Superstring theory and integration over
the moduli space
Kantaro Ohmori
Hongo,The University of Tokyo
June, 24th, 2013@Nagoya
arXiv:1303.7299 [KO,Yuji Tachikawa]
Section 1
introduction
Simplified history of the superstring theory
Simplified history of the superstring theory
∼70’s:Bosonic string theory→ supersymmetrize
Simplified history of the superstring theory
∼70’s:Bosonic string theory→ supersymmetrize
’84∼:1st superstring revolution
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]
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]
’87 “Multiloop calculations in covariant superstring
theory” [E.Verlinde, H.Verlinde ’87]
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]
’87 “Multiloop calculations in covariant superstring
theory” [E.Verlinde, H.Verlinde ’87]
’94∼:2nd superstring revolution
⇒ nonperturbative perspective
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]
’87 “Multiloop calculations in covariant superstring
theory” [E.Verlinde, H.Verlinde ’87]
’94∼:2nd superstring revolution
⇒ nonperturbative perspective
’12: “Superstring Perturbation Theory Revisited”
[E.Witten]:
Superstring perturbation theory and supergeometry
The work of [KO,Tachikawa]
The work of [KO,Tachikawa]
FMS:ambiguity
Formulation using supergeometry
⇒ 2 examples where ambiguity does not appear
The work of [KO,Tachikawa]
FMS:ambiguity
Formulation using supergeometry
⇒ 2 examples where ambiguity does not appear
N = 0 ⊂ N = 1 embedding [N. Berkovits,C. Vafa
’94]
The work of [KO,Tachikawa]
FMS:ambiguity
Formulation using supergeometry
⇒ 2 examples where ambiguity does not appear
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 moduli
Section 2
Supergeometry and Superstring
Bosonic string theory
Bosonic string theory
Action:S =∫Σ d
2zGij∂Xi∂Xj
Bosonic string theory
Action:S =∫Σ d
2zGij∂Xi∂Xj
⇒ Riemann Surface = 2d surface + complex
structure
Bosonic string theory
Action:S =∫Σ d
2zGij∂Xi∂Xj
⇒ Riemann Surface = 2d surface + complex
structure
Conformal symmetry⇒ bc ghost
Bosonic string theory
Action:S =∫Σ d
2zGij∂Xi∂Xj
⇒ Riemann Surface = 2d surface + complex
structure
Conformal symmetry⇒ bc ghost
⇒ A =∫Mbos,g
dmdmF(m, m)
Superstring theory
Superstring theory
Action:S =∫Σ d
2zd2θ GijDθXiDθX
j
Superstring theory
Action:S =∫Σ d
2zd2θ GijDθXiDθX
j
⇒ Super Riemann Surface
= 2d surface + supercomplex structure
= 2d surface + complex structure + gravitinoθ
θ θ
θθ
θ θθθ
Superstring theory
Action:S =∫Σ d
2zd2θ GijDθXiDθX
j
⇒ Super Riemann Surface
= 2d surface + supercomplex structure
= 2d surface + complex structure + gravitinoθ
θ θ
θθ
θ θθθ
Superconformal symmetry
⇒ BC(bcβγ) superghost
Superstring theory
Action:S =∫Σ d
2zd2θ GijDθXiDθX
j
⇒ Super Riemann Surface
= 2d surface + supercomplex structure
= 2d surface + complex structure + gravitinoθ
θ θ
θθ
θ θθθ
Superconformal symmetry
⇒ BC(bcβγ) superghost
⇒ A =∫Msuper,g
dmdηdmdηF(m, m, η, η)
Superstring theory
Action:S =∫Σ d
2zd2θ GijDθXiDθX
j
⇒ Super Riemann Surface
= 2d surface + supercomplex structure
= 2d surface + complex structure + gravitinoθ
θ θ
θθ
θ θθθ
Superconformal symmetry
⇒ BC(bcβγ) superghost
⇒ A =∫Msuper,g
dmdηdmdηF(m, m, η, η)?⇒ A =
∫Mbos,g
dmdmF′(m, m)
Supermanifold
Supermanifold
supermanifold = patched superspaces
Supermanifold
supermanifold = patched superspaces
M : supermanifold of dimension p|qM ∋ (x1, · · · , xp|θ1, · · · , θq)θiθj = −θjθi
Supermanifold
supermanifold = patched superspaces
M : supermanifold of dimension p|qM ∋ (x1, · · · , xp|θ1, · · · , θq)θiθj = −θjθi
x′i = fi(x|θ)θ′µ = ψµ(x|θ)
Uα
Supermanifold
supermanifold = patched superspaces
M : supermanifold of dimension p|qM ∋ (x1, · · · , xp|θ1, · · · , θq)θiθj = −θjθi
x′i = fi(x|θ)θ′µ = ψµ(x|θ)Mred = {(xi, · · · , xp)} −→ M
Uα
Super Riemann Surface (SRS)
Super Riemann Surface (SRS)
Complex supermanifold Σ of dimension 1|1:Σ ∋ (z|θ)
Super Riemann Surface (SRS)
Complex supermanifold Σ of dimension 1|1:Σ ∋ (z|θ)Patches are glued by superconformal transf.
Super Riemann Surface (SRS)
Complex supermanifold Σ of dimension 1|1:Σ ∋ (z|θ)Patches are glued by superconformal transf.
z′ = u(z|η) + θζ(z|η)√
u(z|η)θ′ = ζ′(z|η) + θ
√∂zu(z|η) + ζ(z|η)∂zζ(z|η)
Super Riemann Surface (SRS)
Complex supermanifold Σ of dimension 1|1:Σ ∋ (z|θ)Patches are glued by superconformal transf.
z′ = u(z|η) + θζ(z|η)√
u(z|η)θ′ = ζ′(z|η) + θ
√∂zu(z|η) + ζ(z|η)∂zζ(z|η)
Dθ = ∂θ + θ∂z = F(z|θ)(∂θ′ + θ′∂z′)
Split Super Riemann Surface
Split Super Riemann Surface
z′ = u(z)
θ′ = θ√∂zu(z)
Split Super Riemann Surface
z′ = u(z)
θ′ = θ√∂zu(z)
θ ∈ H0(Σ,TΣ∗1/2red )
⇒ split SRS↔ Riemann surface with spin str.
Supermoduli spaceMsuper
Supermoduli spaceMsuper
Space of deformations of SRS
Supermoduli spaceMsuper
Space of deformations of SRS
even deformation: µzz ∈ H1(Σred,TΣred)
Supermoduli spaceMsuper
Space of deformations of SRS
even deformation: µzz ∈ H1(Σred,TΣred)
odd deformation: χθz ∈ H1(Σred,TΣ1/2red )
Supermoduli spaceMsuper
Space of deformations of SRS
even deformation: µzz ∈ H1(Σred,TΣred)
odd deformation: χθz ∈ H1(Σred,TΣ1/2red )
dimMsuper = ∆e|∆o = 3g− 3|2g− 2 (g ≥ 2)
Supermoduli spaceMsuper
Space of deformations of SRS
even deformation: µzz ∈ H1(Σred,TΣred)
odd deformation: χθz ∈ H1(Σred,TΣ1/2red )
dimMsuper = ∆e|∆o = 3g− 3|2g− 2 (g ≥ 2)
Msuper,red ≃Mspin
Supermoduli space of SRS with punctures
Supermoduli space of SRS with punctures
NS puncture
Supermoduli space of SRS with punctures
NS puncture
R puncture
Supermoduli space of SRS with punctures
NS puncture
R puncture
dimMsuper = ∆e|∆o
= 3g−3+nNS+nR|2g−2+nNS+nR/2 (g ≥ 2)
Integration cycle for string theory
MR 6≃ ML w/ NS-R or R-NS vertex
Integration cycle for string theory
MR 6≃ ML w/ NS-R or R-NS vertex
(t1 · · · t2∆e|ζ1 · · · ζ∆o+∆o) ∈ Γ ⊂MR ×ML
Real subsupermanifold
Integration cycle for string theory
MR 6≃ ML w/ NS-R or R-NS vertex
(t1 · · · t2∆e|ζ1 · · · ζ∆o+∆o) ∈ Γ ⊂MR ×ML
Real subsupermanifold
MR ∋ (m, η), ML ∋ (m, η)
Integration cycle for string theory
MR 6≃ ML w/ NS-R or R-NS vertex
(t1 · · · t2∆e|ζ1 · · · ζ∆o+∆o) ∈ Γ ⊂MR ×ML
Real subsupermanifold
MR ∋ (m, η), ML ∋ (m, η)
m1 = t1 + it2 + nilp., barm1 = t1 − it2 + nilp.,
ζ i = ηi, ζ i+∆o = η i
Integration cycle for string theory
MR 6≃ ML w/ NS-R or R-NS vertex
(t1 · · · t2∆e|ζ1 · · · ζ∆o+∆o) ∈ Γ ⊂MR ×ML
Real subsupermanifold
MR ∋ (m, η), ML ∋ (m, η)
m1 = t1 + it2 + nilp., barm1 = t1 − it2 + nilp.,
ζ i = ηi, ζ i+∆o = η i
Integration over Γ is unique
[E.Witten , arXiv:1209.2199]
Integration cycle for string theory
MR 6≃ ML w/ NS-R or R-NS vertex
(t1 · · · t2∆e|ζ1 · · · ζ∆o+∆o) ∈ Γ ⊂MR ×ML
Real subsupermanifold
MR ∋ (m, η), ML ∋ (m, η)
m1 = t1 + it2 + nilp., barm1 = t1 − it2 + nilp.,
ζ i = ηi, ζ i+∆o = η i
Integration over Γ is unique
[E.Witten , arXiv:1209.2199]
if we fix the behavior of
Γ near boundary.
Odd coordinates ofMsuper
Odd coordinates ofMsuper
Σ0 : Split SRS = (m|0) ∈Msuper,red ⊂Msuper
Odd coordinates ofMsuper
Σ0 : Split SRS = (m|0) ∈Msuper,red ⊂Msuper
Σ : SRS obtained by odd deformation of Σ0
Odd coordinates ofMsuper
Σ0 : Split SRS = (m|0) ∈Msuper,red ⊂Msuper
Σ : SRS obtained by odd deformation of Σ0
deformation χ :gravitino background
Odd coordinates ofMsuper
Σ0 : Split SRS = (m|0) ∈Msuper,red ⊂Msuper
Σ : SRS obtained by odd deformation of Σ0
deformation χ :gravitino background
{χσ} : a (bosonic) basis of gravitino background
χ = ησχσ
Odd coordinates ofMsuper
Σ0 : Split SRS = (m|0) ∈Msuper,red ⊂Msuper
Σ : SRS obtained by odd deformation of Σ0
deformation χ :gravitino background
{χσ} : a (bosonic) basis of gravitino background
χ = ησχσ
Odd coordinates ofMsuper
Σ0 : Split SRS = (m|0) ∈Msuper,red ⊂Msuper
Σ : SRS obtained by odd deformation of Σ0
deformation χ :gravitino background
{χσ} : a (bosonic) basis of gravitino background
χ = ησχσ
Σ = (m|η1, · · · , η∆o)
Coupling between gravitino and SCFT
Coupling between gravitino and SCFT
Sχ = 12π
∫Σred
d2zχTF
Coupling between gravitino and SCFT
Sχ = 12π
∫Σred
d2zχTF
Sχ = 12π
∫Σred
d2zχTF
Coupling between gravitino and SCFT
Sχ = 12π
∫Σred
d2zχTF
Sχ = 12π
∫Σred
d2zχTF
Sχχ =∫ΣredχχA
A ∝ ψµψµ (Flat background)
Coupling between gravitino and SCFT
Sχ = 12π
∫Σred
d2zχTF
Sχ = 12π
∫Σred
d2zχTF
Sχχ =∫ΣredχχA
A ∝ ψµψµ (Flat background)
c.f. Dµφ†Dµφ ∋ AµA
µφ†φ
Scattering amplitudes of superstring
Scattering amplitudes of superstring
Vi : Vertex op. pic. number −1 (NS) or −1/2 (R)
Scattering amplitudes of superstring
Vi : Vertex op. pic. number −1 (NS) or −1/2 (R)
AV,g =∫Γ dmdηdmdη FV,g(m, η, m, η)
Scattering amplitudes of superstring
Vi : Vertex op. pic. number −1 (NS) or −1/2 (R)
AV,g =∫Γ dmdηdmdη FV,g(m, η, m, η)
FV,g(m, η, m, η) =⟨∏i Vi
∏∆e
a=1(∫Σred
d2zbµa)
∏∆e
b=1(∫Σred
d2zbµb)
∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(−Sχ − Sχ − Sχχ)〉m
Scattering amplitudes of superstring
Vi : Vertex op. pic. number −1 (NS) or −1/2 (R)
AV,g =∫Γ dmdηdmdη FV,g(m, η, m, η)
FV,g(m, η, m, η) =⟨∏i Vi
∏∆e
a=1(∫Σred
d2zbµa)
∏∆e
b=1(∫Σred
d2zbµb)
∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(−Sχ − Sχ − Sχχ)〉mSχ =
∑∆o
σ=1ησ2π
∫Σred
d2zχσTF
Picture changing formalism
Picture changing formalism
χσ = δ(z− pσ) pσ ∈ Σred
Picture changing formalism
χσ = δ(z− pσ) pσ ∈ Σred
Factors including χ :∫dη
∏σ(∫Σredβχσ) exp(
∑∆o
σ=1−ησ2π
∫ΣredχσTF)
Picture changing formalism
χσ = δ(z− pσ) pσ ∈ Σred
Factors including χ :∫dη
∏σ(∫Σredβχσ) exp(
∑∆o
σ=1−ησ2π
∫ΣredχσTF)
=∏∆o
σ Y(pσ)
Y(pσ) = δ(β(pσ))TF(pσ)
Picture changing formalism
χσ = δ(z− pσ) pσ ∈ Σred
Factors including χ :∫dη
∏σ(∫Σredβχσ) exp(
∑∆o
σ=1−ησ2π
∫ΣredχσTF)
=∏∆o
σ Y(pσ)
Y(pσ) = δ(β(pσ))TF(pσ)
χχ term vanishes if pσ 6= pτ
⇒ Formulation of FMS
Picture changing formalism
χσ = δ(z− pσ) pσ ∈ Σred
Factors including χ :∫dη
∏σ(∫Σredβχσ) exp(
∑∆o
σ=1−ησ2π
∫ΣredχσTF)
=∏∆o
σ Y(pσ)
Y(pσ) = δ(β(pσ))TF(pσ)
χχ term vanishes if pσ 6= pτ
⇒ Formulation of FMS
Valid where [δ(z− pσ)] spans odd deformations
⇒ Coordinates given by picture changing formalism
is not global!
Integration over a supermanifold
Integration over a supermanifold
M: dim1|2 supermanifold,
M = U1 ∪ U2,U1 ∋ (m|η1, η2), U2 ∋(m′|η′1, η′2)M = V1
∐V2,Ui ⊃ Vi
Integration over a supermanifold
M: dim1|2 supermanifold,
M = U1 ∪ U2,U1 ∋ (m|η1, η2), U2 ∋(m′|η′1, η′2)M = V1
∐V2,Ui ⊃ Vi
m′ = m + aη1η2
η′i = ηi
Integration over a supermanifold
M: dim1|2 supermanifold,
M = U1 ∪ U2,U1 ∋ (m|η1, η2), U2 ∋(m′|η′1, η′2)M = V1
∐V2,Ui ⊃ Vi
m′ = m + aη1η2
η′i = ηi
ω|U1= (f0(m) + f2(m)η1η2)dmdη1dη2
Integration over a supermanifold
M: dim1|2 supermanifold,
M = U1 ∪ U2,U1 ∋ (m|η1, η2), U2 ∋(m′|η′1, η′2)M = V1
∐V2,Ui ⊃ Vi
m′ = m + aη1η2
η′i = ηi
ω|U1= (f0(m) + f2(m)η1η2)dmdη1dη2
ω|U2=
(f0(m′) + (f2(m
′)− a∂f0(m′))η1η2)dm′dη′1dη
′2
Integration over a supermanifold
M: dim1|2 supermanifold,
M = U1 ∪ U2,U1 ∋ (m|η1, η2), U2 ∋(m′|η′1, η′2)M = V1
∐V2,Ui ⊃ Vi
m′ = m + aη1η2
η′i = ηi
ω|U1= (f0(m) + f2(m)η1η2)dmdη1dη2
ω|U2=
(f0(m′) + (f2(m
′)− a∂f0(m′))η1η2)dm′dη′1dη
′2∫
M ω =∫V1,red
f2dm +∫V2,red
(f2 − a∂f0)dm′
Projectedness of supermanifolds
Projectedness of supermanifolds
M:projected⇔an atlas in which x′ = f(x) for all gluing
Projectedness of supermanifolds
M:projected⇔an atlas in which x′ = f(x) for all gluing
⇔ p : M→ Mred exists
Projectedness of supermanifolds
M:projected⇔an atlas in which x′ = f(x) for all gluing
⇔ p : M→ Mred exists∫M ω =
∫Mred
p∗(ω)
Projectedness of supermanifolds
M:projected⇔an atlas in which x′ = f(x) for all gluing
⇔ p : M→ Mred exists∫M ω =
∫Mred
p∗(ω)
All supermanifolds are projected in smooth sense.
Projectedness of supermanifolds
M:projected⇔an atlas in which x′ = f(x) for all gluing
⇔ p : M→ Mred exists∫M ω =
∫Mred
p∗(ω)
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′ = f(x) for all gluing
⇔ p : M→ Mred exists∫M ω =
∫Mred
p∗(ω)
All supermanifolds are projected in smooth sense.
Complex supermanifolds are not projected in
holomorphic sense in general.
non-holomorphic projection destroys holomorphic
factorization.
Supermoduli Space is not projected[R.Donagi, E.Witten, arXiv:1304.7798]
Supermoduli Space is not projected[R.Donagi, E.Witten, arXiv:1304.7798]
Msuper,red ≃Mspin i :Mspin −→Msuper
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)
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 onMsuper
6⇒ Global integration onMspin
(with holomorphic factorization)
Superstring Perturbation Theory Revisited[E.Witten ’12]
Superstring Perturbation Theory Revisited[E.Witten ’12]
Global integration onMsuper
6⇒ Global integration onMspin
Superstring Perturbation Theory Revisited[E.Witten ’12]
Global integration onMsuper
6⇒ Global integration onMspin
PCO: not globally valid
Superstring Perturbation Theory Revisited[E.Witten ’12]
Global integration onMsuper
6⇒ Global integration onMspin
PCO: not globally valid
Movement of PCO⇒ exact form
Superstring Perturbation Theory Revisited[E.Witten ’12]
Global integration onMsuper
6⇒ Global integration onMspin
PCO: not globally valid
Movement of PCO⇒ exact form
E.Witten described the way which does not rely on
the reduction
Superstring Perturbation Theory Revisited[E.Witten ’12]
Global integration onMsuper
6⇒ Global integration onMspin
PCO: not globally valid
Movement of PCO⇒ exact form
E.Witten described the way which does not rely on
the reduction
Infrared regularization
Section 3
Reduction to integration over moduli
Reduction condition
Reduction condition
∫M ω =
∫V1,red
f2dm +∫V2,red
(f2 − a∂f0)dm′
Reduction condition
∫M ω =
∫V1,red
f2dm +∫V2,red
f2dm′
Global integration onMsuper
⇒ Global integration onMspin in special cases
Reduction condition
∫M ω =
∫V1,red
f2dm +∫V2,red
f2dm′
Global integration onMsuper
⇒ Global integration onMspin in special cases
ω = F(m, m)∏
dmdm∏
all ηiηi∏
dηdηi
Reduction condition
∫M ω =
∫V1,red
f2dm +∫V2,red
f2dm′
Global integration onMsuper
⇒ Global integration onMspin in special cases
ω = F(m, m)∏
dmdm∏
all ηiηi∏
dηdηi
ω = i∗α ∧ P.D.[Mspin]
Reduction condition
∫M ω =
∫V1,red
f2dm +∫V2,red
f2dm′
Global integration onMsuper
⇒ Global integration onMspin in special cases
ω = F(m, m)∏
dmdm∏
all ηiηi∏
dηdηi
ω = i∗α ∧ P.D.[Mspin]
ω 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]
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
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 effective field theory is related to
topological string.
Topological string
Twisted N = (2, 2) string theory
Topological string
Twisted N = (2, 2) string theory
Amplitude =∫Mbos· · ·
Topological string
Twisted N = (2, 2) string theory
Amplitude =∫Mbos· · ·
Fg : g loop vacuum amplitude
Reduction to topological string
Type II string on CY(6d)M ×R1,3
⇒ 4d N = 2 supergravity model
Reduction to topological string
Type II string on CY(6d)M ×R1,3
⇒ 4d N = 2 supergravity model
Ag : zero momenta limit of g loop amplitude of
2g − 2 graviphotons (RR) and 2 gravitons (NSNS)
Reduction to topological string
Type II string on CY(6d)M ×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 =∫Γ · · ·
Reduction to topological string
Type II string on CY(6d)M ×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 =∫Γ · · ·
dimR Γ = 6g − 6|6g − 6
Reduction to topological string
Type II string on CY(6d)M ×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 =∫Γ · · ·
dimR Γ = 6g − 6|6g − 6
Ag = (g!)2Fg =∫Mbos· · ·
Vertex operators
Vertex operators
Graviphoton: VT = Σ× spacetime× (ghost)
Σ = exp(i√32(H(z)∓ H(z))) H : U(1)R boson
Σ : U(1)R charge (3/2,∓ 3/2)
Vertex operators
Graviphoton: VT = Σ× spacetime× (ghost)
Σ = exp(i√32(H(z)∓ H(z))) H : U(1)R boson
Σ : U(1)R charge (3/2,∓ 3/2)
Graviton: VR =∫Σred
spacetime,ghost
Calculation of amplitude
Calculation of amplitudeAg =∫Γ
⟨∏2g−2i=1 Σ(xi)
×(spacetime)× (bc)×∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(−Sχ − Sχ − Sχχ)〉m
Calculation of amplitudeAg =∫Γ
⟨∏2g−2i=1 Σ(xi)← U(1)R : (3g − 3,∓3g − 3)
×(spacetime)× (bc)×∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(− ∫Σred
(TFχ+ TFχ+ Aχχ))⟩m
Calculation of amplitudeAg =∫Γ
⟨∏2g−2i=1 Σ(xi)← U(1)R : (3g − 3,∓3g − 3)
×(spacetime)× (bc)×∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(− ∫Σred
(TFχ+ TFχ+ Aχχ))⟩m
TF = TF,spacetime,ghost + G+ + G−
Calculation of amplitudeAg =∫Γ
⟨∏2g−2i=1 Σ(xi)← U(1)R : (3g − 3,∓3g − 3)
×(spacetime)× (bc)×∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(− ∫Σred
(TFχ+ TFχ+ Aχχ))⟩m
TF = TF,spacetime,ghost + G+ + G−
G± : U(1)R charge ±1
Calculation of amplitudeAg =∫Γ
⟨∏2g−2i=1 Σ(xi)← U(1)R : (3g − 3,∓3g − 3)
×(spacetime)× (bc)×∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(− ∫Σred
(TFχ+ TFχ+ Aχχ))⟩m
TF = TF,spacetime,ghost + G+ + G−
G± : U(1)R charge ±1A = A++ + A+− + A−+ + A−− + Aspacetime
A±±: U(1)R charge (±1,±1)⇐ Coupling A to N = (2, 2) supergravity:∫Σred
A±±χ∓χ∓ → χ = χ+ = χ−
Saturation of η’s
Saturation of η’s
Ag =∫Γ
⟨∏2g−2i=1 Σ(xi)← U(1)R : (3g − 3,∓3g − 3)
×(spacetime)× (bc)×∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(− ∫Σred
(TFχ+ TFχ+ Aχχ))⟩m
Saturation of η’s
Ag =∫Γ
⟨∏2g−2i=1 Σ(xi)← U(1)R : (3g − 3,∓3g − 3)
×(spacetime)× (bc)×∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(− ∫Σred
(TFχ+ TFχ+ Aχχ))⟩m
nonzero contribution
⇒ 3g − 3χ and 3g − 3χ
Saturation of η’s
Ag =∫Γ
⟨∏2g−2i=1 Σ(xi)← U(1)R : (3g − 3,∓3g − 3)
×(spacetime)× (bc)×∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(− ∫Σred
(TFχ+ TFχ+ Aχχ))⟩m
nonzero contribution
⇒ 3g − 3η and 3g − 3η
dimΓ = 6g − 6|6g − 6⇒ saturation η’s
Saturation of η’s
Ag =∫Γ
⟨∏2g−2i=1 Σ(xi)← U(1)R : (3g − 3,∓3g − 3)
×(spacetime)× (bc)×∏∆o
σ=1 δ(∫Σred
d2zβχσ)
∏∆o
τ=1 δ(∫Σred
d2zβχτ )
exp(− ∫Σred
(TFχ+ TFχ+ Aχχ))⟩m
nonzero contribution
⇒ 3g − 3η and 3g − 3η
dimΓ = 6g − 6|6g − 6⇒ saturation η’s
The correlation function does not depend of the
choice of χσ
Conclusion
Conclusion
Superstring amplitude cannot be represent as
integration overMbos in general
Conclusion
Superstring amplitude cannot be represent as
integration overMbos in general
Special cases exist.
Conclusion
Superstring amplitude cannot be represent as
integration overMbos in general
Special cases exist.
Amplitudes which is equivalent topological
amplitudes are the case.
Prospects
Prospects
More nontrivial example of calculation
[E.Witten, arXiv:1304.2832],[E.Witten, arXiv:1306.3621]
Prospects
More nontrivial example of calculation
[E.Witten, arXiv:1304.2832],[E.Witten, arXiv:1306.3621]
Superstring field theory
Prospects
More nontrivial example of calculation
[E.Witten, arXiv:1304.2832],[E.Witten, arXiv:1306.3621]
Superstring field theory
Another application of supergeometry
End.
Berkovits and Vafa embedding[N.Berkovits, C.Vafa ’94]
Worldsheet supersymmetric model equivalent to
bosonic string theory
⇒ Bosonic string : An (unstable) vacuum of
superstring theory
〈V1 · · ·Vn〉bos =⟨V′1 · · ·V′n
⟩super
⇔ ∫Mbos· · · = ∫
Msuper· · ·
Supermoduli space of SRS with punctures
Supermoduli space of SRS with punctures
even deformation:
H1(Σred,TΣred)+ positions of punctures
Supermoduli space of SRS with punctures
even deformation:
H1(Σred,TΣred ⊗O(∑
NS pi +∑
R qi))
Supermoduli space of SRS with punctures
even deformation:
H1(Σred,TΣred ⊗O(∑
NS pi +∑
R qi))
odd deformation:
H1(Σred,R⊗O(∑
NS pi))
R2 ≃ TΣred ⊗O(∑
R qi)
Supermoduli space of SRS with punctures
even deformation:
H1(Σred,TΣred ⊗O(∑
NS pi +∑
R qi))
odd deformation:
H1(Σred,R⊗O(∑
NS pi))
R2 ≃ TΣred ⊗O(∑
R qi)
dimMsuper = ∆e|∆o
= 3g−3+nNS+nR|2g−2+nNS+nR/2 (g ≥ 2)
Superconformal transformation
νf = f(z)(∂θ − θ∂z)
Vg = g(z)∂z + 1/2∂zgθ∂θ
Gn = νzn+1/2
Lm = −Vzm+1
Superconformal transformation near R
vertex
νf = f(z)(∂θ − zθ∂z)
Vg = z(g(z)∂z + 1/2∂zgθ∂θ)
Gr = νzr
Lm = −Vzm
Integration over odd variables
Integration over odd variables
ηi:odd variable
Integration over odd variables
ηi:odd variable
ηiηj = −ηjηi η2i = 0
Integration over odd variables
ηi:odd variable
ηiηj = −ηjηi η2i = 0
f(η1, η2) = a + bη1 + cη2 + dη1η2
Integration over odd variables
ηi:odd variable
ηiηj = −ηjηi η2i = 0
f(η1, η2) = a + bη1 + cη2 + dη1η2∫dη2dη1f(η1, η2) = d
SRS with punctures
SRS with punctures
NS puncture: (z0|θ0) ∈ Σ
θ : periodic around z0
SRS with punctures
NS puncture: (z0|θ0) ∈ Σ
θ : periodic around z0
R puncture: {z = z1} ⊂ Σ : submanifold of Σ
SRS with punctures
NS puncture: (z0|θ0) ∈ Σ
θ : periodic around z0
R puncture: {z = z1} ⊂ Σ : submanifold of Σ
D∗θ = ∂θ + (z− z1)θ∂z
SRS with punctures
NS puncture: (z0|θ0) ∈ Σ
θ : periodic around z0
R puncture: {z = z1} ⊂ Σ : submanifold of Σ
D∗θ = ∂θ + (z− z1)θ∂z
θ′ =√z− z1θ ⇒ D∗θ =
√z(∂θ′ + θ′∂z)
θ′:antiperiodic around z1
Subtlety for picture changing formalism
Subtlety for picture changing formalism
Valid where [δ(pσ)] spans odd deformations
⇒ Coordinates given by picture changing formalism
is not global!
Subtlety for picture changing formalism
Valid where [δ(pσ)] spans odd deformations
⇒ Coordinates given by picture changing formalism
is not global!
1 Δ{Y(q�),...,Y(q�)}
1 Δ{Y(q�),...,Y(q�)}
M super
{Y(p�),...,Y(p�)}1 Δ
{Y(p�),...,Y(p�)}
{Y(p�),...,Y(p�)}1 Δ
{Y(p�),...,Y(p�)}
1 Δ{Y(q�),...,Y(q�)}
1 Δ{Y(q�),...,Y(q�)}
Subtlety for picture changing formalism
Valid where [δ(pσ)] spans odd deformations
⇒ Coordinates given by picture changing formalism
is not global!
1 Δ{Y(q�),...,Y(q�)}
1 Δ{Y(q�),...,Y(q�)}
M super
{Y(p�),...,Y(p�)}1 Δ
{Y(p�),...,Y(p�)}
{Y(p�),...,Y(p�)}1 Δ
{Y(p�),...,Y(p�)}
1 Δ{Y(q�),...,Y(q�)}
1 Δ{Y(q�),...,Y(q�)}
movement of PCOs⇒ BRST exact term⇔ exact
form on patches
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