idempotency equation and boundary states in …2004/6/2 seminar@riken 33 summary and discussion...
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2004/6/2 seminar@RIKEN 1
Idempotency Equation and
Boundary States
in Closed String Field Theory
Isao Kishimoto(Univ. of Tokyo)
[KMW1] I.K., Y. Matsuo, E. Watanabe, PRD68 (2003) 126006
[KMW2] I.K., Y. Matsuo, E. Watanabe, PTP111 (2004) 433
[KM] I.K., Y. Matsuo, PLB590(2004)303
(+ I.K., Y. Matsuo, H. Isono, E. Watanabe, work in progress )
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2004/6/2 seminar@RIKEN 2
Introduction
• Sen’s conjecture:Witten’s open SFT
∃ tachyon vacuum
• Vacuum String Field Theory (VSFT)[Rastelli-Sen-Zwiebach(2000)]
D-brane~ Projector with respect to Witten’s * product.
(Sliver, Butterfly,… )
D25-brane
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2004/6/2 seminar@RIKEN 3
D-brane ~ Boundary state ← closed string
Closed SFT description is more natural (!?)
HIKKO cubic CSFT (Nonpolynomial CSFT)
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2004/6/2 seminar@RIKEN 4
Contents
• Introduction
• Star product in closed SFT
• Star product of boundary states [KMW1]
• Idempotency equation [KMW1,KMW2]
• Fluctuations [KMW1,KMW2]
• Cardy states and idempotents [KM]
• TD,TD/Z2 compactification
• Summary and discussion
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2004/6/2 seminar@RIKEN 5
Star product in closed SFT
* product is defined by 3-string vertex:
• HIKKO (Hata-Itoh-Kugo-Kunitomo-Ogawa) type
and ghost sector (to be compatible with BRST invariance)
with projection:
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2004/6/2 seminar@RIKEN 6
α3
α1α2
Overlapping condition for 3 closed strings
Interaction point
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• Explicit representation of the 3-string vertex:
solution to overlapping condition [HIKKO]
( Gaussian !)
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2004/6/2 seminar@RIKEN 8
: Neumann coefficients of light-cone type
We can prove various relations. [Mandelstam, Green-Schwarz,…]
[Yoneya(1987)]
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2004/6/2 seminar@RIKEN 9
Star product of boundary stateThe boundary state for Dp-brane with constant flux:
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2004/6/2 seminar@RIKEN 10
“idempotency equation”
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2004/6/2 seminar@RIKEN 11
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Idempotency equation
The boundary state which corresponds to Dp-brane is
a solution to this equation in the following sense.
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• Boundary state as an “idempotent” :
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Fluctuations
By straightforward computation in oscillator language, we found scalar
and vector type “solutions”:
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2004/6/2 seminar@RIKEN 15
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2004/6/2 seminar@RIKEN 16
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A sufficient condition for this solution : primary with weight 1
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2004/6/2 seminar@RIKEN 18
By modifying the vector type fluctuation [Murakami-Nakatsu(2002)] :
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2004/6/2 seminar@RIKEN 19
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2004/6/2 seminar@RIKEN 20
Cardy states and idempotents• On the flat ( Rd ) background, we have *product formula
for Ishibashi states :
Conjecture
Cardy states ~ idempotents in closed SFT
even on nontrivial backgrounds.
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2004/6/2 seminar@RIKEN 21
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2004/6/2 seminar@RIKEN 22
• Orbifold (M/Γ)
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[cf. Billo et al.(2001)]
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2004/6/2 seminar@RIKEN 24
• Fusion ring of RCFT
[Verlinde(1988)]
[T.Kawai (1989)]
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2004/6/2 seminar@RIKEN 25
TD,TD/Z2 compactification
Explicit formulation of closed SFT on TD,TD/Z2is known. [HIKKO(1987), Itoh-Kunitomo(1988)]
We can compute *product of Ishibashi states directly.
3-string vertex is modified:
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*products of these states are not diagonal.
→ We consider following linear combinations:
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2004/6/2 seminar@RIKEN 27
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2004/6/2 seminar@RIKEN 28
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2004/6/2 seminar@RIKEN 29
doubling
~ doubling of open string 1-loop
1
2
3
1
2
3
T/2
regularize
T
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2004/6/2 seminar@RIKEN 30
[Asakawa-Kugo-Takahashi(1999)]
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2004/6/2 seminar@RIKEN 31
cut
Boundary 1
Boundary 2
Ratio of 1-loop amplitude:
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(※) Neumann type idempotents are obtained from Dirichlet type by T-duality :
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2004/6/2 seminar@RIKEN 33
Summary and discussion
• Cardy states satisfy idempotency equation
in closed SFT (on RD,TD,TD/Z2 ).
• Variation around idempotents gives open string
spectrum.
• Idempotents ~ Cardy states
: detailed correspondence?
• Closed version of VSFT? (Veneziano amplitude,…)
• More nontrivial background? (other orbifolds,…)
• Supersymmetric extension? (HIKKO’s NSR vertex,…)
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2004/6/2 seminar@RIKEN 34
• 3-string vertex in Nonpolynomial CSFT
• n-string vertices (n≧4) in nonpolynomial CSFT?
We can also prove idempotency straightforwardly:
(Computation is simplified by closed sting version of MSFT. [Bars-Kishimoto-Matsuo] )
← closed string version of Witten’s * product