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D-branes and Closed String Field Theory. N. Ishibashi (University of Tsukuba). In collaboration with Y. Baba(Tsukuba) and K. Murakami(KEK). JHEP 05(2006):029 JHEP 05(2007):020 JHEP 10(2007):008. 理研シンポジウム「弦の場の理論 07 」 Oct. 7, 2007. §1 Introduction. D-branes in string field theory.

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slide1

D-branes and Closed String Field Theory

N. Ishibashi (University of Tsukuba)

In collaboration with

Y. Baba(Tsukuba) and K. Murakami(KEK)

JHEP 05(2006):029

JHEP 05(2007):020

JHEP 10(2007):008

理研シンポジウム「弦の場の理論07」

Oct. 7, 2007

slide2

§1Introduction

D-branes in string field theory

  • 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 BRS invariant source term

♦tension of the brane cannot be fixed

slide3

=

Tensions of D-branes

Can one fix the normalization of the boundary states

without using open strings?

For noncritical strings, the answer is yes.

Fukuma and Yahikozawa

Let us describe their construction using SFT

for noncritical strings.

slide4

l

c=0 noncritical strings

+

+

double scaling

limit

String Field

Describing the matrix model in terms of

this field, we obtain the string field theory

for c=0 noncritical strings

Kawai, N.I.

slide5

Stochastic quantization of the matrix model

Jevicki and Rodrigues

joining-splitting

interactions

slide6

correlation functions

Virasoro constraints

FKN, DVV

Virasoro constraints

Schwinger-Dyson equations

for the correlation functions

~

various vacua

slide7

Fukuma and Yahikozawa

Hanada, Hayakawa, Kawai, Kuroki,

Matsuo, Tada and N.I.

Solitonic operators

These coefficients are chosen so that

From one solution to the Virasoro constraint,

one can generate another by the solitonic operator.

slide8

These solitonic operators correspond to D-branes

amplitudes with

ZZ-branes

state with D(-1)-brane

ghost D-brane

=

Okuda,Takayanagi

slide9

critical strings?

noncritical case is simple

idempotency equation

Kishimoto, Matsuo, Watanabe

For boundary states, things may not be so complicated

slide10

If we have

・SFT with length variable

・nonlinear equation like the Virasoro constraint

for critical strings, similar construction will be possible.

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.

♦the BRS invariant observables

→BRS invariant source terms~D-branes

♦the tensions of the branes are fixed

slide11

Plan of the talk

§2 OSp Invariant String Field Theory

§3 Observables and Correlation Functions

§4 D-brane States

§5 Disk Amplitudes

§6 Conclusion and Discussion

slide12

§2 OSp Invariant String Field Theory

light-cone gauge SFT

O(25,1) symmetry

OSp invariant SFT=light-cone SFT with

Grassmann

Siegel, Uehara,Neveu, West,

Zwiebach, Kugo, Kawano,….

OSp(27,1|2) symmetry

slide17

The “action” cannot be considered as the usual action

looks like the action for stochastic quantization

BRS symmetry

the string field Hamiltonian is BRS exact

slide19

S-matrix elements in 26D

?

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

OSp invariant SFT 〜stochastic formulation of string theory ?

Green’s functions of BRS

invariant observables

Green’s functions in 26D

▪Wick rotation

▪1 particle pole

slide20

§3 Observables and Correlation Functions

observables

ignoring the multi-string

contribution

if

we need the cohomology of

slide21

cohomology of

(on-shell)

This state corresponds to a particle with mass

slide23

Free propagator

two point function

We would like to show that the lowest order contribution

to this two point function coincides with the free propagator

of the particle corresponding to this state.

slide26

the Hamiltonian is BRS exact

a representative of the class

slide27

N point functions

we would like to look for the singularity

light-cone diagram

slide30

Rem.

higher order corrections

・two point function

・N point function

can also be treated in the same way

at least formally

slide31

・LHS is an analytic function of

・Wick rotation + OSp(27,1|2) trans.

Wick rotation

reproduces the light-cone gauge result

slide32

§3 D-brane States

Let us construct off-shell BRS invariant states,

imitating the construction of the solitonic operator

in the noncritical case.

Boundary states in OSp theory

boundary state for a flat Dp-brane

slide33

BRS invariant regularization

states with one soliton

will be fixed by requiring

slide36

Using these, we obtain

We need the idempotency equation.

slide37

Idempotency equations

leading order in

corrections

slide40

can also be evaluated from the Neumann

function

: quadratic in

can be evaluated from the Neumann

function

slide41

can be evaluated in the same way

we can fix the normalization in the limit

slide43

States with N solitons

Imposing the BRS invariance, we obtain

looks like the matrix model

slide44

may be identified with the eigenvalues

of the matrix valued tachyon

can be identified with the open string tachyon

open string tachyon

slide45

More generally we obtain

・we can also construct

slide46

§4 Disk Amplitudes

BRS invariant source term

Let us calculate amplitudes in the presence of

such a source term.

slide47

saddle point approximation

generate boundaries on the

worldsheet

slide48

correlation functions

=

+

+…

+

+…

Let us calculate the disk amplitude

slide49

Rem.

In our first paper, we calculated the vacuum amplitude

▪we implicitly introduced two D-branes by

taking the bra and ket

→states with even number of D-branes

▪vacuum amplitude is problematic in

light-cone type formulations

▪we overlooked a factor of 2

but

slide51

tachyon-tachyon

coincides with the disk amplitude up to normalization

・more general disk amplitudes can be calculated in the

same way

slide52

low-energy effective action

Comparing this with the action for D-branes

slide53

§5 Conclusion and Discussion

BRS invariant source term

♦D-brane

♦other amplitudes

♦similar construction for superstrings

we need to construct OSp SFT for superstrings

♦more fundamental formulation?