ALCW Source WG Goal M. Kuriki(Hiroshima U./KEK) ALCW2015, 20-24 April 2015,Tsukuba/Tokyo, Japan.
In collaboration with Y. Baba(Tsukuba) and K. Murakami(KEK)
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Transcript of In collaboration with Y. Baba(Tsukuba) and K. Murakami(KEK)
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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)
理研シンポジウム「弦の場の理論 07」Oct. 7, 2007
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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 BRS invariant source term♦tension of the brane cannot be fixed
§1 Introduction
D-branes in string field theory
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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.
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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 YahikozawaHanada, Hayakawa, Kawai, Kuroki, Matsuo, Tada and N.I.
From one solution to the Virasoro constraint, one can generate another by the solitonic operator.
These coefficients are chosen so that
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These solitonic operators correspond to D-branes
state with D(-1)-brane ghost D-brane=
amplitudes withZZ-branes
Okuda,Takayanagi
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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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If we have ・ SFT with length variable ・ nonlinear equation like the Virasoro constraintfor 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-branesimitating 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
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§2 OSp Invariant String Field Theory
§3 Observables and Correlation Functions
§4 D-brane States
§5 Disk Amplitudes
§6 Conclusion and Discussion
Plan of the talk
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§2 OSp Invariant String Field Theory
light-cone gauge SFT
O(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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Green’s functions of BRS invariant observables
Green’s functions in 26D
∥
OSp invariant SFT 〜 stochastic formulation of string theory ?
S-matrix elements in 26D
S-matrix elements derived from the light-cone gauge SFT
∥ ?
▪Wick rotation
▪1 particle pole
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§3 Observables and Correlation Functions
observables
ignoring the multi-string contribution
if
we need the cohomology of
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(on-shell)
cohomology of
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Observables
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Free propagator
two point function
We would like to show that the lowest order contribution to this two point function coincides with the free propagatorof the particle corresponding to this state.
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light-cone quantization
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the Hamiltonian is BRS exact
a representative of the class
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N point functions
we would like to look for the singularity
light-cone diagram
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the singularity comes from
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Repeating this for we get
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Rem.
higher order corrections
can also be treated in the same way at least formally
・ two point function
・ N point function
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Wick rotation
・ LHS is an analytic function of
・Wick rotation + OSp(27,1|2) trans.
・
reproduces the light-cone gauge result
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§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
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BRS invariant regularization
states with one soliton
will be fixed by requiring
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creation
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shorthand notation
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Using these, we obtain
We need the idempotency equation.
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Idempotency equations
leading order in
corrections
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Derivation of the idempotency equations
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・ can also be evaluated from the Neumann function
・
: quadratic in can be evaluated from the Neumann function
・
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can be evaluated in the same way
we can fix the normalization in the limit
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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 the 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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§4 Disk Amplitudes
BRS invariant source term
Let us calculate amplitudes in the presence of such a source term.
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saddle point approximation
generate boundaries on the worldsheet
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correlation functions
= + +…
+ +…
Let us calculate the disk amplitude
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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
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disk two point function
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tachyon-tachyon
coincides with the disk amplitude up to normalization
・more general disk amplitudes can be calculated in the same way
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low-energy effective action
Comparing this with the action for D-branes
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§5 Conclusion and Discussion
♦similar construction for superstrings
♦other amplitudes
we need to construct OSp SFT for superstrings
♦D-brane BRS invariant source term
♦more fundamental formulation?