superstring theory and integration over the moduli space · ’84⇠:1st superstring revolution...

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.