1. 2004/10/15seminar@Hokkaido Univ.1 Idempotency Equation and Boundary States in Closed String Field Theory Isao Kishimoto (Univ. of Tokyo → KEK) Collaboration with Y. Matsuo, E. Watanabe (Univ. of Tokyo) KMW1: Phys.Rev.D68 (2003) 126006[hep-th/0306189], KMW2: Prog.Theor.Phys. 111 (2004) 433[hep-th/0312122], KM1: Phys.Lett. B590 (2004) 303 [hep-th/0402107], KM2: hep-th/0409069.

2. 2004/10/15seminar@Hokkaido Univ.2 Introduction and motivation Sen’s conjecture : Witten’s open SFT ∃ tachyon vacuum Vacuum String Field Theory (VSFT) [(Gaiotto)-Rastelli-Sen-Zwiebach(2000/2001)] D25-brane : Pure ghost BRST operator

3. 2004/10/15seminar@Hokkaido Univ.3 VSFT: D-brane ～ classical solution of ～ Projector with respect to Witten’s ＊ product in the matter sector: Sliver, Butterfly,… are constructed explicitly. Essentially, they are the same as noncommutative solitons because Witten’s ＊ can be expressed as the Moyal product.

4. 2004/10/15seminar@Hokkaido Univ.4 Around the sliver solution, we can solve As solutions for, open string spectrum is included and D-brane tension is reproduced: However, there are more solutions which give different tension. [Hata-Kawano(2001)] [Okawa(2002)]

5. 2004/10/15seminar@Hokkaido Univ.5 On the other hand, D-brane ～ Boundary state ← closed string Closed SFT description is more natural (!?) HIKKO cubic closed SFT (Nonpolynomial closed SFT)

6. 2004/10/15seminar@Hokkaido Univ.6 Contents Introduction Star product in closed SFT Star product of boundary states [KMW1] Idempotency equation [KMW1,KMW2] Fluctuations [KMW1,KMW2] Overall factor [KM2] Cardy states and idempotents [KM1,2] T D,T D /Z 2 compactification [KM2] Comment on the Seiberg-Witten limit [KM2] Summary and discussion [KMW3(?)]

7. 2004/10/15seminar@Hokkaido Univ.7 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:

8. 2004/10/15seminar@Hokkaido Univ.8 α3α3 α1α1 α2α2 Overlapping condition for 3 closed strings Interaction point

9. 2004/10/15seminar@Hokkaido Univ.9 Explicit representation of the 3-string vertex: solution to overlapping condition [HIKKO] ( Gaussian !)

10. 2004/10/15seminar@Hokkaido Univ.10 : Neumann coefficients of light-cone type We can prove various relations. [Mandelstam, Green-Schwarz,…] In particular, Yoneya formulae are essential to computation of B ＊ B.

11. 2004/10/15seminar@Hokkaido Univ.11 Star product of boundary state The boundary state for Dp-brane with constant flux: (Neumann) (Dirichlet)

12. 2004/10/15seminar@Hokkaido Univ.12 Comment on the ghost sector The ghost sector of conventional boundary state: ↓ ⇒ Note 1: which follows from Note 2: φ ： “physical sector” i.e.,

13. 2004/10/15seminar@Hokkaido Univ.13 “idempotency equation”

14. 2004/10/15seminar@Hokkaido Univ.14 By algebraic calculation, we obtain the differential equation: where These are evaluated by identifying Neumann coefficients for zero modes in 4-string vertex:

15. 2004/10/15seminar@Hokkaido Univ.15

16. 2004/10/15seminar@Hokkaido Univ.16

17. 2004/10/15seminar@Hokkaido Univ.17 Idempotency equation The boundary state which corresponds to Dp-brane is a solution to this equation in the following sense.

18. 2004/10/15seminar@Hokkaido Univ.18 Boundary state as an “idempotent” :

19. 2004/10/15seminar@Hokkaido Univ.19 Fluctuations By straightforward computation in oscillator language, we found scalar and vector type “solutions”:

20. 2004/10/15seminar@Hokkaido Univ.20

21. 2004/10/15seminar@Hokkaido Univ.21

22. 2004/10/15seminar@Hokkaido Univ.22 Let us consider LPP formulation [LeClair-Peskin-Preitshopf(1989)], which refers to CFT correlation function to define 3-string vertex: where

23. 2004/10/15seminar@Hokkaido Univ.23 A sufficient condition for this solution : primary with weight 1

24. 2004/10/15seminar@Hokkaido Univ.24 By modifying the vector type fluctuation [Murakami-Nakatsu(2002)] :

25. 2004/10/15seminar@Hokkaido Univ.25

26. 2004/10/15seminar@Hokkaido Univ.26 Overall factor Let us reconsider B ＊ B. Note: regularization

27. 2004/10/15seminar@Hokkaido Univ.27 Conversely, it corresponds to a particular case of degeneration of a Riemann surface: Generally, this process is described by factorization: In the case of B ＊ B, it roughly implies

28. 2004/10/15seminar@Hokkaido Univ.28 More precisely, we should consider modulus in terms of regulator and ghost structure in computation of ＊ product: Mandelstam mapping: Modulus: c.f. [Asakawa-Kugo-Takahashi(1999)]

29. 2004/10/15seminar@Hokkaido Univ.29 Evaluation of the coefficient Using the idempotency equation, we get in the matter sector and in the ghost sector. From the previous figure b), we compute the numerator as

30. 2004/10/15seminar@Hokkaido Univ.30 The denominator is give by Combining matter and ghost contribution, the numerator is evaluated as: Namely, for c=26 and this is consistent with the correspondence of regularizations:

31. 2004/10/15seminar@Hokkaido Univ.31 Cardy states and idempotents On the flat ( R d ) background, we have ＊ product formula for Ishibashi states : Conjecture Cardy states ～ idempotents in closed SFT even on nontrivial backgrounds.

32. 2004/10/15seminar@Hokkaido Univ.32

33. 2004/10/15seminar@Hokkaido Univ.33 Orbifold （ M/Γ ）

34. 2004/10/15seminar@Hokkaido Univ.34 [cf. Billo et al.(2001)]

35. 2004/10/15seminar@Hokkaido Univ.35 Fusion ring of RCFT [ Verlinde(1988) ] [T.Kawai (1989)]

36. 2004/10/15seminar@Hokkaido Univ.36 Conversely, suppose the idempotency relation Then, the algebra of the Ishibashi states becomes where is generalized Cardy state s.t. [cf. Behrend et al.(1999) ] Can we check the above algebra in closed SFT on nontrivial background?

37. 2004/10/15seminar@Hokkaido Univ.37 T D,T D /Z 2 compactification Explicit formulation of closed SFT on T D,T D /Z 2 is known. [HIKKO(1987), Itoh-Kunitomo(1988)] We can compute ＊ product of Ishibashi states directly. 3-string vertex is modified:

38. 2004/10/15seminar@Hokkaido Univ.38 ＊ products of these states are not diagonal. → We consider following linear combinations:

39. 2004/10/15seminar@Hokkaido Univ.39 In the above formulae, twisted sector is not diagonalized. We should take linear combinations of untwisted and twisted sector in order to get idempotents which include twisted sector. (Here we have included flat R d sector and ghost sector.)

40. 2004/10/15seminar@Hokkaido Univ.40 Neumann coefficients in the twisted sector We have used the above relations to compute ＊ product. Note:

41. 2004/10/15seminar@Hokkaido Univ.41 We can conclude that

42. 2004/10/15seminar@Hokkaido Univ.42 Ratio of 1-loop amplitude : :degenerating limit

43. 2004/10/15seminar@Hokkaido Univ.43 （※） Neumann type idempotents are obtained from Dirichlet type by T-duality :

44. 2004/10/15seminar@Hokkaido Univ.44 Comment on the Seiberg-Witten limit KT operator which was introduced to represent noncommutativety in SFT on constant B-field background : [ Kawano-Takahashi ] More precisely, we find the identity: Dirichlet type Ishibashi stateNeumann type boundary state

45. 2004/10/15seminar@Hokkaido Univ.45 i.e., projector eq. with respect to the Strachan product (or one of the generalized star product: ＊ 2 ) which is commutative and non-associative. feature of the HIKKO closed SFT ＊ product In terms of coefficients function:

46. 2004/10/15seminar@Hokkaido Univ.46 Summary and discussion Cardy states satisfy idempotency equation in closed SFT (on R D,T D,T D /Z 2 ). Variation around idempotents gives open string spectrum. Idempotents ～ Cardy states more detailed and general correspondence? (Proof of necessary and sufficient conditions) Closed version of VSFT? (Veneziano amplitude,…) Supersymmetric extension? (HIKKO’s NSR vertex, Green-Schwarz LCSFT, Witten type, Berkovits type…)

47. 2004/10/15seminar@Hokkaido Univ.47 3-string vertex in Nonpolynomial closed SFT n-string vertices (n ≧ 4) in nonpolynomial closedSFT? 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 [Saadi-Zwiebach,Kugo-Kunitomo-Suehiro,Kugo-Suehiro,Kaku,… ]

48. 2004/10/15seminar@Hokkaido Univ.48 Green-Schwarz-Brink’s closed light-cone super SFT We will be able to compute straightforwardly as We have checked the above relation for “D-instanton” (up to zero mode dependence) by direct computation. where G comes from the prefactor of interaction vertex and the determinant of Neumann matrices. The boundary state |B 〉 is given in terms of Green-Schwarz formulation, which is constructed in [Green-Gutperle(1996)]: What is the explicit form and meaning of G?

49. 2004/10/15seminar@Hokkaido Univ.49 super SFT in terms of NSR formulation There are explicit representations of 3-string vertex for both HIKKO and Witten type in open super SFT [HIKKO, Gross-Jevicki, Suehiro, Samuel…] Boundary states are known : Define 3-string vertex for closed super SFT. Compute ＊ product of boundary states: Can we find a universal relation for boundary states ? Divergence of the coefficients would vanish and the idempotency equation might become regular.