Post on 21-Jul-2020
transcript
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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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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����
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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�
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��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