superstring theory and integration over the moduli space
TRANSCRIPT
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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]
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Section 1
introduction
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Simplified history of the superstring theory
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Simplified history of the superstring theory
∼70’s:Bosonic string theory→ supersymmetrize
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Simplified history of the superstring theory
∼70’s:Bosonic string theory→ supersymmetrize
’84∼:1st superstring revolution
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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]
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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]
’87 “Multiloop calculations in covariant superstring
theory” [E.Verlinde, H.Verlinde ’87]
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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]
’87 “Multiloop calculations in covariant superstring
theory” [E.Verlinde, H.Verlinde ’87]
’94∼:2nd superstring revolution
⇒ nonperturbative perspective
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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]
’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
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The work of [KO,Tachikawa]
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The work of [KO,Tachikawa]
FMS:ambiguity
Formulation using supergeometry
⇒ 2 examples where ambiguity does not appear
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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]
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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]
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1 introduction
2 Supergeometry and Superstring
3 Reduction to integration over moduli
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Section 2
Supergeometry and Superstring
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Bosonic string theory
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Bosonic string theory
Action:S =∫Σ d
2zGij∂Xi∂Xj
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Bosonic string theory
Action:S =∫Σ d
2zGij∂Xi∂Xj
⇒ Riemann Surface = 2d surface + complex
structure
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Bosonic string theory
Action:S =∫Σ d
2zGij∂Xi∂Xj
⇒ Riemann Surface = 2d surface + complex
structure
Conformal symmetry⇒ bc ghost
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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)
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Superstring theory
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Superstring theory
Action:S =∫Σ d
2zd2θ GijDθXiDθX
j
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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
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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
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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)
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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)
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Supermanifold
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Supermanifold
supermanifold = patched superspaces
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Supermanifold
supermanifold = patched superspaces
M : supermanifold of dimension p|qM ∋ (x1, · · · , xp|θ1, · · · , θq)θiθj = −θjθi
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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
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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
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Super Riemann Surface (SRS)
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Super Riemann Surface (SRS)
Complex supermanifold Σ of dimension 1|1:Σ ∋ (z|θ)
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Super Riemann Surface (SRS)
Complex supermanifold Σ of dimension 1|1:Σ ∋ (z|θ)Patches are glued by superconformal transf.
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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)
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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′)
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Split Super Riemann Surface
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Split Super Riemann Surface
z′ = u(z)
θ′ = θ√∂zu(z)
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Split Super Riemann Surface
z′ = u(z)
θ′ = θ√∂zu(z)
θ ∈ H0(Σ,TΣ∗1/2red )
⇒ split SRS↔ Riemann surface with spin str.
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Supermoduli spaceMsuper
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Supermoduli spaceMsuper
Space of deformations of SRS
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Supermoduli spaceMsuper
Space of deformations of SRS
even deformation: µzz ∈ H1(Σred,TΣred)
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Supermoduli spaceMsuper
Space of deformations of SRS
even deformation: µzz ∈ H1(Σred,TΣred)
odd deformation: χθz ∈ H1(Σred,TΣ1/2red )
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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)
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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
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Supermoduli space of SRS with punctures
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Supermoduli space of SRS with punctures
NS puncture
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Supermoduli space of SRS with punctures
NS puncture
R puncture
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Supermoduli space of SRS with punctures
NS puncture
R puncture
dimMsuper = ∆e|∆o
= 3g−3+nNS+nR|2g−2+nNS+nR/2 (g ≥ 2)
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Integration cycle for string theory
MR 6≃ ML w/ NS-R or R-NS vertex
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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
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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)
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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
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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]
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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.
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Odd coordinates ofMsuper
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Odd coordinates ofMsuper
Σ0 : Split SRS = (m|0) ∈Msuper,red ⊂Msuper
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Odd coordinates ofMsuper
Σ0 : Split SRS = (m|0) ∈Msuper,red ⊂Msuper
Σ : SRS obtained by odd deformation of Σ0
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Odd coordinates ofMsuper
Σ0 : Split SRS = (m|0) ∈Msuper,red ⊂Msuper
Σ : SRS obtained by odd deformation of Σ0
deformation χ :gravitino background
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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
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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
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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)
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Coupling between gravitino and SCFT
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Coupling between gravitino and SCFT
Sχ = 12π
∫Σred
d2zχTF
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Coupling between gravitino and SCFT
Sχ = 12π
∫Σred
d2zχTF
Sχ = 12π
∫Σred
d2zχTF
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Coupling between gravitino and SCFT
Sχ = 12π
∫Σred
d2zχTF
Sχ = 12π
∫Σred
d2zχTF
Sχχ =∫ΣredχχA
A ∝ ψµψµ (Flat background)
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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
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Scattering amplitudes of superstring
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Scattering amplitudes of superstring
Vi : Vertex op. pic. number −1 (NS) or −1/2 (R)
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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)
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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
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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
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Picture changing formalism
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Picture changing formalism
χσ = δ(z− pσ) pσ ∈ Σred
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Picture changing formalism
χσ = δ(z− pσ) pσ ∈ Σred
Factors including χ :∫dη
∏σ(∫Σredβχσ) exp(
∑∆o
σ=1−ησ2π
∫ΣredχσTF)
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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)
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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
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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!
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Integration over a supermanifold
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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
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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
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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
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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
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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′
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Projectedness of supermanifolds
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Projectedness of supermanifolds
M:projected⇔an atlas in which x′ = f(x) for all gluing
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Projectedness of supermanifolds
M:projected⇔an atlas in which x′ = f(x) for all gluing
⇔ p : M→ Mred exists
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Projectedness of supermanifolds
M:projected⇔an atlas in which x′ = f(x) for all gluing
⇔ p : M→ Mred exists∫M ω =
∫Mred
p∗(ω)
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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.
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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.
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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.
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Supermoduli Space is not projected[R.Donagi, E.Witten, arXiv:1304.7798]
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Supermoduli Space is not projected[R.Donagi, E.Witten, arXiv:1304.7798]
Msuper,red ≃Mspin i :Mspin −→Msuper
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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)
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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)
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Superstring Perturbation Theory Revisited[E.Witten ’12]
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Superstring Perturbation Theory Revisited[E.Witten ’12]
Global integration onMsuper
6⇒ Global integration onMspin
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Superstring Perturbation Theory Revisited[E.Witten ’12]
Global integration onMsuper
6⇒ Global integration onMspin
PCO: not globally valid
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Superstring Perturbation Theory Revisited[E.Witten ’12]
Global integration onMsuper
6⇒ Global integration onMspin
PCO: not globally valid
Movement of PCO⇒ exact form
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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
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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
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Section 3
Reduction to integration over moduli
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Reduction condition
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Reduction condition
∫M ω =
∫V1,red
f2dm +∫V2,red
(f2 − a∂f0)dm′
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Reduction condition
∫M ω =
∫V1,red
f2dm +∫V2,red
f2dm′
Global integration onMsuper
⇒ Global integration onMspin in special cases
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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
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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]
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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
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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]
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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
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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.
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Topological string
Twisted N = (2, 2) string theory
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Topological string
Twisted N = (2, 2) string theory
Amplitude =∫Mbos· · ·
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Topological string
Twisted N = (2, 2) string theory
Amplitude =∫Mbos· · ·
Fg : g loop vacuum amplitude
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Reduction to topological string
Type II string on CY(6d)M ×R1,3
⇒ 4d N = 2 supergravity model
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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)
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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 · · ·
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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
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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· · ·
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Vertex operators
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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)
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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
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Calculation of amplitude
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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
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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
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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−
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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
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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−
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Saturation of η’s
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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
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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
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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
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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
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Conclusion
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Conclusion
Superstring amplitude cannot be represent as
integration overMbos in general
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Conclusion
Superstring amplitude cannot be represent as
integration overMbos in general
Special cases exist.
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Conclusion
Superstring amplitude cannot be represent as
integration overMbos in general
Special cases exist.
Amplitudes which is equivalent topological
amplitudes are the case.
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Prospects
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Prospects
More nontrivial example of calculation
[E.Witten, arXiv:1304.2832],[E.Witten, arXiv:1306.3621]
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Prospects
More nontrivial example of calculation
[E.Witten, arXiv:1304.2832],[E.Witten, arXiv:1306.3621]
Superstring field theory
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Prospects
More nontrivial example of calculation
[E.Witten, arXiv:1304.2832],[E.Witten, arXiv:1306.3621]
Superstring field theory
Another application of supergeometry
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End.
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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· · ·
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Supermoduli space of SRS with punctures
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Supermoduli space of SRS with punctures
even deformation:
H1(Σred,TΣred)+ positions of punctures
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Supermoduli space of SRS with punctures
even deformation:
H1(Σred,TΣred ⊗O(∑
NS pi +∑
R qi))
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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)
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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)
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Superconformal transformation
νf = f(z)(∂θ − θ∂z)
Vg = g(z)∂z + 1/2∂zgθ∂θ
Gn = νzn+1/2
Lm = −Vzm+1
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Superconformal transformation near R
vertex
νf = f(z)(∂θ − zθ∂z)
Vg = z(g(z)∂z + 1/2∂zgθ∂θ)
Gr = νzr
Lm = −Vzm
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Integration over odd variables
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Integration over odd variables
ηi:odd variable
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Integration over odd variables
ηi:odd variable
ηiηj = −ηjηi η2i = 0
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Integration over odd variables
ηi:odd variable
ηiηj = −ηjηi η2i = 0
f(η1, η2) = a + bη1 + cη2 + dη1η2
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Integration over odd variables
ηi:odd variable
ηiηj = −ηjηi η2i = 0
f(η1, η2) = a + bη1 + cη2 + dη1η2∫dη2dη1f(η1, η2) = d
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SRS with punctures
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SRS with punctures
NS puncture: (z0|θ0) ∈ Σ
θ : periodic around z0
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SRS with punctures
NS puncture: (z0|θ0) ∈ Σ
θ : periodic around z0
R puncture: {z = z1} ⊂ Σ : submanifold of Σ
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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
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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
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Subtlety for picture changing formalism
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Subtlety for picture changing formalism
Valid where [δ(pσ)] spans odd deformations
⇒ Coordinates given by picture changing formalism
is not global!
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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�)}
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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