Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution...
Transcript of Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution...
![Page 1: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/1.jpg)
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
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Section 1
introduction
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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
![Page 4: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/4.jpg)
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
![Page 5: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/5.jpg)
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
![Page 6: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/6.jpg)
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
![Page 7: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/7.jpg)
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
![Page 8: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/8.jpg)
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
![Page 9: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/9.jpg)
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)
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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)
![Page 11: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/11.jpg)
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)
![Page 12: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/12.jpg)
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)
![Page 13: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/13.jpg)
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)
![Page 14: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/14.jpg)
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)
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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)
![Page 16: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/16.jpg)
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)
![Page 17: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/17.jpg)
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)
![Page 18: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/18.jpg)
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)
![Page 19: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/19.jpg)
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)
![Page 20: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/20.jpg)
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)
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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)
![Page 22: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/22.jpg)
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
![Page 23: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/23.jpg)
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
![Page 24: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/24.jpg)
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
![Page 25: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/25.jpg)
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]
![Page 26: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/26.jpg)
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]
![Page 27: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/27.jpg)
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]
![Page 28: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/28.jpg)
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]
![Page 29: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/29.jpg)
1 introduction
2 Supergeometry and Superstring
3 Reduction to integration over bosonic moduli
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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
![Page 31: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/31.jpg)
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
![Page 32: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/32.jpg)
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
![Page 33: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/33.jpg)
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
![Page 34: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/34.jpg)
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
![Page 35: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/35.jpg)
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
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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
![Page 37: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/37.jpg)
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
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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
![Page 39: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/39.jpg)
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
![Page 40: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/40.jpg)
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
![Page 41: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/41.jpg)
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
![Page 42: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/42.jpg)
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)
![Page 43: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/43.jpg)
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)
![Page 44: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/44.jpg)
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)
![Page 45: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/45.jpg)
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�†�
![Page 46: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/46.jpg)
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�†�
![Page 47: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/47.jpg)
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�†�
![Page 48: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/48.jpg)
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�†�
![Page 49: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/49.jpg)
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�†�
![Page 50: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/50.jpg)
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.
![Page 51: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/51.jpg)
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.
![Page 52: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/52.jpg)
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.
![Page 53: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/53.jpg)
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.
![Page 54: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/54.jpg)
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.
![Page 55: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/55.jpg)
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.
![Page 56: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/56.jpg)
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
![Page 57: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/57.jpg)
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
![Page 58: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/58.jpg)
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
![Page 59: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/59.jpg)
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
![Page 60: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/60.jpg)
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.
![Page 61: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/61.jpg)
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.
![Page 62: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/62.jpg)
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.
![Page 63: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/63.jpg)
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.
![Page 64: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/64.jpg)
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.
![Page 65: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/65.jpg)
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.
![Page 66: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/66.jpg)
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)
![Page 67: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/67.jpg)
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)
![Page 68: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/68.jpg)
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)
![Page 69: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/69.jpg)
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)
![Page 70: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/70.jpg)
Section 3
Reduction to integration over bosonic
moduli
![Page 71: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/71.jpg)
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
![Page 72: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/72.jpg)
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
![Page 73: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/73.jpg)
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
![Page 74: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/74.jpg)
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
![Page 75: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/75.jpg)
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.
![Page 76: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/76.jpg)
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.
![Page 77: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/77.jpg)
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.
![Page 78: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/78.jpg)
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
· · ·
![Page 79: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/79.jpg)
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
· · ·
![Page 80: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/80.jpg)
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
· · ·
![Page 81: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/81.jpg)
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
· · ·
![Page 82: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/82.jpg)
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
![Page 83: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/83.jpg)
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
![Page 84: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/84.jpg)
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
![Page 85: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/85.jpg)
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 ��
![Page 86: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/86.jpg)
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 ��
![Page 87: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/87.jpg)
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 ��
![Page 88: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/88.jpg)
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 ��
![Page 89: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/89.jpg)
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 ��
![Page 90: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/90.jpg)
Conclusion
Superstring amplitude cannot be represent as
integration over Mspin in general
Special cases exist.
Amplitudes which is equivalent topological
amplitudes are the case.
![Page 91: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/91.jpg)
Conclusion
Superstring amplitude cannot be represent as
integration over Mspin in general
Special cases exist.
Amplitudes which is equivalent topological
amplitudes are the case.
![Page 92: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/92.jpg)
Conclusion
Superstring amplitude cannot be represent as
integration over Mspin in general
Special cases exist.
Amplitudes which is equivalent topological
amplitudes are the case.
![Page 93: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/93.jpg)
Conclusion
Superstring amplitude cannot be represent as
integration over Mspin in general
Special cases exist.
Amplitudes which is equivalent topological
amplitudes are the case.
![Page 94: Superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution ’86: ”Conformal Invariance, Supersymmetry, and String Theory” [D.Friedan, E.Martinec,](https://reader035.fdocuments.in/reader035/viewer/2022070711/5ec8041943fef409d772d738/html5/thumbnails/94.jpg)
End.