The effective theory of type IIA AdS4 compactifications on...
Transcript of The effective theory of type IIA AdS4 compactifications on...
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
The effective theory of type IIA AdS4 compactifications onnilmanifolds and cosets
Based on: 0804.0614 (PK, Tsimpis, Lust), 0806.3458 (Caviezel, PK, Kors, Lust, Tsimpis, Zagermann)
Paul Koerber
Max-Planck-Institut fur Physik, Munich
Zurich, 11 August 2008
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 2: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/2.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Motivation
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 3: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/3.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Motivation
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
stabilization Kahler moduli through non-perturbative effectsuplift susy vacuum to dS, models of inflation
2 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 4: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/4.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Motivation
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
stabilization Kahler moduli through non-perturbative effectsuplift susy vacuum to dS, models of inflation
Type IIA compactifications with AdS4 space-time
2 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 5: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/5.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Motivation
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
stabilization Kahler moduli through non-perturbative effectsuplift susy vacuum to dS, models of inflation
Type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds:Derendinger et al., DeWolfe et al., Camara et al.
All moduli can be stabilized at tree level
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 6: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/6.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Motivation
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
stabilization Kahler moduli through non-perturbative effectsuplift susy vacuum to dS, models of inflation
Type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds:Derendinger et al., DeWolfe et al., Camara et al.
All moduli can be stabilized at tree levelUplift susy vacuum to dS, models of inflation: problems
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 7: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/7.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Motivation
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
stabilization Kahler moduli through non-perturbative effectsuplift susy vacuum to dS, models of inflation
Type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds:Derendinger et al., DeWolfe et al., Camara et al.
All moduli can be stabilized at tree levelUplift susy vacuum to dS, models of inflation: problemsNo-go theorem modular inflation: fluxes, D6/O6 Hertzberg,
Kachru, Taylor, Tegmark
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 8: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/8.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Motivation
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
stabilization Kahler moduli through non-perturbative effectsuplift susy vacuum to dS, models of inflation
Type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds:Derendinger et al., DeWolfe et al., Camara et al.
All moduli can be stabilized at tree levelUplift susy vacuum to dS, models of inflation: problemsNo-go theorem modular inflation: fluxes, D6/O6 Hertzberg,
Kachru, Taylor, Tegmark
Way-out: geometric fluxes, NS5-branes, non-geometric fluxes
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 9: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/9.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Motivation
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
stabilization Kahler moduli through non-perturbative effectsuplift susy vacuum to dS, models of inflation
Type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds:Derendinger et al., DeWolfe et al., Camara et al.
All moduli can be stabilized at tree levelUplift susy vacuum to dS, models of inflation: problemsNo-go theorem modular inflation: fluxes, D6/O6 Hertzberg,
Kachru, Taylor, Tegmark
Way-out: geometric fluxes, NS5-branes, non-geometric fluxes
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 10: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/10.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Motivation
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
stabilization Kahler moduli through non-perturbative effectsuplift susy vacuum to dS, models of inflation
Type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds:Derendinger et al., DeWolfe et al., Camara et al.
All moduli can be stabilized at tree levelUplift susy vacuum to dS, models of inflation: problemsNo-go theorem modular inflation: fluxes, D6/O6 Hertzberg,
Kachru, Taylor, Tegmark
Way-out: geometric fluxes, NS5-branes, non-geometric fluxes
Geometric fluxes: deviation from Calabi-Yau
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 11: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/11.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Motivation
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
stabilization Kahler moduli through non-perturbative effectsuplift susy vacuum to dS, models of inflation
Type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds:Derendinger et al., DeWolfe et al., Camara et al.
All moduli can be stabilized at tree levelUplift susy vacuum to dS, models of inflation: problemsNo-go theorem modular inflation: fluxes, D6/O6 Hertzberg,
Kachru, Taylor, Tegmark
Way-out: geometric fluxes, NS5-branes, non-geometric fluxes
Geometric fluxes: deviation from Calabi-Yau
Other application: type IIA on AdS4 × CP3
AdS/CFT Aharony, Bergman, Jafferis, Maldacena M2-branes
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Type IIA susy vacua with AdS4 space-time
Lust, Tsimpis
RR-forms: Romans mass F0 = m, F2, F4,NSNS 3-form: H , dilaton: Φ
SU(3)-structure: J , Ω
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 13: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/13.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Type IIA susy vacua with AdS4 space-time
Lust, Tsimpis
Solution susy equations:
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 14: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/14.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Type IIA susy vacua with AdS4 space-time
Lust, Tsimpis
Solution susy equations:
Constant warp factor
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 15: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/15.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Type IIA susy vacua with AdS4 space-time
Lust, Tsimpis
Solution susy equations:
Constant warp factorGeometric flux i.e. non-zero torsion classes:
dJ =3
2Im(W1Ω
∗) + W4 ∧ J + W3
dΩ = W1J ∧ J + W2 ∧ J + W∗
5 ∧ Ω
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 16: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/16.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Type IIA susy vacua with AdS4 space-time
Lust, Tsimpis
Solution susy equations:
Constant warp factorGeometric flux i.e. non-zero torsion classes:
dJ =3
2Im(W1Ω
∗)
dΩ = W1J ∧ J + W2 ∧ Jwith
W1 = −4i
9eΦf
W2 = −ieΦF ′
2
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 17: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/17.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Type IIA susy vacua with AdS4 space-time
Lust, Tsimpis
Solution susy equations:
Constant warp factorGeometric flux i.e. non-zero torsion classes:
dJ =3
2Im(W1Ω
∗)
dΩ = W1J ∧ J + W2 ∧ Jwith
W1 = −4i
9eΦf
W2 = −ieΦF ′
2
Form-fluxes: AdS4 superpotential W :
H =2m
5eΦReΩ
F2 =f
9J + F ′
2
F4 = fvol4 +3m
10J ∧ J
∇µζ− =1
2Wγµζ+ definition
Weiθ = −1
5eΦm +
i
3eΦf
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Bianchi identities
Automatically satisfied except for
dF2 + HF0 = −j6
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Bianchi identities
Automatically satisfied except for
dF2 + HF0 = −j6
Source j6 (O6/D6) must be calibrated (here SLAG):
j6 ∧ J = 0 j6 ∧ ReΩ = 0
4 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 20: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/20.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Bianchi identities
Automatically satisfied except for
dF2 + HF0 = −j6
Source j6 (O6/D6) must be calibrated (here SLAG):
j6 ∧ J = 0 j6 ∧ ReΩ = 0 ⇒ j6 = −2
5e−ΦµReΩ + w3
w3 simple (1,2)+(2,1)
4 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 21: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/21.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Bianchi identities
Automatically satisfied except for
dF2 + HF0 = −j6
Source j6 (O6/D6) must be calibrated (here SLAG):
j6 ∧ J = 0 j6 ∧ ReΩ = 0 ⇒ j6 = −2
5e−ΦµReΩ + w3
w3 simple (1,2)+(2,1)
µ > 0: net orientifold charge, µ < 0: net D-brane charge
4 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 22: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/22.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Bianchi identities
Automatically satisfied except for
dF2 + HF0 = −j6
Source j6 (O6/D6) must be calibrated (here SLAG):
j6 ∧ J = 0 j6 ∧ ReΩ = 0 ⇒ j6 = −2
5e−ΦµReΩ + w3
w3 simple (1,2)+(2,1)
µ > 0: net orientifold charge, µ < 0: net D-brane charge
Bianchi:
e2Φm2 = µ +5
16
(3|W1|2 − |W2|2
)≥ 0
w3 = −ie−ΦdW2
∣∣∣(2,1)+(1,2)
4 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 23: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/23.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Smeared orientifolds
Warp factor constant ⇒ only smeared orientifolds
5 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 24: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/24.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Smeared orientifolds
Warp factor constant ⇒ only smeared orientifolds
Solutions without orientifolds:
5 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 25: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/25.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Smeared orientifolds
Warp factor constant ⇒ only smeared orientifolds
Solutions without orientifolds: possible!
No-go theorem Maldacena,Nunez only for Minkowski, not AdS4
5 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 26: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/26.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Smeared orientifolds
Warp factor constant ⇒ only smeared orientifolds
Solutions without orientifolds: possible!
No-go theorem Maldacena,Nunez only for Minkowski, not AdS4
We will introduce orientifolds to obtain N = 1
5 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 27: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/27.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Smeared orientifolds
Warp factor constant ⇒ only smeared orientifolds
Solutions without orientifolds: possible!
No-go theorem Maldacena,Nunez only for Minkowski, not AdS4
We will introduce orientifolds to obtain N = 1
Smearing:
j6 = TO6 δ(x4, x5, x6) dx4 ∧ dx5 ∧ dx6 → TOp c dx4 ∧ dx5 ∧ dx6
5 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 28: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/28.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Smeared orientifolds
Warp factor constant ⇒ only smeared orientifolds
Solutions without orientifolds: possible!
No-go theorem Maldacena,Nunez only for Minkowski, not AdS4
We will introduce orientifolds to obtain N = 1
Smearing:
j6 = TO6 δ(x4, x5, x6) dx4 ∧ dx5 ∧ dx6 → TOp c dx4 ∧ dx5 ∧ dx6
We will still associate orientifold involution:
O6 : dx4 → −dx4 , dx5 → −dx5 , dx6 → −dx6
5 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
![Page 29: The effective theory of type IIA AdS4 compactifications on ...conf.itp.phys.ethz.ch/prestrings08/talks/adseffective.pdfBased on: 0804.0614 (PK, Tsimpis, Lu¨st), 0806.3458 (Caviezel,](https://reader033.fdocuments.in/reader033/viewer/2022041415/5e1a9f73a06b877d807a9223/html5/thumbnails/29.jpg)
Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Smeared orientifolds
Warp factor constant ⇒ only smeared orientifolds
Solutions without orientifolds: possible!
No-go theorem Maldacena,Nunez only for Minkowski, not AdS4
We will introduce orientifolds to obtain N = 1
Smearing:
j6 = TO6 δ(x4, x5, x6) dx4 ∧ dx5 ∧ dx6 → TOp c dx4 ∧ dx5 ∧ dx6
We will still associate orientifold involution:
O6 : dx4 → −dx4 , dx5 → −dx5 , dx6 → −dx6
Write j6 as sum of decomposable forms
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Smeared orientifolds
Warp factor constant ⇒ only smeared orientifolds
Solutions without orientifolds: possible!
No-go theorem Maldacena,Nunez only for Minkowski, not AdS4
We will introduce orientifolds to obtain N = 1
Smearing:
j6 = TO6 δ(x4, x5, x6) dx4 ∧ dx5 ∧ dx6 → TOp c dx4 ∧ dx5 ∧ dx6
We will still associate orientifold involution:
O6 : dx4 → −dx4 , dx5 → −dx5 , dx6 → −dx6
Write j6 as sum of decomposable formsB, H,F2, ReΩ odd, F0, F4, ImΩ even
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Calabi-Yau solution
Calabi-Yau solution Acharya, Benini, Valandro
j6 = −2
5e−ΦµReΩ + w3
e2Φm2 = µ +5
16
(3|W1|2 − |W2|2
)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Calabi-Yau solution
Calabi-Yau solution Acharya, Benini, Valandro
Put W1 = 0,W2 = 0 ⇒ w3 = 0
j6 = −2
5e−ΦµReΩ + w3
e2Φm2 = µ +5
16
(3|W1|2 − |W2|2
)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Calabi-Yau solution
Calabi-Yau solution Acharya, Benini, Valandro
Put W1 = 0,W2 = 0 ⇒ w3 = 0
j6 = −2
5e−ΦµReΩ
e2Φm2 = µ
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Calabi-Yau solution
Calabi-Yau solution Acharya, Benini, Valandro
Put W1 = 0,W2 = 0 ⇒ w3 = 0
j6 = −2
5e−ΦµReΩ
e2Φm2 = µ
Torus orientifolds
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
What about equations of motion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
With sources: PK, Tsimpis 0706.1244Under mild conditions (subtleties time direction):
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
What about equations of motion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
With sources: PK, Tsimpis 0706.1244Under mild conditions (subtleties time direction):
Bulk supersymmetry conditions
Bianchi identities form-fields with source
Supersymmetry conditions source = generalized calibrationconditions PK; Martucci, Smyth
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
What about equations of motion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
With sources: PK, Tsimpis 0706.1244Under mild conditions (subtleties time direction):
Bulk supersymmetry conditions
Bianchi identities form-fields with source
Supersymmetry conditions source = generalized calibrationconditions PK; Martucci, Smyth
imply
Einstein equations with source
Dilaton equation of motion with source
Form field equations of motion
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Some older solutions
Nearly-Kahler: only W1 6= 0 Behrndt, Cvetic
SU(2)×SU(2) and the coset spaces G2
SU(3) ,Sp(2)
S(U(2)×U(1)) ,SU(3)
U(1)×U(1)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Some older solutions
Nearly-Kahler: only W1 6= 0 Behrndt, Cvetic
SU(2)×SU(2) and the coset spaces G2
SU(3) ,Sp(2)
S(U(2)×U(1)) ,SU(3)
U(1)×U(1)
Iwasawa manifold Lust, Tsimpis:
A certain nilmanifold or twisted torus= a group manifold associated to a nilpotent algebraLeft-invariant forms ei obeying Maurer-Cartan relation:
dei = −1
2f i
jkej ∧ ek
Singular limit of T2 bundle over K3
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Group manifolds
SU(3)-structures with J,Ω constant in terms of left-invariant formson a group manifold: fertile source of examples
dei = −1
2f i
jkej ∧ ek
⇒ algebraic relations
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Group manifolds
SU(3)-structures with J,Ω constant in terms of left-invariant formson a group manifold: fertile source of examples
dei = −1
2f i
jkej ∧ ek
⇒ algebraic relations
SU(2)×SU(2)
Nilmanifolds (divide discrete group: compact)
Only Iwasawa & torusNeeds smeared orientifolds
Solvmanifolds (compact?): no solution
Extend to cosets
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Coset manifolds
Tomasiello: W2 6= 0 on two examples Sp(2)S(U(2)×U(1)) ,
SU(3)U(1)×U(1)
PK, Lust, Tsimpis: classification
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Coset manifolds
Tomasiello: W2 6= 0 on two examples Sp(2)S(U(2)×U(1)) ,
SU(3)U(1)×U(1)
PK, Lust, Tsimpis: classification
Maurer-Cartan more complicated:
dei = −1
2f i
jkej ∧ ek − f iajω
a ∧ ej
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Coset manifolds
Tomasiello: W2 6= 0 on two examples Sp(2)S(U(2)×U(1)) ,
SU(3)U(1)×U(1)
PK, Lust, Tsimpis: classification
Maurer-Cartan more complicated:
dei = −1
2f i
jkej ∧ ek − f iajω
a ∧ ej
Condition left-invariance p-form φ: constant and components
f ja[i1φi2...ip]j = 0
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Type IIA AdS4 susy vacua on coset manifolds
PK, Lust, Tsimpis 0804.0614
SU(2)×SU(2) SU(3)U(1)×U(1)
Sp(2)S(U(2)×U(1))
G2SU(3)
SU(3)×U(1)SU(2)
# of parameters 2 4 4 3 2 4W2 6= 0 No Yes Yes Yes No Yes
j6 ∝ ReΩ Yes No Yes Yes Yes No
Note: geometric flux: W1 6= 0,W2 6= 0
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Type IIA AdS4 susy vacua on coset manifolds
PK, Lust, Tsimpis 0804.0614
SU(2)×SU(2) SU(3)U(1)×U(1)
Sp(2)S(U(2)×U(1))
G2SU(3)
SU(3)×U(1)SU(2)
# of parameters 2 4 4 3 2 4W2 6= 0 No Yes Yes Yes No Yes
j6 ∝ ReΩ Yes No Yes Yes Yes No
Note: geometric flux: W1 6= 0,W2 6= 0
Parameters:
Geometric: scale and shape
J = ae12 + be34 + ce56
Scale a and shape ρ = b/a, σ = c/a:
Orientifold charge µ
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalization
SU(3)×SU(3) susy ansatz: 10d → 4d
ǫ1 =ζ+ ⊗ η(1)+ + (c.c.)
ǫ2 =ζ+ ⊗ η(2)∓ + (c.c.)
for IIA/IIB
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalization
SU(3)×SU(3) susy ansatz: 10d → 4d
ǫ1 =ζ+ ⊗ η(1)+ + (c.c.)
ǫ2 =ζ+ ⊗ η(2)∓ + (c.c.)
for IIA/IIB
Strict SU(3)-structure: η(2)+ = eiθη
(1)+ Type: (0,3)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalization
SU(3)×SU(3) susy ansatz: 10d → 4d
ǫ1 =ζ+ ⊗ η(1)+ + (c.c.)
ǫ2 =ζ+ ⊗ η(2)∓ + (c.c.)
for IIA/IIB
Strict SU(3)-structure: η(2)+ = eiθη
(1)+ Type: (0,3)
Static SU(2)-structure: η(1), η(2) orthogonal everywhereType: (2,1)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalization
SU(3)×SU(3) susy ansatz: 10d → 4d
ǫ1 =ζ+ ⊗ η(1)+ + (c.c.)
ǫ2 =ζ+ ⊗ η(2)∓ + (c.c.)
for IIA/IIB
Strict SU(3)-structure: η(2)+ = eiθη
(1)+ Type: (0,3)
Static SU(2)-structure: η(1), η(2) orthogonal everywhereType: (2,1)intermediate SU(2)-structure: η(1), η(2) fixed angle,but neither a zero angle nor a right angle Type: (0,1)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalization
SU(3)×SU(3) susy ansatz: 10d → 4d
ǫ1 =ζ+ ⊗ η(1)+ + (c.c.)
ǫ2 =ζ+ ⊗ η(2)∓ + (c.c.)
for IIA/IIB
Strict SU(3)-structure: η(2)+ = eiθη
(1)+ Type: (0,3)
Static SU(2)-structure: η(1), η(2) orthogonal everywhereType: (2,1)intermediate SU(2)-structure: η(1), η(2) fixed angle,but neither a zero angle nor a right angle Type: (0,1)dynamic SU(3)×SU(3)-structure: angle changes, type may change
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
SU(3)×SU(3)-structure susy equations
Grana, Minasian, Petrini, Tomasiello
dH
(e4A−ΦImΨ1
)= 3e3A−ΦIm(W ∗Ψ2) + e4AF
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
dH
[e3A−ΦIm(W ∗Ψ2)
]= 0
dH
(e2A−ΦReΨ1
)= 0 .
with
dH = d + H∧Ψ1 = Ψ∓ , Ψ2 = Ψ±
/Ψ+ =8
|η(1)||η(2)|η(1)+ ⊗ η
(2)†+ , /Ψ− =
8
|η(1)||η(2)|η(1)+ ⊗ η
(2)†− .
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
SU(3)×SU(3)-structure susy equations
Grana, Minasian, Petrini, Tomasiello
dH
(e4A−ΦImΨ1
)= 3e3A−ΦIm(W ∗Ψ2) + e4AF
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
dH
[e3A−ΦIm(W ∗Ψ2)
]= 0
dH
(e2A−ΦReΨ1
)= 0 .
Example: strict SU(3)-structure in IIA
Ψ1 = Ψ− = −Ω , Ψ2 = Ψ+ = e−iθeiJ
leads to susy equations Lust, Tsimpis
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
SU(3)×SU(3)-structure susy equations
Grana, Minasian, Petrini, Tomasiello
dH
(e4A−ΦImΨ1
)= 3e3A−ΦIm(W ∗Ψ2) + e4AF
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
dH
[e3A−ΦIm(W ∗Ψ2)
]= 0
dH
(e2A−ΦReΨ1
)= 0 .
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: no-go theorems
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: no-go theorems
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: no-go theorems
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: no-go theorems
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0But we also need 〈Ψ1, Ψ1〉 = −8ivol 6= 0with the Mukai pairing 〈φ1, φ2〉 = φ1 ∧ α(φ2)|top
and α reverses the indices of a form
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: no-go theorems
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0But we also need 〈Ψ1, Ψ1〉 = −8ivol 6= 0⇒ impossible
14 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: no-go theorems
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0But we also need 〈Ψ1, Ψ1〉 = −8ivol 6= 0⇒ impossible
Intermediate SU(2)-structure:
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: no-go theorems
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0But we also need 〈Ψ1, Ψ1〉 = −8ivol 6= 0⇒ impossible
Intermediate SU(2)-structure:Type: (0,1) ⇒ general form e2A−ΦΨ1 = ceiω+b
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: no-go theorems
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0But we also need 〈Ψ1, Ψ1〉 = −8ivol 6= 0⇒ impossible
Intermediate SU(2)-structure:Type: (0,1) ⇒ general form e2A−ΦΨ1 = ceiω+b
⇒ Rec = 0 , cω exact
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: no-go theorems
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0But we also need 〈Ψ1, Ψ1〉 = −8ivol 6= 0⇒ impossible
Intermediate SU(2)-structure:Type: (0,1) ⇒ general form e2A−ΦΨ1 = ceiω+b
⇒ Rec = 0 , cω exactImc〈e2A−ΦΨ1, e
2A−Φ, Ψ1〉 = 83! (cω)3
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: no-go theorems
dH
[e3A−ΦRe(W ∗Ψ2)
]= 2|W |2e2A−ΦReΨ1
Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0But we also need 〈Ψ1, Ψ1〉 = −8ivol 6= 0⇒ impossible
Intermediate SU(2)-structure:Type: (0,1) ⇒ general form e2A−ΦΨ1 = ceiω+b
⇒ Rec = 0 , cω exactImc〈e2A−ΦΨ1, e
2A−Φ, Ψ1〉 = 83! (cω)3
⇒ c not everywhere non-vanishing
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: conclusion of no-go theorems
Type IIB: only static SU(2) or dynamic SU(3)×SU(3) that changestype to static SU(2) at least somewhere is possible
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: conclusion of no-go theorems
Type IIB: only static SU(2) or dynamic SU(3)×SU(3) that changestype to static SU(2) at least somewhere is possible
Example: type IIB static SU(2) on nilmanifold 5.1 (table of Grana,
Minasian, Petrini, Tomasiello)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Generalizations: conclusion of no-go theorems
Type IIB: only static SU(2) or dynamic SU(3)×SU(3) that changestype to static SU(2) at least somewhere is possible
Example: type IIB static SU(2) on nilmanifold 5.1 (table of Grana,
Minasian, Petrini, Tomasiello)
Type IIA: only strict SU(3) or dynamic SU(3)×SU(3) for whichd
(e2A−Φη(2)†η(1)
)6= 0
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Low energy effective theory: nilmanifolds
Torus (IIA), nilmanifold 5.1 (IIB), Iwasawa (IIA): T-dual to eachother
Calculation mass spectrum
Using effective 4D sugra techniquesFor Iwasawa & torus: direct KK reduction of equations of motionPerfect agreement
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Low energy effective theory: nilmanifolds
Torus (IIA), nilmanifold 5.1 (IIB), Iwasawa (IIA): T-dual to eachother
Calculation mass spectrum
Using effective 4D sugra techniquesFor Iwasawa & torus: direct KK reduction of equations of motionPerfect agreement
Result for M2/|W |2
Complex structure −2, −2, −2Kahler & dilaton 70, 18, 18, 18
Three axions of δC3 0, 0, 0δB & one more axion 88, 10, 10, 10
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Issue: decoupling KK tower
Consistency approximation: gs ≪ 1, ls/Lint ≪ 1
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Issue: decoupling KK tower
Consistency approximation: gs ≪ 1, ls/Lint ≪ 1√
17 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Issue: decoupling KK tower
Consistency approximation: gs ≪ 1, ls/Lint ≪ 1√
Decoupling KK modes: |W |2L2int =
e2Φm2L2
int
25 +e2Φf2L2
int
9 ≪ 1
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Issue: decoupling KK tower
Consistency approximation: gs ≪ 1, ls/Lint ≪ 1√
Decoupling KK modes: |W |2L2int =
e2Φm2L2
int
25 +e2Φf2L2
int
9 ≪ 1
Can be tuned with orientifold charge µ:
e2Φm2 = µ +5
16
(3|W1|
2 − |W2|2) ≈ 0
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Issue: decoupling KK tower
Consistency approximation: gs ≪ 1, ls/Lint ≪ 1√
Decoupling KK modes: |W |2L2int =
e2Φm2L2
int
25 +e2Φf2L2
int
9 ≪ 1
eΦfLint ∝ W1Lint ≪ 1:
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Issue: decoupling KK tower
Consistency approximation: gs ≪ 1, ls/Lint ≪ 1√
Decoupling KK modes: |W |2L2int =
e2Φm2L2
int
25 +e2Φf2L2
int
9 ≪ 1
eΦfLint ∝ W1Lint ≪ 1:nearly Calabi-Yau
17 / 24
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Issue: decoupling KK tower
Consistency approximation: gs ≪ 1, ls/Lint ≪ 1√
Decoupling KK modes: |W |2L2int =
e2Φm2L2
int
25 +e2Φf2L2
int
9 ≪ 1
eΦfLint ∝ W1Lint ≪ 1:nearly Calabi-YauCan be arranged for torus, Iwasawa, nilmanifold 5.1
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Issue: decoupling KK tower
Consistency approximation: gs ≪ 1, ls/Lint ≪ 1√
Decoupling KK modes: |W |2L2int =
e2Φm2L2
int
25 +e2Φf2L2
int
9 ≪ 1
eΦfLint ∝ W1Lint ≪ 1:nearly Calabi-YauCan be arranged for torus, Iwasawa, nilmanifold 5.1Harder for coset examples
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Low energy theory: superpotential and Kahlerpotential
We use the superpotential of Grana, Louis, Waldram;
Benmachiche, Grimm; PK, Martucci
Generalizes Gukov, Vafa, Witten
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Low energy theory: superpotential and Kahlerpotential
We use the superpotential of Grana, Louis, Waldram;
Benmachiche, Grimm; PK, Martucci
Generalizes Gukov, Vafa, Witten
WE =−i
4κ210
∫
M
〈Ψ2eδB
︸ ︷︷ ︸
Z
, F + i dH
(eδBe−ΦImΨ1 − iδC︸ ︷︷ ︸
T
)〉
Z,T holomorphic coordinates,
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Low energy theory: superpotential and Kahlerpotential
We use the superpotential of Grana, Louis, Waldram;
Benmachiche, Grimm; PK, Martucci
Generalizes Gukov, Vafa, Witten
WE =−i
4κ210
∫
M
〈Ψ2eδB
︸ ︷︷ ︸
Z
, F + i dH
(eδBe−ΦImΨ1 − iδC︸ ︷︷ ︸
T
)〉
Z,T holomorphic coordinates, and the Kahler potential
K = − ln i
∫
M
〈Z, Z〉 − 2 ln i
∫
M
〈e−ΦΨ1, e−ΦΨ1〉 + 3 ln(8κ2
10M2P )
where e−ΦeδBΨ1: function of ReT = e−ΦeδBImΨ1 Hitchin.
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Superpotential and Kahler potential: SU(3)
Specialized to SU(3): Ψ1 = −Ω , Ψ2 = e−iθeiJ
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Superpotential and Kahler potential: SU(3)
Specialized to SU(3): Ψ1 = −Ω , Ψ2 = e−iθeiJ
Superpotential:
WE =−ie−iθ
4κ210
∫
M
〈ei(J−iδB), F − idH
(e−ΦImΩ + iδC3
)〉
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Superpotential and Kahler potential: SU(3)
Specialized to SU(3): Ψ1 = −Ω , Ψ2 = e−iθeiJ
Superpotential:
WE =−ie−iθ
4κ210
∫
M
〈ei(J−iδB), F − idH
(e−ΦImΩ + iδC3
)〉
Kahler potential:
K = − ln
∫
M
4
3J3−2 ln
∫
M
2 e−ΦImΩ∧e−ΦReΩ+3 ln(8κ210M
2P ) ,
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Superpotential and Kahler potential: SU(3)
Specialized to SU(3): Ψ1 = −Ω , Ψ2 = e−iθeiJ
Superpotential:
WE =−ie−iθ
4κ210
∫
M
〈ei(J−iδB), F − idH
(e−ΦImΩ + iδC3
)〉
Kahler potential:
K = − ln
∫
M
4
3J3−2 ln
∫
M
2 e−ΦImΩ∧e−ΦReΩ+3 ln(8κ210M
2P ) ,
Expansion:
Jc = J − iδB = (ki − ibi)Y(2−)i = tiY
(2−)i
e−ΦImΩ + iδC3 = (ui + ici)e−ΦY(3+)i = zie−ΦY
(3+)i
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Comments
We used the same superpotential and Kahler potential for staticSU(2)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Comments
We used the same superpotential and Kahler potential for staticSU(2)
F-flatness conditions:
DiWE = ∂iWE + ∂iKWE = 0
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Comments
We used the same superpotential and Kahler potential for staticSU(2)
F-flatness conditions:
DiWE = ∂iWE + ∂iKWE = 0
PK, Martucci:
If W = M−1P 〈WE〉 6= 0 i.e. AdS4 compactification
F-flatness conditions ⇔ 10D susy conditions vacuum
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Comments
We used the same superpotential and Kahler potential for staticSU(2)
F-flatness conditions:
DiWE = ∂iWE + ∂iKWE = 0
PK, Martucci:
If W = M−1P 〈WE〉 6= 0 i.e. AdS4 compactification
F-flatness conditions ⇔ 10D susy conditions vacuumIf W = 0 i.e. Minkowski compactificationF-flatness & D-flatness conditions ⇔ 10D susy conditions vacuum
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Comments
We used the same superpotential and Kahler potential for staticSU(2)
F-flatness conditions:
DiWE = ∂iWE + ∂iKWE = 0
PK, Martucci:
If W = M−1P 〈WE〉 6= 0 i.e. AdS4 compactification
F-flatness conditions ⇔ 10D susy conditions vacuumIf W = 0 i.e. Minkowski compactificationF-flatness & D-flatness conditions ⇔ 10D susy conditions vacuum
The scalar potential is
V = M−2P eK
(KiDiWEDW∗
E − 3|WE|2)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Comments
We used the same superpotential and Kahler potential for staticSU(2)
F-flatness conditions:
DiWE = ∂iWE + ∂iKWE = 0
PK, Martucci:
If W = M−1P 〈WE〉 6= 0 i.e. AdS4 compactification
F-flatness conditions ⇔ 10D susy conditions vacuumIf W = 0 i.e. Minkowski compactificationF-flatness & D-flatness conditions ⇔ 10D susy conditions vacuum
The scalar potential is
V = M−2P eK
(KiDiWEDW∗
E − 3|WE|2)
Mass spectrum from quadratic terms
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Low energy theory: cosets
For all cosets (but not for SU(2)×SU(2)): all moduli stabilized attree level
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Low energy theory: cosets
For all cosets (but not for SU(2)×SU(2)): all moduli stabilized attree level
Example Sp(2)S(U(2)×U(1))
2 4 6 8 10
5
10
15
20
25
µ
M2/|W |2
(a) σ = 1
2 4 6 8 10 12
5
10
15
20
25
µ
M2/|W |2
(b) σ = 25
Movie
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Low energy theory: cosets
For all cosets (but not for SU(2)×SU(2)): all moduli stabilized attree level
Example Sp(2)S(U(2)×U(1))
2 4 6 8 10
5
10
15
20
25
µ
M2/|W |2
(a) σ = 1
2 4 6 8 10 12
5
10
15
20
25
µ
M2/|W |2
(b) σ = 25
Movie
µ big enough: all mass-squared positive
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Nearly-Calabi Yau limit
Important decoupling KK-modes
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Nearly-Calabi Yau limit
Important decoupling KK-modes
Example Sp(2)S(U(2)×U(1)) ⇒ analytic continuation to negative σ = −2
Twistor bundle description more appropriate Xu
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Nearly-Calabi Yau limit
Important decoupling KK-modes
Example Sp(2)S(U(2)×U(1)) ⇒ analytic continuation to negative σ = −2
Twistor bundle description more appropriate Xu
10 20 30 40 50
20
40
60
80
100
120
µ
M2/|W |2
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Nearly-Calabi Yau limit
Important decoupling KK-modes
Example Sp(2)S(U(2)×U(1)) ⇒ analytic continuation to negative σ = −2
Twistor bundle description more appropriate Xu
10 20 30 40 50
20
40
60
80
100
120
µ
M2/|W |2
Light modes: M2/|W |2 = (−38/49, 130/49)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Inflation
No-go theorem modular inflation IIAHertzberg, Kachru, Taylor, Tegmark
Dependence on volume-modulus ρ and dilaton-modulus τ
Ingredients: form-fluxes, D6-branes & O6-planes
ǫ > 27/13
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Inflation
No-go theorem modular inflation IIAHertzberg, Kachru, Taylor, Tegmark
Dependence on volume-modulus ρ and dilaton-modulus τ
Ingredients: form-fluxes, D6-branes & O6-planes
ǫ > 27/13
Way out: geometric flux, NS5-branes, other branes, non-geometricflux
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Inflation
No-go theorem modular inflation IIAHertzberg, Kachru, Taylor, Tegmark
Dependence on volume-modulus ρ and dilaton-modulus τ
Ingredients: form-fluxes, D6-branes & O6-planes
ǫ > 27/13
Way out: geometric flux, NS5-branes, other branes, non-geometricflux
potential geometric flux Vf > 0 ⇔ R < 0
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Inflation
No-go theorem modular inflation IIAHertzberg, Kachru, Taylor, Tegmark
Dependence on volume-modulus ρ and dilaton-modulus τ
Ingredients: form-fluxes, D6-branes & O6-planes
ǫ > 27/13
Way out: geometric flux, NS5-branes, other branes, non-geometricflux
potential geometric flux Vf > 0 ⇔ R < 0not possible for G2
SU(3), possible for all other cosets and SU(2)×SU(2)
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Conclusions
Tractable models with geometric flux: nilmanifold & coset models
Other models: e.g. twistor bundles with negative σ
Uplift: uplifting term or look for dS minimum e.g. Silverstein
Study inflation
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)
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Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions
Conclusions
Tractable models with geometric flux: nilmanifold & coset models
Other models: e.g. twistor bundles with negative σ
Uplift: uplifting term or look for dS minimum e.g. Silverstein
Study inflation
The
end. ..T
he end. . .The end
.
. .
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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)