idempotency equation and boundary states in closed string ... · 2004/6/22 seminar@osaka univ. 1...
TRANSCRIPT
![Page 1: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/1.jpg)
2004/6/22 seminar@Osaka Univ. 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
![Page 2: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/2.jpg)
2004/6/22 seminar@Osaka Univ. 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
![Page 3: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/3.jpg)
2004/6/22 seminar@Osaka Univ. 3
D-brane ~ Boundary state ← closed string
Closed SFT description is more natural (!?)
HIKKO cubic CSFT (Nonpolynomial CSFT)
![Page 4: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/4.jpg)
2004/6/22 seminar@Osaka Univ. 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
![Page 5: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/5.jpg)
2004/6/22 seminar@Osaka Univ. 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:
![Page 6: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/6.jpg)
2004/6/22 seminar@Osaka Univ. 6
α3
α1α2
Overlapping condition for 3 closed strings
Interaction point
![Page 7: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/7.jpg)
2004/6/22 seminar@Osaka Univ. 7
• Explicit representation of the 3-string vertex:
solution to overlapping condition [HIKKO]
( Gaussian !)
![Page 8: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/8.jpg)
2004/6/22 seminar@Osaka Univ. 8
: Neumann coefficients of light-cone type
We can prove various relations. [Mandelstam, Green-Schwarz,…]
[Yoneya(1987)]
![Page 9: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/9.jpg)
2004/6/22 seminar@Osaka Univ. 9
Star product of boundary stateThe boundary state for Dp-brane with constant flux:
![Page 10: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/10.jpg)
2004/6/22 seminar@Osaka Univ. 10
“idempotency equation”
![Page 11: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/11.jpg)
2004/6/22 seminar@Osaka Univ. 11
![Page 12: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/12.jpg)
2004/6/22 seminar@Osaka Univ. 12
Idempotency equation
The boundary state which corresponds to Dp-brane is
a solution to this equation in the following sense.
![Page 13: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/13.jpg)
2004/6/22 seminar@Osaka Univ. 13
• Boundary state as an “idempotent” :
![Page 14: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/14.jpg)
2004/6/22 seminar@Osaka Univ. 14
• “Non-associative” product for coefficient functions
in “non-commutative” background.
[Kawano-Takahashi (1999)]
cf. [Murakami-Nakatsu(2002)]
![Page 15: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/15.jpg)
2004/6/22 seminar@Osaka Univ. 15
![Page 16: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/16.jpg)
2004/6/22 seminar@Osaka Univ. 16
In particular, one of the two α-parameter becomes zero,
this induced product becomes Strachan product :
Projector eq. with respect to the Strachan product
which is commutative and non-associative.
Feature of the HIKKO * product
![Page 17: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/17.jpg)
2004/6/22 seminar@Osaka Univ. 17
Fluctuations
By straightforward computation in oscillator language, we found scalar
and vector type “solutions”:
cf. [Hata-Kawano(2001)]
![Page 18: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/18.jpg)
2004/6/22 seminar@Osaka Univ. 18
![Page 19: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/19.jpg)
2004/6/22 seminar@Osaka Univ. 19
![Page 20: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/20.jpg)
2004/6/22 seminar@Osaka Univ. 20
A sufficient condition for this solution : primary with weight 1
cf.[Okawa(2002)]
![Page 21: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/21.jpg)
2004/6/22 seminar@Osaka Univ. 21
By modifying the vector type fluctuation [Murakami-Nakatsu(2002)] :
![Page 22: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/22.jpg)
2004/6/22 seminar@Osaka Univ. 22
![Page 23: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/23.jpg)
2004/6/22 seminar@Osaka Univ. 23
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.
![Page 24: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/24.jpg)
2004/6/22 seminar@Osaka Univ. 24
![Page 25: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/25.jpg)
2004/6/22 seminar@Osaka Univ. 25
• Orbifold (M/Γ)
![Page 26: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/26.jpg)
2004/6/22 seminar@Osaka Univ. 26
[cf. Billo et al.(2001)]
![Page 27: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/27.jpg)
2004/6/22 seminar@Osaka Univ. 27
• Fusion ring of RCFT
[Verlinde(1988)]
[T.Kawai (1989)]
![Page 28: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/28.jpg)
2004/6/22 seminar@Osaka Univ. 28
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: cf. [Maeno-Takano(1989)]
![Page 29: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/29.jpg)
2004/6/22 seminar@Osaka Univ. 29
*products of these states are not diagonal.
→ We consider following linear combinations:
![Page 30: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/30.jpg)
2004/6/22 seminar@Osaka Univ. 30
![Page 31: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/31.jpg)
2004/6/22 seminar@Osaka Univ. 31
![Page 32: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/32.jpg)
2004/6/22 seminar@Osaka Univ. 32
doubling
~ doubling of open string 1-loop
1
2
3
1
2
3
T/2
regularize
T
![Page 33: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/33.jpg)
2004/6/22 seminar@Osaka Univ. 33
[Asakawa-Kugo-Takahashi(1999)]
![Page 34: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/34.jpg)
2004/6/22 seminar@Osaka Univ. 34
cut
Boundary 2
Boundary 1
Ratio of 1-loop amplitude:
![Page 35: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/35.jpg)
2004/6/22 seminar@Osaka Univ. 35
(※) Neumann type idempotents are obtained from Dirichlet type by T-duality :
![Page 36: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/36.jpg)
2004/6/22 seminar@Osaka Univ. 36
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,…)
• Relation to the original HIKKO theory?
• More nontrivial background? (other orbifolds,…)
• Supersymmetric extension? (HIKKO’s NSR vertex,…)
![Page 37: Idempotency Equation and Boundary States in Closed String ... · 2004/6/22 seminar@Osaka Univ. 1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto](https://reader034.vdocument.in/reader034/viewer/2022042313/5edc8d26ad6a402d666742f4/html5/thumbnails/37.jpg)
2004/6/22 seminar@Osaka Univ. 37
• 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