superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution...
TRANSCRIPT
Superstring theory and integration over
the moduli space
Kantaro Ohmori
Hongo,The University of Tokyo
Oct, 23rd, 2013
arXiv:1303.7299 [KO,Yuji Tachikawa]
Todai/Riken joint workshop on Super Yang-Mills,
solvable systems and related subjects
Section 1
introduction
Simplified history of the superstring theory
⇠70’s:Bosonic string theory! supersymmetrize
’84⇠:1st superstring revolution
’86: ”Conformal Invariance, Supersymmetry, and
String Theory” [D.Friedan, E.Martinec, S.Shenker]
’94⇠:2nd superstring revolution
) nonperturbative perspective
’12: “Superstring Perturbation Theory Revisited”
[E.Witten]:
Superstring perturbation theory and supergeometry
Simplified history of the superstring theory
⇠70’s:Bosonic string theory! supersymmetrize
’84⇠:1st superstring revolution
’86: ”Conformal Invariance, Supersymmetry, and
String Theory” [D.Friedan, E.Martinec, S.Shenker]
’94⇠:2nd superstring revolution
) nonperturbative perspective
’12: “Superstring Perturbation Theory Revisited”
[E.Witten]:
Superstring perturbation theory and supergeometry
Simplified history of the superstring theory
⇠70’s:Bosonic string theory! supersymmetrize
’84⇠:1st superstring revolution
’86: ”Conformal Invariance, Supersymmetry, and
String Theory” [D.Friedan, E.Martinec, S.Shenker]
’94⇠:2nd superstring revolution
) nonperturbative perspective
’12: “Superstring Perturbation Theory Revisited”
[E.Witten]:
Superstring perturbation theory and supergeometry
Simplified history of the superstring theory
⇠70’s:Bosonic string theory! supersymmetrize
’84⇠:1st superstring revolution
’86: ”Conformal Invariance, Supersymmetry, and
String Theory” [D.Friedan, E.Martinec, S.Shenker]
’94⇠:2nd superstring revolution
) nonperturbative perspective
’12: “Superstring Perturbation Theory Revisited”
[E.Witten]:
Superstring perturbation theory and supergeometry
Simplified history of the superstring theory
⇠70’s:Bosonic string theory! supersymmetrize
’84⇠:1st superstring revolution
’86: ”Conformal Invariance, Supersymmetry, and
String Theory” [D.Friedan, E.Martinec, S.Shenker]
’94⇠:2nd superstring revolution
) nonperturbative perspective
’12: “Superstring Perturbation Theory Revisited”
[E.Witten]:
Superstring perturbation theory and supergeometry
Simplified history of the superstring theory
⇠70’s:Bosonic string theory! supersymmetrize
’84⇠:1st superstring revolution
’86: ”Conformal Invariance, Supersymmetry, and
String Theory” [D.Friedan, E.Martinec, S.Shenker]
’94⇠:2nd superstring revolution
) nonperturbative perspective
’12: “Superstring Perturbation Theory Revisited”
[E.Witten]:
Superstring perturbation theory and supergeometry
Bosonic string theory
Action:S =
R⌃
d2zGij
@X i
¯@X j
) Riemann Surface = 2d surface + metric
Conformal symmetry) bc ghost
) A =
RMbos,g
dmdm̄F (m, m̄)
Bosonic string theory
Action:S =
R⌃
d2zGij
@X i
¯@X j
) Riemann Surface = 2d surface + metric
Conformal symmetry) bc ghost
) A =
RMbos,g
dmdm̄F (m, m̄)
Bosonic string theory
Action:S =
R⌃
d2zGij
@X i
¯@X j
) Riemann Surface = 2d surface + metric
Conformal symmetry) bc ghost
) A =
RMbos,g
dmdm̄F (m, m̄)
Bosonic string theory
Action:S =
R⌃
d2zGij
@X i
¯@X j
) Riemann Surface = 2d surface + metric
Conformal symmetry) bc ghost
) A =
RMbos,g
dmdm̄F (m, m̄)
Bosonic string theory
Action:S =
R⌃
d2zGij
@X i
¯@X j
) Riemann Surface = 2d surface + metric
Conformal symmetry) bc ghost
) A =
RMbos,g
dmdm̄F (m, m̄)
Superstring theory
Action:S =
R⌃
d2zd2✓ Gij
D✓X iD¯✓X
j
) Super Riemann Surface
= 2d surface + supermetric
= 2d surface + metric + gravitino
Superconformal symmetry
) BC (bc��) superghost) A =
RMsuper,g
dmd⌘dm̄d⌘̄F (m, m̄, ⌘, ⌘̄)
) A =
RMspin,g
dmdm̄F 0(m, m̄)
Superstring theory
Action:S =
R⌃
d2zd2✓ Gij
D✓X iD¯✓X
j
) Super Riemann Surface
= 2d surface + supermetric
= 2d surface + metric + gravitino
Superconformal symmetry
) BC (bc��) superghost) A =
RMsuper,g
dmd⌘dm̄d⌘̄F (m, m̄, ⌘, ⌘̄)
) A =
RMspin,g
dmdm̄F 0(m, m̄)
Superstring theory
Action:S =
R⌃
d2zd2✓ Gij
D✓X iD¯✓X
j
) Super Riemann Surface
= 2d surface + supermetric
= 2d surface + metric + gravitinoθ
θ θ
θθ
θ θθθ
Superconformal symmetry
) BC (bc��) superghost) A =
RMsuper,g
dmd⌘dm̄d⌘̄F (m, m̄, ⌘, ⌘̄)
) A =
RMspin,g
dmdm̄F 0(m, m̄)
Superstring theory
Action:S =
R⌃
d2zd2✓ Gij
D✓X iD¯✓X
j
) Super Riemann Surface
= 2d surface + supermetric
= 2d surface + metric + gravitinoθ
θ θ
θθ
θ θθθ
Superconformal symmetry
) BC (bc��) superghost
) A =
RMsuper,g
dmd⌘dm̄d⌘̄F (m, m̄, ⌘, ⌘̄)
) A =
RMspin,g
dmdm̄F 0(m, m̄)
Superstring theory
Action:S =
R⌃
d2zd2✓ Gij
D✓X iD¯✓X
j
) Super Riemann Surface
= 2d surface + supermetric
= 2d surface + metric + gravitinoθ
θ θ
θθ
θ θθθ
Superconformal symmetry
) BC (bc��) superghost) A =
RMsuper,g
dmd⌘dm̄d⌘̄F (m, m̄, ⌘, ⌘̄)
) A =
RMspin,g
dmdm̄F 0(m, m̄)
Superstring theory
Action:S =
R⌃
d2zd2✓ Gij
D✓X iD¯✓X
j
) Super Riemann Surface
= 2d surface + supermetric
= 2d surface + metric + gravitinoθ
θ θ
θθ
θ θθθ
Superconformal symmetry
) BC (bc��) superghost) A =
RMsuper,g
dmd⌘dm̄d⌘̄F (m, m̄, ⌘, ⌘̄)?) A =
RMspin,g
dmdm̄F 0(m, m̄)
The bosonic and super moduli space
Msuper 6=Mspin⇥ odd direction
Msuper �Mspin
Global integration on Msuper
6) Global integration on Mspin
(preserving holomorphic factorization)
The bosonic and super moduli space
Msuper 6=Mspin⇥ odd direction
Msuper �Mspin
Global integration on Msuper
6) Global integration on Mspin
(preserving holomorphic factorization)
The work of [KO,Tachikawa]
Global integration on Msuper
6) Global integration on Mspin
We found special cases where
Global integration on Msuper
) Global integration on Mspin
The work of [KO,Tachikawa]
Global integration on Msuper
6) Global integration on Mspin
We found special cases where
Global integration on Msuper
) Global integration on Mspin
The work of [KO,Tachikawa]
Global integration on Msuper
6) Global integration on Mspin
We found special cases where
Global integration on Msuper
) Global integration on Mspin
The work of [KO,Tachikawa] (2)
2 concrete examples
N = 0 ⇢ N = 1 embedding [N. Berkovits,C. Vafa
’94]
Topological amplitudes
[I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor ’94]
[M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa ’94]
The work of [KO,Tachikawa] (2)
2 concrete examples
N = 0 ⇢ N = 1 embedding [N. Berkovits,C. Vafa
’94]
Topological amplitudes
[I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor ’94]
[M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa ’94]
The work of [KO,Tachikawa] (2)
2 concrete examples
N = 0 ⇢ N = 1 embedding [N. Berkovits,C. Vafa
’94]
Topological amplitudes
[I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor ’94]
[M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa ’94]
The work of [KO,Tachikawa] (2)
2 concrete examples
N = 0 ⇢ N = 1 embedding [N. Berkovits,C. Vafa
’94]
Topological amplitudes
[I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor ’94]
[M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa ’94]
1 introduction
2 Supergeometry and Superstring
3 Reduction to integration over bosonic moduli
Supermanifold
supermanifold = patched superspaces
M : supermanifold of dimension p|qM 3 (x
1
, · · · , xp
|✓1
, · · · , ✓q
)
✓i
✓j
= �✓j
✓i
x 0i
= fi
(x|✓)✓0µ = µ(x|✓)Mred = {(x
i
, · · · , xp
)} ,�! M
Supermanifold
supermanifold = patched superspaces
M : supermanifold of dimension p|qM 3 (x
1
, · · · , xp
|✓1
, · · · , ✓q
)
✓i
✓j
= �✓j
✓i
x 0i
= fi
(x|✓)✓0µ = µ(x|✓)Mred = {(x
i
, · · · , xp
)} ,�! M
Supermanifold
supermanifold = patched superspaces
M : supermanifold of dimension p|qM 3 (x
1
, · · · , xp
|✓1
, · · · , ✓q
)
✓i
✓j
= �✓j
✓i
x 0i
= fi
(x|✓)✓0µ = µ(x|✓)Mred = {(x
i
, · · · , xp
)} ,�! M
Supermanifold
supermanifold = patched superspaces
M : supermanifold of dimension p|qM 3 (x
1
, · · · , xp
|✓1
, · · · , ✓q
)
✓i
✓j
= �✓j
✓i
x 0i
= fi
(x|✓)✓0µ = µ(x|✓)
Mred = {(xi
, · · · , xp
)} ,�! M
Uα
Supermanifold
supermanifold = patched superspaces
M : supermanifold of dimension p|qM 3 (x
1
, · · · , xp
|✓1
, · · · , ✓q
)
✓i
✓j
= �✓j
✓i
x 0i
= fi
(x|✓)✓0µ = µ(x|✓)Mred = {(x
i
, · · · , xp
)} ,�! M
Uα
Supermoduli space Msuper
Space of deformations of SRS (Super Riemann
Surface)
even deformation: µz
z̃
2 H1
(⌃red,T⌃red)
odd deformation: �✓z̃
2 H1
(⌃red,T⌃
1/2red )
{��}: basis of gravitino b.g) � = ⌘���
dim Msuper = �
e
|�o
= 3g � 3|2g � 2 (g � 2)
Msuper,red 'Mspin
Supermoduli space Msuper
Space of deformations of SRS (Super Riemann
Surface)
even deformation: µz
z̃
2 H1
(⌃red,T⌃red)
odd deformation: �✓z̃
2 H1
(⌃red,T⌃
1/2red )
{��}: basis of gravitino b.g) � = ⌘���
dim Msuper = �
e
|�o
= 3g � 3|2g � 2 (g � 2)
Msuper,red 'Mspin
Supermoduli space Msuper
Space of deformations of SRS (Super Riemann
Surface)
even deformation: µz
z̃
2 H1
(⌃red,T⌃red)
odd deformation: �✓z̃
2 H1
(⌃red,T⌃
1/2red )
{��}: basis of gravitino b.g) � = ⌘���
dim Msuper = �
e
|�o
= 3g � 3|2g � 2 (g � 2)
Msuper,red 'Mspin
Supermoduli space Msuper
Space of deformations of SRS (Super Riemann
Surface)
even deformation: µz
z̃
2 H1
(⌃red,T⌃red)
odd deformation: �✓z̃
2 H1
(⌃red,T⌃
1/2red )
{��}: basis of gravitino b.g) � = ⌘���
dim Msuper = �
e
|�o
= 3g � 3|2g � 2 (g � 2)
Msuper,red 'Mspin
Supermoduli space Msuper
Space of deformations of SRS (Super Riemann
Surface)
even deformation: µz
z̃
2 H1
(⌃red,T⌃red)
odd deformation: �✓z̃
2 H1
(⌃red,T⌃
1/2red )
{��}: basis of gravitino b.g) � = ⌘���
dim Msuper = �
e
|�o
= 3g � 3|2g � 2 (g � 2)
Msuper,red 'Mspin
Supermoduli space Msuper
Space of deformations of SRS (Super Riemann
Surface)
even deformation: µz
z̃
2 H1
(⌃red,T⌃red)
odd deformation: �✓z̃
2 H1
(⌃red,T⌃
1/2red )
{��}: basis of gravitino b.g) � = ⌘���
dim Msuper = �
e
|�o
= 3g � 3|2g � 2 (g � 2)
Msuper,red 'Mspin
Supermoduli space Msuper
Space of deformations of SRS (Super Riemann
Surface)
even deformation: µz
z̃
2 H1
(⌃red,T⌃red)
odd deformation: �✓z̃
2 H1
(⌃red,T⌃
1/2red )
{��}: basis of gravitino b.g) � = ⌘���
dim Msuper = �
e
|�o
= 3g � 3|2g � 2 (g � 2)
Msuper,red 'Mspin
Supermoduli space of SRS with punctures
nNS NS punctures and nR R punctures
dim Msuper = �
e
|�o
= 3g�3+nNS+nR|2g�2+nNS+nR/2 (g � 2)
Supermoduli space of SRS with punctures
nNS NS punctures and nR R punctures
dim Msuper = �
e
|�o
= 3g�3+nNS+nR|2g�2+nNS+nR/2 (g � 2)
Supermoduli space of SRS with punctures
nNS NS punctures and nR R punctures
dim Msuper = �
e
|�o
= 3g�3+nNS+nR|2g�2+nNS+nR/2 (g � 2)
Coupling between gravitino and SCFT
S� =
1
2⇡
Rd2z�T
F
Se� =
1
2⇡
Rd2z e� eT
F
S�e� =
Rd2z�e�A
A / µ e µ (Flat background)
c.f. Dµ�†Dµ� 3 AµAµ�†�
Coupling between gravitino and SCFT
S� =
1
2⇡
Rd2z�T
F
Se� =
1
2⇡
Rd2z e� eT
F
S�e� =
Rd2z�e�A
A / µ e µ (Flat background)
c.f. Dµ�†Dµ� 3 AµAµ�†�
Coupling between gravitino and SCFT
S� =
1
2⇡
Rd2z�T
F
Se� =
1
2⇡
Rd2z e� eT
F
S�e� =
Rd2z�e�A
A / µ e µ (Flat background)
c.f. Dµ�†Dµ� 3 AµAµ�†�
Coupling between gravitino and SCFT
S� =
1
2⇡
Rd2z�T
F
Se� =
1
2⇡
Rd2z e� eT
F
S�e� =
Rd2z�e�A
A / µ e µ (Flat background)
c.f. Dµ�†Dµ� 3 AµAµ�†�
Coupling between gravitino and SCFT
S� =
1
2⇡
Rd2z�T
F
Se� =
1
2⇡
Rd2z e� eT
F
S�e� =
Rd2z�e�A
A / µ e µ (Flat background)
c.f. Dµ�†Dµ� 3 AµAµ�†�
Scattering amplitudes of superstring
Vi
: Vertex op. pic. number �1 (NS) or �1/2 (R)
AV,g =
RMsuper
dmd⌘dm̄d⌘̄ FV,g(m, ⌘, m̄, ⌘̄)
FV,g(m, ⌘, m̄, ⌘̄) =DQi
Vi
Q�
e
a=1
(
R⌃red
d2zbµa
)
Q�
e
b=1
(
R⌃red
d2z ebeµb
)
Q�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�S� � Se� � S�e�)i
m
S� =
P�
o
�=1
⌘�2⇡
R⌃red
d2z��TF
We get result of FMS when �� = �(z � p�)and int. out. ⌘’s.
Scattering amplitudes of superstring
Vi
: Vertex op. pic. number �1 (NS) or �1/2 (R)
AV,g =
RMsuper
dmd⌘dm̄d⌘̄ FV,g(m, ⌘, m̄, ⌘̄)
FV,g(m, ⌘, m̄, ⌘̄) =DQi
Vi
Q�
e
a=1
(
R⌃red
d2zbµa
)
Q�
e
b=1
(
R⌃red
d2z ebeµb
)
Q�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�S� � Se� � S�e�)i
m
S� =
P�
o
�=1
⌘�2⇡
R⌃red
d2z��TF
We get result of FMS when �� = �(z � p�)and int. out. ⌘’s.
Scattering amplitudes of superstring
Vi
: Vertex op. pic. number �1 (NS) or �1/2 (R)
AV,g =
RMsuper
dmd⌘dm̄d⌘̄ FV,g(m, ⌘, m̄, ⌘̄)
FV,g(m, ⌘, m̄, ⌘̄) =DQi
Vi
Q�
e
a=1
(
R⌃red
d2zbµa
)
Q�
e
b=1
(
R⌃red
d2z ebeµb
)
Q�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�S� � Se� � S�e�)i
m
S� =
P�
o
�=1
⌘�2⇡
R⌃red
d2z��TF
We get result of FMS when �� = �(z � p�)and int. out. ⌘’s.
Scattering amplitudes of superstring
Vi
: Vertex op. pic. number �1 (NS) or �1/2 (R)
AV,g =
RMsuper
dmd⌘dm̄d⌘̄ FV,g(m, ⌘, m̄, ⌘̄)
FV,g(m, ⌘, m̄, ⌘̄) =DQi
Vi
Q�
e
a=1
(
R⌃red
d2zbµa
)
Q�
e
b=1
(
R⌃red
d2z ebeµb
)
Q�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�S� � Se� � S�e�)i
m
S� =
P�
o
�=1
⌘�2⇡
R⌃red
d2z��TF
We get result of FMS when �� = �(z � p�)and int. out. ⌘’s.
Scattering amplitudes of superstring
Vi
: Vertex op. pic. number �1 (NS) or �1/2 (R)
AV,g =
RMsuper
dmd⌘dm̄d⌘̄ FV,g(m, ⌘, m̄, ⌘̄)
FV,g(m, ⌘, m̄, ⌘̄) =DQi
Vi
Q�
e
a=1
(
R⌃red
d2zbµa
)
Q�
e
b=1
(
R⌃red
d2z ebeµb
)
Q�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�S� � Se� � S�e�)i
m
S� =
P�
o
�=1
⌘�2⇡
R⌃red
d2z��TF
We get result of FMS when �� = �(z � p�)and int. out. ⌘’s.
Scattering amplitudes of superstring
Vi
: Vertex op. pic. number �1 (NS) or �1/2 (R)
AV,g =
RMsuper
dmd⌘dm̄d⌘̄ FV,g(m, ⌘, m̄, ⌘̄)
FV,g(m, ⌘, m̄, ⌘̄) =DQi
Vi
Q�
e
a=1
(
R⌃red
d2zbµa
)
Q�
e
b=1
(
R⌃red
d2z ebeµb
)
Q�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�S� � Se� � S�e�)i
m
S� =
P�
o
�=1
⌘�2⇡
R⌃red
d2z��TF
We get result of FMS when �� = �(z � p�)and int. out. ⌘’s.
Integration over a supermanifold
E = {(m|⌘1
, ⌘2
)}/ ⇠m ⇠ m + ⌧ + ⌘
1
⌘2
⇠ m + 1
RE
(a + b⌘1
⌘2
)dmdmd⌘1
d⌘2
= b=⌧ +
p�1a/2
⌘1
⌘2
mixed with m
Integration over a supermanifold
E = {(m|⌘1
, ⌘2
)}/ ⇠m ⇠ m + ⌧ + ⌘
1
⌘2
⇠ m + 1
RE
(a + b⌘1
⌘2
)dmdmd⌘1
d⌘2
= b=⌧ +
p�1a/2
⌘1
⌘2
mixed with m
Integration over a supermanifold
E = {(m|⌘1
, ⌘2
)}/ ⇠m ⇠ m + ⌧ + ⌘
1
⌘2
⇠ m + 1
RE
(a + b⌘1
⌘2
)dmdmd⌘1
d⌘2
= b=⌧ +
p�1a/2
⌘1
⌘2
mixed with m
Integration over a supermanifold
E = {(m|⌘1
, ⌘2
)}/ ⇠m ⇠ m + ⌧ + ⌘
1
⌘2
⇠ m + 1
RE
(a + b⌘1
⌘2
)dmdmd⌘1
d⌘2
= b=⌧ +
p�1a/2
⌘1
⌘2
mixed with m
Projectedness of supermanifolds
M :projected,an atlas in which x 0 = f (x) for all gluing
) naive integrations with ⌘’s are allowed
, p : M ! Mred existsRM
! =
RMred
p⇤(!)
All supermanifolds are projected in smooth sense.
Complex supermanifolds are not projected in
holomorphic sense in general.
Projectedness of supermanifolds
M :projected,an atlas in which x 0 = f (x) for all gluing
) naive integrations with ⌘’s are allowed
, p : M ! Mred existsRM
! =
RMred
p⇤(!)
All supermanifolds are projected in smooth sense.
Complex supermanifolds are not projected in
holomorphic sense in general.
Projectedness of supermanifolds
M :projected,an atlas in which x 0 = f (x) for all gluing
) naive integrations with ⌘’s are allowed
, p : M ! Mred exists
RM
! =
RMred
p⇤(!)
All supermanifolds are projected in smooth sense.
Complex supermanifolds are not projected in
holomorphic sense in general.
Projectedness of supermanifolds
M :projected,an atlas in which x 0 = f (x) for all gluing
) naive integrations with ⌘’s are allowed
, p : M ! Mred existsRM
! =
RMred
p⇤(!)
All supermanifolds are projected in smooth sense.
Complex supermanifolds are not projected in
holomorphic sense in general.
Projectedness of supermanifolds
M :projected,an atlas in which x 0 = f (x) for all gluing
) naive integrations with ⌘’s are allowed
, p : M ! Mred existsRM
! =
RMred
p⇤(!)
All supermanifolds are projected in smooth sense.
Complex supermanifolds are not projected in
holomorphic sense in general.
Projectedness of supermanifolds
M :projected,an atlas in which x 0 = f (x) for all gluing
) naive integrations with ⌘’s are allowed
, p : M ! Mred existsRM
! =
RMred
p⇤(!)
All supermanifolds are projected in smooth sense.
Complex supermanifolds are not projected in
holomorphic sense in general.
Supermoduli Space is not projected[R.Donagi, E.Witten, arXiv:1304.7798]
Msuper,red 'Mspin i : Mspin ,�!Msuper
p : Msuper !Mspin does not exists.
(g � 5, non-holomorphic one exists)
Global integration on Msuper
6) Global integration on Mspin
(preserving holomorphic factorization)
Supermoduli Space is not projected[R.Donagi, E.Witten, arXiv:1304.7798]
Msuper,red 'Mspin i : Mspin ,�!Msuper
p : Msuper !Mspin does not exists.
(g � 5, non-holomorphic one exists)
Global integration on Msuper
6) Global integration on Mspin
(preserving holomorphic factorization)
Supermoduli Space is not projected[R.Donagi, E.Witten, arXiv:1304.7798]
Msuper,red 'Mspin i : Mspin ,�!Msuper
p : Msuper !Mspin does not exists.
(g � 5, non-holomorphic one exists)
Global integration on Msuper
6) Global integration on Mspin
(preserving holomorphic factorization)
Supermoduli Space is not projected[R.Donagi, E.Witten, arXiv:1304.7798]
Msuper,red 'Mspin i : Mspin ,�!Msuper
p : Msuper !Mspin does not exists.
(g � 5, non-holomorphic one exists)
Global integration on Msuper
6) Global integration on Mspin
(preserving holomorphic factorization)
Section 3
Reduction to integration over bosonic
moduli
Reduction condition
Global integration on Msuper
) Global integration on Mspin in special cases
F (m, m̄, ⌘, ⌘̄) = G(m, m̄)
Qall ⌘i
⌘̄i
(saturations of ⌘’s)
mixing of m and ⌘ (m0 = m + ⌘1
⌘2
) does not
changes F
F does not depend on the locations of picture
changing operators
Reduction condition
Global integration on Msuper
) Global integration on Mspin in special cases
F (m, m̄, ⌘, ⌘̄) = G(m, m̄)
Qall ⌘i
⌘̄i
(saturations of ⌘’s)
mixing of m and ⌘ (m0 = m + ⌘1
⌘2
) does not
changes F
F does not depend on the locations of picture
changing operators
Reduction condition
Global integration on Msuper
) Global integration on Mspin in special cases
F (m, m̄, ⌘, ⌘̄) = G(m, m̄)
Qall ⌘i
⌘̄i
(saturations of ⌘’s)
mixing of m and ⌘ (m0 = m + ⌘1
⌘2
) does not
changes F
F does not depend on the locations of picture
changing operators
Reduction condition
Global integration on Msuper
) Global integration on Mspin in special cases
F (m, m̄, ⌘, ⌘̄) = G(m, m̄)
Qall ⌘i
⌘̄i
(saturations of ⌘’s)
mixing of m and ⌘ (m0 = m + ⌘1
⌘2
) does not
changes F
F does not depend on the locations of picture
changing operators
Topological amplitudes in string theory[I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor ’94]
[M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa ’94]
Type II string on CY (N = (2, 2)SCFT) ⇥R1,3
) 4d N = 2 supergravity model
F-term of 4d e↵ective field theory is related to
topological string.
Topological amplitudes in string theory[I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor ’94]
[M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa ’94]
Type II string on CY (N = (2, 2)SCFT) ⇥R1,3
) 4d N = 2 supergravity model
F-term of 4d e↵ective field theory is related to
topological string.
Topological amplitudes in string theory[I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor ’94]
[M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa ’94]
Type II string on CY (N = (2, 2)SCFT) ⇥R1,3
) 4d N = 2 supergravity model
F-term of 4d e↵ective field theory is related to
topological string.
Reduction to topological string
Type II string on CY(6d) ⇥R1,3
) 4d N = 2 supergravity model
Ag
: zero momenta limit of g loop amplitude of
2g � 2 graviphotons (RR) and 2 gravitons (NSNS)
Ag
=
RMsuper
· · ·A
g
= (g !)2Fg
=
RMbos
· · ·
Reduction to topological string
Type II string on CY(6d) ⇥R1,3
) 4d N = 2 supergravity model
Ag
: zero momenta limit of g loop amplitude of
2g � 2 graviphotons (RR) and 2 gravitons (NSNS)
Ag
=
RMsuper
· · ·A
g
= (g !)2Fg
=
RMbos
· · ·
Reduction to topological string
Type II string on CY(6d) ⇥R1,3
) 4d N = 2 supergravity model
Ag
: zero momenta limit of g loop amplitude of
2g � 2 graviphotons (RR) and 2 gravitons (NSNS)
Ag
=
RMsuper
· · ·
Ag
= (g !)2Fg
=
RMbos
· · ·
Reduction to topological string
Type II string on CY(6d) ⇥R1,3
) 4d N = 2 supergravity model
Ag
: zero momenta limit of g loop amplitude of
2g � 2 graviphotons (RR) and 2 gravitons (NSNS)
Ag
=
RMsuper
· · ·A
g
= (g !)2Fg
=
RMbos
· · ·
Vertex operators
Graviphoton: VT
= ⌃⇥ spacetime⇥ (ghost)
⌃ = exp(ip3
2
(H(z)⌥ eH(z))) H : U(1)
R
boson
⌃ : U(1)
R
charge (3/2,⌥ 3/2)
Graviton: VR
=
R⌃red
spacetime,ghost
Vertex operators
Graviphoton: VT
= ⌃⇥ spacetime⇥ (ghost)
⌃ = exp(ip3
2
(H(z)⌥ eH(z))) H : U(1)
R
boson
⌃ : U(1)
R
charge (3/2,⌥ 3/2)
Graviton: VR
=
R⌃red
spacetime,ghost
Vertex operators
Graviphoton: VT
= ⌃⇥ spacetime⇥ (ghost)
⌃ = exp(ip3
2
(H(z)⌥ eH(z))) H : U(1)
R
boson
⌃ : U(1)
R
charge (3/2,⌥ 3/2)
Graviton: VR
=
R⌃red
spacetime,ghost
Saturation of ⌘’s
Ag
=
R DQ2g�2i=1
⌃(xi
) U(1)
R
: (3g � 3,⌥3g � 3)
⇥(spacetime)⇥ (bc)⇥Q
�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�
R⌃red
(TF
�+
eTF
e�+ Ae��))E
m
nonzero contribution
) 3g � 3 and 3g � 3eThere are only 3g � 3 ⌘ and 3g � 3
e⌘The correlation function does not depend of the
choice of ��
Saturation of ⌘’s
Ag
=
R DQ2g�2i=1
⌃(xi
) U(1)
R
: (3g � 3,⌥3g � 3)
⇥(spacetime)⇥ (bc)⇥Q
�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�
R⌃red
(TF
�+
eTF
e�+ Ae��))E
m
nonzero contribution
) 3g � 3 and 3g � 3eThere are only 3g � 3 ⌘ and 3g � 3
e⌘The correlation function does not depend of the
choice of ��
Saturation of ⌘’s
Ag
=
R DQ2g�2i=1
⌃(xi
) U(1)
R
: (3g � 3,⌥3g � 3)
⇥(spacetime)⇥ (bc)⇥Q
�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�
R⌃red
(TF
�+
eTF
e�+ Ae��))E
m
nonzero contribution
) 3g � 3� and 3g � 3
e�
There are only 3g � 3 ⌘ and 3g � 3
e⌘The correlation function does not depend of the
choice of ��
Saturation of ⌘’s
Ag
=
R DQ2g�2i=1
⌃(xi
) U(1)
R
: (3g � 3,⌥3g � 3)
⇥(spacetime)⇥ (bc)⇥Q
�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�
R⌃red
(TF
�+
eTF
e�+ Ae��))E
m
nonzero contribution
) 3g � 3⌘ and 3g � 3
e⌘There are only 3g � 3 ⌘ and 3g � 3
e⌘
The correlation function does not depend of the
choice of ��
Saturation of ⌘’s
Ag
=
R DQ2g�2i=1
⌃(xi
) U(1)
R
: (3g � 3,⌥3g � 3)
⇥(spacetime)⇥ (bc)⇥Q
�
o
�=1
�(R⌃red
d2z���)Qe
�
o
⌧=1
�(R⌃red
d2z e� e�⌧ )exp(�
R⌃red
(TF
�+
eTF
e�+ Ae��))E
m
nonzero contribution
) 3g � 3⌘ and 3g � 3
e⌘There are only 3g � 3 ⌘ and 3g � 3
e⌘The correlation function does not depend of the
choice of ��
Conclusion
Superstring amplitude cannot be represent as
integration over Mspin in general
Special cases exist.
Amplitudes which is equivalent topological
amplitudes are the case.
Conclusion
Superstring amplitude cannot be represent as
integration over Mspin in general
Special cases exist.
Amplitudes which is equivalent topological
amplitudes are the case.
Conclusion
Superstring amplitude cannot be represent as
integration over Mspin in general
Special cases exist.
Amplitudes which is equivalent topological
amplitudes are the case.
Conclusion
Superstring amplitude cannot be represent as
integration over Mspin in general
Special cases exist.
Amplitudes which is equivalent topological
amplitudes are the case.
End.