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���

References:

T. Kawano, I.K., T. Takahashi,

arXiv:0804.1541(accepted for publication in NPB), arXiv0804.4414

seminar@nara-wu, 6/6 (2008)

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�� ���$%��&'��(

�� )*�+,-���.$%�/01(234

56 �7����&8��(

��9:;<=>�7����[Witten (1986)]

��

Q =∮

dz

2πi

(cT m + bc∂c +

3

2∂2c

)

S[Ψ] = − 1

g2

(1

2〈Ψ, QΨ〉 +

1

3〈Ψ, Ψ ∗ Ψ〉

)����

����

�-��BRST ����

Ψ[X(σ), b(σ), c(σ)] = 〈X(σ), b(σ), c(σ)|Ψ〉

?@AB��BPZ����

�������

�������

QΨ + Ψ∗Ψ =0

δΛΨ = QΛ + Ψ∗Λ − Λ∗Ψ

〈A, QB〉 = −(−1)|A|〈QA, B〉Q(A ∗ B) = (QA) ∗ B + (−1)|A|A ∗ (QB)

〈A, B〉 = (−1)|A||B|〈B, A〉 〈A, B ∗ C〉 = 〈A ∗ B, C〉

A ∗ (B ∗ C) = (A ∗ B) ∗ C : associative�CD,?@AB-.EF�

δΛS[Ψ] = 0

:GHBH�

:?@AB,IJKderivation�

6�

Q2 = 0 L526MNOP�

�� OP�QH7�RST��U�

|Ψ〉 = φ(x)c1|0〉 + Aμ(x)αμ−1c1|0〉 + iB(x)c0|0〉 + · · ·

Xμ(σ) = xμ + i√

α′/2∑n�=0

1

nαμ

n cos nσ , · · ·

V%W��

〈Ψ, QΨ〉 =∫

d26x

(φ(−α′� − 1)φ

− α′Aμ�Aμ + 2√

2α′B∂μAμ + 2B2 + · · ·)

@XYZ7�

�V%[\1]^_`ab�B(x)FμνF μν (Fμν ≡ ∂μAν − ∂νAμ) massivec7�

��9:;<=c>���dD25-branee���]-$%�/,-@XYZ2fghi0�L�.$%�/

,-i0cjP2faU(?)

Sen�kl�(1999)�.$%�/jP]-D25-brane2^mJK*K>�nop2c*U

��>�7����q*KSen�kl�rst

������

�� 1999u2002 Sen-Zwiebach, …,Gaiotto-Rastelli

Oη(ΨN)L0 -0.68461612 -0.94855344 -0.98640346 -0.99477278 -0.997779510 -0.999116112 -0.999790714 -1.000158016 -1.000367818 -1.00049

L0 -0.6846161OOηη((ΨΨNN

2 -0.95937664 -0.98782186 -0.99517718 -0.997930210 -0.999182512 -0.999822314 -1.000173716 -1.000375418 -1.0004937

−2π2g2S[ΨN]/V26 −2π2g2S[ΨN]/V26

(L,2L)vw� (L,3L)vw�

SiegelxAy5555555�+�level truncationvw]z{| potential�}~���L�

b0|ΨN〉 = 0potential����D25-brane tension

− S[Ψ]/V26

Ψλ=1 Ψ

− 1

2π2g2

Schnabl�: �� �!"#$�

potential����D25-brane tension

[Schnabl(2005),…]

��%�&BRST cohomology'()

[Ellwood-Schnabl(2006)]�

phantom*+,�

��%�&-�.,�

2005�11�Schnabl2 “SchnablxAy” �+� +�/,�QJ�U�

B0|Ψλ=1〉 = 0

Ψλ=1

S[Ψλ]/V26 =

{ 12π2g2 (λ = 1)

0 (|λ| < 1)

��/0��1�2�34��.�5"678�943:��

Ψλ =λ∂r

λe∂r − 1ψr|r=0 =

∞∑n=0

fn(λ)

n!∂n

r ψr|r=0

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

limN→∞

(ψN+1 −

N∑n=0

∂rψr|r=n

)(λ = 1)

−∞∑

n=0

λn+1∂rψr|r=n (λ �= 1)

ψr ≡ 2

πU†

r+2Ur+2

[− 1

π(B0 + B†

0)c(πr

4)c(−πr

4) +

1

2(c(−πr

4) + c(

πr

4))]|0〉

λ

−S[Ψλ]/(V26T25)

1.0 1.0

1.0

0.5

0

L0 -0.577922 -1.065184 -1.047986 -1.032878 -1.0232610 -1.01705

−S[Ψλ=1]/(V26T25)

“(L,3L)”vw�

level truncationvw]-“phantomW”-��c*�

[Schnabl(2005),Takahashi(2007)]�

Ψλ = −∑n≥0

λn+1(∂rψr|r=n)L

(−1 ≤ λ ≤ 1)

�� >�7���,��a|�q���xAyh��bJKonshell closed string state������2fa�[Zwiebach,…]�

V (i) = c(i)c(−i)Vm(i,−i)

〈I|V (i)

QΦV = 0, 〈ΦV , Ψ ∗ Λ〉 = 〈ΦV , Λ ∗ Ψ〉

matter primary, dim (1,1)�

on-shell� midpoint�

∴ OV (δΛΨ) = 0

;6pure gauge�6<=4.>?�� OV (e−ΛQeΛ) = 0

OV (Ψ) = 〈I|V (i)|Ψ〉 = 〈ΦV , Ψ〉

@A�BBB2BBBC=�

2�34D�=EF6BBBBB�����CG:������������������

ΦV =∑m,n

ζmnc(i)Vm(i)c(−i)Vn(−i)|I〉

=∑m,n

ζmnU†1U1c(i∞)Vm(i∞)c(−i∞)Vn(−i∞)|0〉

±i∞ ±iM

ΦV,M ≡∑m,n

ζmnU†1U1c(iM)Vm(iM)c(−iM)Vn(−iM)|0〉

i

i∞

arctan z = z

z

z

M → +∞

�CH Vm(y)Vn(z) ∼ vmn

(y − z)2+ finite (y → z)

∴ 〈ΦV , ψr〉 = limM→+∞

〈ΦV,M , ψr〉 =CV

2πi

OV (Ψλ) =∞∑

k=0

fk(λ)

k!∂k

r 〈ΦV , ψr〉|r=0 = f0(λ)〈ΦV , ψ0〉 =

{CV

2πi(λ = 1)

0 (λ �= 1)

��/0��1�2�3:Cphantom*BBB����IJK��ψN+1

〈ΦV,M , ψr〉 =CV

2πi

(sinh

4M

r + 1− 4M

πsin

π

r + 1

)(cosh

4M

r + 1− cos

π

r + 1

)(sinh

4M

r + 1

)−2

,

CV = mat〈0|0〉mat

∑m,n

ζmnvmn .

Φη =1

52α′iημν lim

θ→π2

c(eiθ)∂Xμ(eiθ)c(e−iθ)∂Xν(e−iθ)|I〉

=

(1

4− 2

13

∞∑n,m=1

mn cos(m − n)π

2α−m·α−n

)eEc0c1|0〉 ,

E =∞∑

n=1

(−1)n

(− 1

2nα−n·α−n + c−nb−n

)

L9.BRSTM�:

;6�� (Lmat2n − Lmat

−2n)|Φη〉 = (−1)n3n|Φη〉Q|Φη〉 = 0

�� NO�P��1�&.

level L�&D��

BBBBBBBBB���������Q�phantom*.JK=R3

Oη(Ψλ,L) = −∞∑

n=0

λn+1∂r〈Φη, ψr,L〉|r=n(−1 ≤ λ ≤ 1)

u2(r) = −r2 − 4

3r2, u4(r) =

r4 − 16

30r4, u6(r) = −16(r2 − 4)(r2 − 1)(r2 + 5)

945r6, . . .

ψr−2 =

⎡⎣ ∞∏

k=1,←eu2k(r)L−2k

⎤⎦[1

πsin

2π

r

(1 − r

2πsin

2π

r

) ∑p≥−1;p:odd

(2

rcot

π

r

)p

c−p|0〉

+r

2π2

(sin

2π

r

)2 ∑s≥2;s:even

(−1)s2+1

s2 − 1

(2

r

)s ∑p,q≥−1;p+q:odd

(−1)q

(2

rcot

π

r

)p+q

b−sc−pc−q|0〉]

ψN+1 = O(N−3) (N → ∞)

O

L=0

L=2L=4L=6L=8

L=10L=12L=14

L0 0.138372 0.149284 0.156866 0.157408 0.1588010 0.1587712 0.1592214 0.15916

Oη(Ψλ=1,L)

Oη(Ψλ) =

⎧⎨⎩

1

2π 0.159155 (λ = 1)

0 (λ �= 1)������

L0 0.1140442 0.1397904 0.1479316 0.1512258 0.15288710 0.15402912 0.154750

Oη(ΨN) L0 0.1140442 0.1416264 0.1483256 0.1513698 0.15297610 0.15408012 -

Oη(ΨN)

(L,3L)vw�(L,2L)vw�

1

2π� 0.159155 ,v*��u97�: Oη(ΨN) � Oη(Ψλ=1)

1�

3�

9�

26�

69�

171�

402�

7����

��on-shell-�ST6<UG:���M�V278=W�

��Schnabl�BB��&.BBBB���X�I�()RY�

B�5"ZQ[\Y"6]^_W�

B` ���78, cohomology�a!CbXc'd:�

��Siegel����\Y�BB&e78=fgh@�Y2

iW�

` +���hY,�

��jk"lm.,open-closed SFT?

Ellwood6Q:C�

Ψλ λ = 1

ΨN

Ψλ=1 ∼ ΨN

OV (Ψ) = AdiskΨ (V ) − Adisk

0 (V )

��xAyh���

��������

ex.) �V%�dilaton��-�

O

ih

Mi

h

OV (Ψ) = 〈γ(1c, 2)|Vc〉1c |Ψ〉2|ΦV 〉3 = 〈γ(1c, 2)|Vc〉1c |R(2, 3)〉

〈γ(1c, 2)|

|Vc〉 =−1

26α−1 ·α−1c1c1|0〉

〈γ(1c, 2)|Ψλ=1〉2Pb−0 =

1

2π〈BN| + 〈γ(1c, 2)|χ〉2Pb−

0

Ψλ=1 = ψ0 +∞∑

n=0

(ψn+1 − ψn − ∂rψr|r=n)

≡ ψ0 + χ

��]Schnabl+�phantomW����{�1������

D25-brane��`�����

Q(−Ψλ=1) + (−Ψλ=1) ∗ (−Ψλ=1) = 0

�.$%�/jP� g����+b¡c`�b�]¢a��

Q ≡ Q + adΨλ=1 �555� g�BRST£¤��Ψλ=1