring

c.f.

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Boundary states in OSp theory

boundary state for a flat Dp-brane

BRS invariant regularization

It is straightforward to include constant

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shorthand notation

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Idempotency equations

leading order in

Kishimoto, Matsuo, Watanabe

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Using these relations we obtain

if

Later we will show

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States with N solitons

Imposing the BRS invariance, we obtain

looks like a matrix model

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can be identified with the open string tachyon

may be identified with the eigenvalues

of the matrix valued tachyon

open string tachyon

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More generally we obtain

・ we can also construct

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amplitudes

saddle point approximation

generate boundaries on the worldsheet

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We have checked that the disk and cylinder amplitudes are reproduced correctly.

etc.

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§4 Conclusion and Discussion

♦similar construction for superstrings

♦other amplitudes

we need to construct OSp SFT for superstrings

♦D-brane ~ BRS invariant source term coherent state

♦more fundamental formulation?

♦ lower order in

contact term?

Yoneya