flux compactifications on coset manifolds and applicationskoerber/utrecht2009.pdf · 2009. 9....
TRANSCRIPT
![Page 1: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/1.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Flux Compactifications on Coset Manifolds and Applications
Based on: 0707.1038, 0710.5530 (PK, Martucci), 0706.1244, 0804.0614 (PK, Tsimpis, Lust), 0806.3458,0812.3551 (Caviezel, PK, Kors, Lust, Tsimpis, Wrase, Zagermann), work in progress (PK)
http://wwwth.mppmu.mpg.de/members/koerber/talks.html
Paul Koerber
Max-Planck-Institut fur Physik, Munich
Utrecht, 19 March 2009
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 2: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/2.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
2 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 3: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/3.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
Type IIB orientifold with D3/D7-branes on conformal CY:well-studied
2 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 4: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/4.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 5: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/5.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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
Generically: fluxes =⇒ not CY
2 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 6: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/6.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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
Generically: fluxes =⇒ not CY geometric fluxes
2 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 7: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/7.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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
Generically: fluxes =⇒ not CY geometric fluxes
Models with SU(3)-structureInteresting class: type IIA compactifications with AdS4 space-time
Models with SU(3)×SU(3)-structure generalized geometry
2 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 8: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/8.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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
Generically: fluxes =⇒ not CY geometric fluxes
Models with SU(3)-structureInteresting class: type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds (DeWolfe et al.): allmoduli can be stabilized at tree level
Models with SU(3)×SU(3)-structure generalized geometry
2 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 9: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/9.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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
Generically: fluxes =⇒ not CY geometric fluxes
Models with SU(3)-structureInteresting class: type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds (DeWolfe et al.): allmoduli can be stabilized at tree levelIn this talk: models on coset manifolds with geometric fluxes
Models with SU(3)×SU(3)-structure generalized geometry
2 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 10: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/10.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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
Generically: fluxes =⇒ not CY geometric fluxes
Models with SU(3)-structureInteresting class: type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds (DeWolfe et al.): allmoduli can be stabilized at tree levelIn this talk: models on coset manifolds with geometric fluxesUplift susy vacuum to dS, models of inflation: problems
Models with SU(3)×SU(3)-structure generalized geometry
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 11: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/11.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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
Generically: fluxes =⇒ not CY geometric fluxes
Models with SU(3)-structureInteresting class: type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds (DeWolfe et al.): allmoduli can be stabilized at tree levelIn this talk: models on coset manifolds with geometric fluxesUplift susy vacuum to dS, models of inflation: problems
New application to AdS4/CFT Aharony, Bergman, Jafferis,
MaldacenaModels with SU(3)×SU(3)-structure generalized geometry
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 12: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/12.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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
Generically: fluxes =⇒ not CY geometric fluxes
Models with SU(3)-structureInteresting class: type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds (DeWolfe et al.): allmoduli can be stabilized at tree levelIn this talk: models on coset manifolds with geometric fluxesUplift susy vacuum to dS, models of inflation: problems
New application to AdS4/CFT Aharony, Bergman, Jafferis,
MaldacenaModels with SU(3)×SU(3)-structure generalized geometry
Many properties can be proven in general for this class
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 13: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/13.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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
Generically: fluxes =⇒ not CY geometric fluxes
Models with SU(3)-structureInteresting class: type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds (DeWolfe et al.): allmoduli can be stabilized at tree levelIn this talk: models on coset manifolds with geometric fluxesUplift susy vacuum to dS, models of inflation: problems
New application to AdS4/CFT Aharony, Bergman, Jafferis,
MaldacenaModels with SU(3)×SU(3)-structure generalized geometry
Many properties can be proven in general for this classHard to find examples ’06: Grana, Minasian, Petrini, Tomasiello,’08: Andriot
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 14: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/14.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Motivation
Compactification: 10D → 4D, add RR, NSNS fluxes
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
Generically: fluxes =⇒ not CY geometric fluxes
Models with SU(3)-structureInteresting class: type IIA compactifications with AdS4 space-time
Early type IIA models on torus orientifolds (DeWolfe et al.): allmoduli can be stabilized at tree levelIn this talk: models on coset manifolds with geometric fluxesUplift susy vacuum to dS, models of inflation: problems
New application to AdS4/CFT Aharony, Bergman, Jafferis,
MaldacenaModels with SU(3)×SU(3)-structure generalized geometry
Many properties can be proven in general for this classHard to find examples ’06: Grana, Minasian, Petrini, Tomasiello,’08: Andriot
Non-geometry: Hull and others
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 15: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/15.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Compactification ansatz
We consider type IIA/IIB supergravity
Metric:
ds2 = e2A(y)g(4)µν(x)dxµdxν + gmn(y)dymdyn ,
with g(4) flat Minkowski or AdS4 metric, A warp factor
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 16: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/16.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Compactification ansatz
We consider type IIA/IIB supergravity
Metric:
ds2 = e2A(y)g(4)µν(x)dxµdxν + gmn(y)dymdyn ,
with g(4) flat Minkowski or AdS4 metric, A warp factor
RR-fluxes:
Democratic formalism: double fields, impose duality conditionCombine forms into one polyform
Ftot =∑
l
F(l) = F + e4Avol4 ∧ Fel , (Fel = ⋆6σ(F ))
with l even/odd in type IIA/IIB
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 17: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/17.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Supersymmetry ansatz I
N = 1 ansatz for the two Majorana-Weyl susy generatorsSU(3)-structure ansatz:
ǫ1 = ζ+ ⊗ η+ + ζ− ⊗ η− ,
ǫ2 = ζ+ ⊗ η∓ + ζ− ⊗ η± ,
ζ: 4d spinor characterizes preserved susy in 4dη: fixed 6d-spinor, property background
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 18: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/18.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Supersymmetry ansatz I
N = 1 ansatz for the two Majorana-Weyl susy generatorsSU(3)-structure ansatz:
ǫ1 = ζ+ ⊗ η+ + ζ− ⊗ η− ,
ǫ2 = ζ+ ⊗ η∓ + ζ− ⊗ η± ,
ζ: 4d spinor characterizes preserved susy in 4dη: fixed 6d-spinor, property background
Define (poly)forms
Ψ+ = −i
||η||2
∑
l even
(−1)l/2 1
l!η†+γi1...il
η+dxi1 ∧ . . . ∧ dxil = ceiJ
Ψ− =i
||η||21
3!η†−γi1...i3η+dx
i1 ∧ dxi2 ∧ dxi3 = iΩ
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 19: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/19.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Supersymmetry ansatz I
N = 1 ansatz for the two Majorana-Weyl susy generatorsSU(3)-structure ansatz:
ǫ1 = ζ+ ⊗ η+ + ζ− ⊗ η− ,
ǫ2 = ζ+ ⊗ η∓ + ζ− ⊗ η± ,
ζ: 4d spinor characterizes preserved susy in 4dη: fixed 6d-spinor, property background
Define (poly)forms
Ψ+ = −i
||η||2
∑
l even
(−1)l/2 1
l!η†+γi1...il
η+dxi1 ∧ . . . ∧ dxil = ceiJ
Ψ− =i
||η||21
3!η†−γi1...i3η+dx
i1 ∧ dxi2 ∧ dxi3 = iΩ
In the absence of fluxes, susy conditions: dJ = 0, dΩ = 0=⇒ SU(3)-holonomy i.e. CY=⇒ J Kahler-form, Ω holomorphic three-form
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 20: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/20.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Supersymmetry ansatz II
Most general N = 1 ansatz for susy generatorsSU(3)×SU(3)-structure ansatz:
ǫ1 = ζ+ ⊗ η(1)+ + ζ− ⊗ η
(1)− ,
ǫ2 = ζ+ ⊗ η(2)∓ + ζ− ⊗ η
(2)± ,
ζ: 4d spinor characterizes preserved susyη(1,2): fixed 6d-spinors, property background
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 21: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/21.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Supersymmetry ansatz II
Most general N = 1 ansatz for susy generatorsSU(3)×SU(3)-structure ansatz:
ǫ1 = ζ+ ⊗ η(1)+ + ζ− ⊗ η
(1)− ,
ǫ2 = ζ+ ⊗ η(2)∓ + ζ− ⊗ η
(2)± ,
ζ: 4d spinor characterizes preserved susyη(1,2): fixed 6d-spinors, property background
Define polyforms
Ψ+ = −i
||η||2
∑
l even
(−1)l/2 1
l!η(2)†+ γi1...il
η(1)+ dxi1 ∧ . . . ∧ dxil
Ψ− = −i
||η||2
∑
l odd
(−1)(l−1)/2 1
l!η(2)†− γi1...il
η(1)+ dxi1 ∧ dxi2 ∧ dxil
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 22: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/22.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Supersymmetry ansatz II
Most general N = 1 ansatz for susy generatorsSU(3)×SU(3)-structure ansatz:
ǫ1 = ζ+ ⊗ η(1)+ + ζ− ⊗ η
(1)− ,
ǫ2 = ζ+ ⊗ η(2)∓ + ζ− ⊗ η
(2)± ,
ζ: 4d spinor characterizes preserved susyη(1,2): fixed 6d-spinors, property background
Define polyforms
Ψ+ = −i
||η||2
∑
l even
(−1)l/2 1
l!η(2)†+ γi1...il
η(1)+ dxi1 ∧ . . . ∧ dxil
Ψ− = −i
||η||2
∑
l odd
(−1)(l−1)/2 1
l!η(2)†− γi1...il
η(1)+ dxi1 ∧ dxi2 ∧ dxil
These polyforms can be considered as spinors of TM ⊕ T ⋆MNot every polyform is related to a spinor bilinear: only pure spinorsType of the pure spinor: lowest dimension of the polyform
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 23: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/23.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Background susy conditions
Grana, Minasian, Petrini, Tomasiello
Susy conditions type II sugra:Gravitino’s
δψ1M =
(
∇M +1
4/HM
)
ǫ1 +1
16eΦ /Ftot ΓMΓ(10)ǫ
2 = 0
δψ2M =
(
∇M −1
4/HM
)
ǫ2 −1
16eΦσ(/Ftot) ΓMΓ(10)ǫ
1 = 0
Dilatino’s
δλ1 =
(
/∂Φ +1
2/H
)
ǫ1 +1
16eΦΓM /Ftot ΓMΓ(10)ǫ
2 = 0
δλ2 =
(
/∂Φ −1
2/H
)
ǫ2 −1
16eΦΓMσ(/Ftot) ΓMΓ(10)ǫ
1 = 0
σ: reverses order indices
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 24: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/24.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Background susy conditions
Grana, Minasian, Petrini, Tomasiello
Susy conditions type II sugra:Gravitino’s
δψ1M =
(
∇M +1
4/HM
)
ǫ1 +1
16eΦ /Ftot ΓMΓ(10)ǫ
2 = 0
δψ2M =
(
∇M −1
4/HM
)
ǫ2 −1
16eΦσ(/Ftot) ΓMΓ(10)ǫ
1 = 0
Dilatino’s
δλ1 =
(
/∂Φ +1
2/H
)
ǫ1 +1
16eΦΓM /Ftot ΓMΓ(10)ǫ
2 = 0
δλ2 =
(
/∂Φ −1
2/H
)
ǫ2 −1
16eΦΓMσ(/Ftot) ΓMΓ(10)ǫ
1 = 0
σ: reverses order indices=⇒ can be concisely rewritten as . . .
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 25: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/25.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Background susy conditions
Grana, Minasian, Petrini, Tomasiello
Susy equations in polyform notation:
dH
(
e4A−ΦReΨ1
)
= e4AFel ,
dH
(
e3A−ΦΨ2
)
= 0 ,
dH(e2A−ΦImΨ1) = 0 ,
for Minkowski.
Fel: external part polyform RR-fluxes, Φ: dilaton, A: warp factor,H NSNS 3-form, dH = d+H∧
Ψ1 = Ψ∓,Ψ2 = Ψ± for IIA/IIB
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 26: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/26.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Background susy conditions
Grana, Minasian, Petrini, Tomasiello
Susy equations in polyform notation:
dH
(
e4A−ΦReΨ1
)
= (3/R) e3A−ΦRe(eiθΨ2) + e4AFel ,
dH
(
e3A−ΦΨ2
)
= (2/R)i e2A−Φe−iθImΨ1 ,
dH(e2A−ΦImΨ1) = 0 ,
for AdS: ∇µζ− = ± e−iθ
2R γµζ+.
Fel: external part polyform RR-fluxes, Φ: dilaton, A: warp factor,H NSNS 3-form, dH = d+H∧
Ψ1 = Ψ∓,Ψ2 = Ψ± for IIA/IIB
6 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 27: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/27.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Generalized calibrations
hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci
Nice interpretation susy conditions in terms of generalizedcalibrations
7 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 28: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/28.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Generalized calibrations
hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci
Nice interpretation susy conditions in terms of generalizedcalibrations
Calibration form:mathematical tool to construct submanifolds with minimal volume
7 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 29: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/29.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Generalized calibrations
hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci
Nice interpretation susy conditions in terms of generalizedcalibrations
Calibration form:mathematical tool to construct submanifolds with minimal volumee.g. CY case:
eiJ ⇒ complex submanifoldsΩ ⇒ special Lagrangian
7 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 30: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/30.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Generalized calibrations
hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci
Nice interpretation susy conditions in terms of generalizedcalibrations
Calibration form:mathematical tool to construct submanifolds with minimal volumee.g. CY case:
eiJ ⇒ complex submanifoldsΩ ⇒ special Lagrangian
Generalized calibration form φ:constructs D-branes with minimal energy
7 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 31: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/31.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Generalized calibrations
hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci
Nice interpretation susy conditions in terms of generalizedcalibrations
Calibration form:mathematical tool to construct submanifolds with minimal volumee.g. CY case:
eiJ ⇒ complex submanifoldsΩ ⇒ special Lagrangian
Generalized calibration form φ:constructs D-branes with minimal energy
Calibrated D-brane (Σ,F) with F the world-volume gauge fieldmust satisfy
e−Φ√
g + F|Σ = φ|Σ ∧ eF
7 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 32: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/32.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Generalized calibrations
hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci
Nice interpretation susy conditions in terms of generalizedcalibrations
Calibration form:mathematical tool to construct submanifolds with minimal volumee.g. CY case:
eiJ ⇒ complex submanifoldsΩ ⇒ special Lagrangian
Generalized calibration form φ:constructs D-branes with minimal energy
Calibrated D-brane (Σ,F) with F the world-volume gauge fieldmust satisfy
e−Φ√
g + F|Σ = φ|Σ ∧ eF
Calibrated D-brane ⇔ supersymmetric D-brane
7 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 33: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/33.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Natural generalized calibration forms
Martucci, Smyth
Calibration forms are the polyforms:
ωsf = e4A−ΦReΨ1 ,
ωDWφ = e3A−ΦRe(eiφΨ2) ,
ωstring = e2A−ΦImΨ1 .
8 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 34: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/34.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Natural generalized calibration forms
Martucci, Smyth
Calibration forms are the polyforms:
ωsf = e4A−ΦReΨ1 ,
ωDWφ = e3A−ΦRe(eiφΨ2) ,
ωstring = e2A−ΦImΨ1 .
Good calibration forms must satisfy differential property, which isexactly provided by the bulk susy equations:
dH
(
e4A−ΦReΨ1
)
= e4AFel , space-filling D-brane
dH
(
e3A−ΦΨ2
)
= 0 , domain wall
dH(e2A−ΦImΨ1) = 0 , string-like D-brane
8 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 35: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/35.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Natural generalized calibration forms
Martucci, Smyth
Calibration forms are the polyforms:
ωsf = e4A−ΦReΨ1 ,
ωDWφ = e3A−ΦRe(eiφΨ2) ,
ωstring = e2A−ΦImΨ1 .
Good calibration forms must satisfy differential property, which isexactly provided by the bulk susy equations:
dH
(
e4A−ΦReΨ1
)
= e4AFel , space-filling D-brane
dH
(
e3A−ΦΨ2
)
= 0 , domain wall
dH(e2A−ΦImΨ1) = 0 , string-like D-brane
Spoiled in the AdS case:interpretation 0710.5530 PK, Martucci
8 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 36: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/36.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Sugra equations of motion
Solve the susy conditions, do we actually have solution equations ofmotion?
9 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 37: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/37.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Sugra equations of motion
Solve the susy conditions, do we actually have solution equations ofmotion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
Under mild conditions: susy and Bianchi & eom form fields =⇒ allothers eom
9 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 38: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/38.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Sugra equations of motion
Solve the susy conditions, do we actually have solution equations ofmotion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
Under mild conditions: susy and Bianchi & eom form fields =⇒ allothers eom
Let’s add sources: dF = jPK, Tsimpis 0706.1244
9 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 39: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/39.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Sugra equations of motion
Solve the susy conditions, do we actually have solution equations ofmotion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
Under mild conditions: susy and Bianchi & eom form fields =⇒ allothers eom
Let’s add sources: dF = jPK, Tsimpis 0706.1244Under mild conditions (subtleties time direction):
9 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 40: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/40.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Sugra equations of motion
Solve the susy conditions, do we actually have solution equations ofmotion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
Under mild conditions: susy and Bianchi & eom form fields =⇒ allothers eom
Let’s add sources: dF = jPK, Tsimpis 0706.1244Under mild conditions (subtleties time direction):
Bulk supersymmetry conditionsBianchi identities form-fields with sourceSupersymmetry conditions source = generalized calibrationconditions
9 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 41: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/41.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Sugra equations of motion
Solve the susy conditions, do we actually have solution equations ofmotion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
Under mild conditions: susy and Bianchi & eom form fields =⇒ allothers eom
Let’s add sources: dF = jPK, Tsimpis 0706.1244Under mild conditions (subtleties time direction):
Bulk supersymmetry conditionsBianchi identities form-fields with sourceSupersymmetry conditions source = generalized calibrationconditions
imply
Einstein equations with sourceDilaton equation of motion with sourceForm field equations of motion
9 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 42: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/42.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Problems constructing solutions
No-go theorem Maldacena, Nunez: Minkowski compactifications →negative-tension sources
10 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 43: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/43.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Problems constructing solutions
No-go theorem Maldacena, Nunez: Minkowski compactifications →negative-tension sources
Negative-tension sources in string theory: orientifolds
10 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 44: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/44.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Problems constructing solutions
No-go theorem Maldacena, Nunez: Minkowski compactifications →negative-tension sources
Negative-tension sources in string theory: orientifolds
Difficult to construct solutions with localized sources
10 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 45: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/45.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Problems constructing solutions
No-go theorem Maldacena, Nunez: Minkowski compactifications →negative-tension sources
Negative-tension sources in string theory: orientifolds
Difficult to construct solutions with localized sources→ smeared orientifolds
10 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 46: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/46.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Problems constructing solutions
No-go theorem Maldacena, Nunez: Minkowski compactifications →negative-tension sources
Negative-tension sources in string theory: orientifolds
Difficult to construct solutions with localized sources→ smeared orientifolds
AdS4 compactifications can avoid the no-go theorem!
10 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 47: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/47.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Problems constructing solutions
No-go theorem Maldacena, Nunez: Minkowski compactifications →negative-tension sources
Negative-tension sources in string theory: orientifolds
Difficult to construct solutions with localized sources→ smeared orientifolds
AdS4 compactifications can avoid the no-go theorem!
Bulk susy conditions for AdS4 compactifications impose: Caviezel,PK, Kors, Lust, Tsimpis, Zagermann
IIB: no strict SU(3)-structureIIA: no static SU(2)-structure (type (1,2))
10 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 48: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/48.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Problems constructing solutions
No-go theorem Maldacena, Nunez: Minkowski compactifications →negative-tension sources
Negative-tension sources in string theory: orientifolds
Difficult to construct solutions with localized sources→ smeared orientifolds
AdS4 compactifications can avoid the no-go theorem!
Bulk susy conditions for AdS4 compactifications impose: Caviezel,PK, Kors, Lust, Tsimpis, Zagermann
IIB: no strict SU(3)-structureIIA: no static SU(2)-structure (type (1,2))strict SU(3)-structure possible, but no non-constant warp factor(and thus no localized sources) unless Romans mass m = 0
10 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 49: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/49.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
SU(3)-structure AdS4 solutions
Ψ− = iΩ, Ψ+ = ceiJ
Susy conditions reduce to conditions of Lust,Tsimpis:
11 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 50: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/50.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
SU(3)-structure AdS4 solutions
Ψ− = iΩ, Ψ+ = ceiJ
Susy conditions reduce to conditions of Lust,Tsimpis:
Constant warp factor
11 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 51: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/51.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
SU(3)-structure AdS4 solutions
Ψ− = iΩ, Ψ+ = ceiJ
Susy conditions reduce to conditions of Lust,Tsimpis:
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
11 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 52: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/52.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
SU(3)-structure AdS4 solutions
Ψ− = iΩ, Ψ+ = ceiJ
Susy conditions reduce to conditions of Lust,Tsimpis:
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
11 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 53: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/53.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
SU(3)-structure AdS4 solutions
Ψ− = iΩ, Ψ+ = ceiJ
Susy conditions reduce to conditions of Lust,Tsimpis:
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
11 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 54: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/54.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
Automatically satisfied except for
dF2 +Hm = −j6
12 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 55: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/55.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
Automatically satisfied except for
dF2 +Hm = −j6
Source j6 (O6/D6) must be calibrated (here SLAG):
j6 ∧ J = 0 j6 ∧ ReΩ = 0
12 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 56: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/56.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
Automatically satisfied except for
dF2 +Hm = −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)
12 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 57: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/57.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
Automatically satisfied except for
dF2 +Hm = −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
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
Automatically satisfied except for
dF2 +Hm = −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)
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Summarizing
We are looking for a geometry that satisfies:
Only non-zero torsion classes W1, W2
dW2 ∧ J = 0e2Φm2 = µ + 5
16
(
3|W1|2 − |W2|
2)
≥ 0
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Summarizing
We are looking for a geometry that satisfies:
Only non-zero torsion classes W1, W2
dW2 ∧ J = 0e2Φm2 = µ + 5
16
(
3|W1|2 − |W2|
2)
≥ 0
If we want a solution without source term, we put µ = 0,dW2 ∝ ReΩ in the above
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Summarizing
We are looking for a geometry that satisfies:
Only non-zero torsion classes W1, W2
dW2 ∧ J = 0e2Φm2 = 5
16
(
3|W1|2 − |W2|
2)
≥ 0
If we want a solution without source term, we put µ = 0,dW2 ∝ ReΩ in the above
13 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Summarizing
We are looking for a geometry that satisfies:
Only non-zero torsion classes W1, W2
dW2 ∧ J = 0e2Φm2 = 5
16
(
3|W1|2 − |W2|
2)
≥ 0
If we want a solution without source term, we put µ = 0,dW2 ∝ ReΩ in the above
Nearly-Kahler solutions Behrndt, Cvetic W2 = 0The only homogeneous examples in six dimensions (and onlyknown):
SU(2)×SU(2), G2
SU(3) = S6, Sp(2)S(U(2)×U(1)) = CP
3, SU(3)U(1)×U(1)
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Coset manifolds
Homogeneous manifolds G/H , where G acts on the left and theisotropy H on the right
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Coset manifolds
Homogeneous manifolds G/H , where G acts on the left and theisotropy H on the right
Structure constants Ha ∈ alg(H), Ki rest of alg(G):
[Ha,Hb] = f cabHc
[Ha,Ki] = f jaiKj + f b
aiHb
[Ki,Kj ] = fkijKk + fa
ijHa
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Coset manifolds
Homogeneous manifolds G/H , where G acts on the left and theisotropy H on the right
Structure constants Ha ∈ alg(H), Ki rest of alg(G):
[Ha,Hb] = f cabHc
[Ha,Ki] = f jaiKj + f b
aiHb
[Ki,Kj ] = fkijKk + fa
ijHa
Decomposition of Lie-algebra valued one-form L
L−1dL = eiKi + ωaHa
defines a coframe ei(y), which satisfies
dei = −1
2f i
jkej ∧ ek−f i
ajωa ∧ ej
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Left-invariant forms
Definition:
Constant coefficients in ei basisf j
a[i1φi2...ip]j = 0
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Left-invariant forms
Definition:
Constant coefficients in ei basisf j
a[i1φi2...ip]j = 0
Globally defined
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Left-invariant forms
Definition:
Constant coefficients in ei basisf j
a[i1φi2...ip]j = 0
Globally defined
Expand all structures and forms in left-invariant forms
15 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Left-invariant forms
Definition:
Constant coefficients in ei basisf j
a[i1φi2...ip]j = 0
Globally defined
Expand all structures and forms in left-invariant forms=⇒ consistent reduction Cassani, Kashani-Poor
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Left-invariant forms
Definition:
Constant coefficients in ei basisf j
a[i1φi2...ip]j = 0
Globally defined
Expand all structures and forms in left-invariant forms=⇒ consistent reduction Cassani, Kashani-Poor
Disadvantage: a genuine SU(3)×SU(3)-structure solution is notpossibleCaviezel, PK, Kors, Lust, Tsimpis, Zagermann
15 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Left-invariant forms
Definition:
Constant coefficients in ei basisf j
a[i1φi2...ip]j = 0
Globally defined
Expand all structures and forms in left-invariant forms=⇒ consistent reduction Cassani, Kashani-Poor
Disadvantage: a genuine SU(3)×SU(3)-structure solution is notpossibleCaviezel, PK, Kors, Lust, Tsimpis, Zagermann
=⇒ strict SU(3)-structure
15 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
AdS4 N = 1 solutions on cosets
Tomasiello; PK, Lust, Tsimpis
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 3 5 5 4 3 5W2 6= 0 No Yes Yes Yes No Yesj6 ∝ ReΩ Yes No Yes Yes Yes No
# of par. j6 = 0 2 / 4 3 2 /
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Motivation Overview of general results Application to coset manifolds Conclusions
AdS4 N = 1 solutions on cosets
Tomasiello; PK, Lust, Tsimpis
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 3 5 5 4 3 5W2 6= 0 No Yes Yes Yes No Yesj6 ∝ ReΩ Yes No Yes Yes Yes No
# of par. j6 = 0 2 / 4 3 2 /
Parameters:
Two parameters for all models: dilaton, scale
Shape
Orientifold charge µ
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Effective theory
Parameters: not massless moduli, since they change flux quanta
17 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Effective theory
Parameters: not massless moduli, since they change flux quanta
Effective theory studied in 0806.3458
Caviezel, PK, Kors, Lust, Tsimpis, Wrase, Zagermann
For all cosets (but not for SU(2)×SU(2)): generically all modulistabilized at tree level
17 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Effective theory
Parameters: not massless moduli, since they change flux quanta
Effective theory studied in 0806.3458
Caviezel, PK, Kors, Lust, Tsimpis, Wrase, Zagermann
For all cosets (but not for SU(2)×SU(2)): generically all modulistabilized at tree 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 (nearly-Kahler)
2 4 6 8 10
5
10
15
20
25
µ
M2/|W |2
(b) σ = 2 (m = 0 Ein-stein)
2 4 6 8 10 12
5
10
15
20
25
µ
M2/|W |2
(c) σ = 25(m = 0)
σ: shape parameter
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Moduli space
For some values of the flux quanta: several susy solutions
For some values: no susy solutions
-4 -2 0 2 4
-4
-2
0
2
4
Figure: Regions one/two supersymmetric solutions (red/yellow)Sp(2)
S(U(2)×U(1)) . X-axis (y-axis): mwµ′2 (f ′m2
µ′3 ).
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Moduli space
For some values of the flux quanta: several susy solutions
For some values: no susy solutions
-4 -2 0 2 4
-4
-2
0
2
4
Figure: Regions one/two supersymmetric solutions (red/yellow)Sp(2)
S(U(2)×U(1)) . X-axis (y-axis): mwµ′2 (f ′m2
µ′3 ).
Many non-supersymmetric AdS4 solutions possibleLust,Marchesano,Martucci,Tsimpis;Lust, Tsimpis;Cassani,Kashani-Poor
18 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
dS vacua and inflation
No-go theorem modular inflation: fluxes, D6/O6 Hertzberg,
Kachru, Taylor, Tegmark
19 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
dS vacua and inflation
No-go theorem modular inflation: fluxes, D6/O6 Hertzberg,
Kachru, Taylor, Tegmark
Way-out: geometric fluxes, NS5-branes, KK-monopoles,non-geometric fluxese.g. Silverstein
19 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
dS vacua and inflation
No-go theorem modular inflation: fluxes, D6/O6 Hertzberg,
Kachru, Taylor, Tegmark
Way-out: geometric fluxes, NS5-branes, KK-monopoles,non-geometric fluxese.g. Silverstein
Above models have geometric fluxes
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
dS vacua and inflation
No-go theorem modular inflation: fluxes, D6/O6 Hertzberg,
Kachru, Taylor, Tegmark
Way-out: geometric fluxes, NS5-branes, KK-monopoles,non-geometric fluxese.g. Silverstein
Above models have geometric fluxes
0812.3551 Caviezel, PK, Kors, Lust, Wrase, Zagermann:
Coset models: do not allow dS vacua nor small ǫSU(2)×SU(2): allows dS vacuum (and small ǫ), but has tachyonicdirection
Related work: Flauger, Paban, Robbins, Wrase;Haque, Shiu, Underwood, Van Riet
19 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
dS vacua and inflation
No-go theorem modular inflation: fluxes, D6/O6 Hertzberg,
Kachru, Taylor, Tegmark
Way-out: geometric fluxes, NS5-branes, KK-monopoles,non-geometric fluxese.g. Silverstein
Above models have geometric fluxes
0812.3551 Caviezel, PK, Kors, Lust, Wrase, Zagermann:
Coset models: do not allow dS vacua nor small ǫSU(2)×SU(2): allows dS vacuum (and small ǫ), but has tachyonicdirection
Related work: Flauger, Paban, Robbins, Wrase;Haque, Shiu, Underwood, Van Riet
dS solutions/inflation seems not natural!
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Solutions without sources: CP3
CP3 : Sp(2)
S(U(2)×U(1))
σ
25
2
bb
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Solutions without sources: CP3
CP3 : Sp(2)
S(U(2)×U(1))
σ
25
2
bb
σ = 2: m = 0 Einstein:
CP3 = SU(4)
S(U(3)×U(1))with standard Fubini-Study metric
Supersymmetry enhances to N = 6Global symmetry enhances to SU(4)Geometry of original ABJM in type IIA limitCareful: standard closed Kahler form is not J of N = 1 susyM-theory lift S7 = SO(8)
SO(7)
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Motivation Overview of general results Application to coset manifolds Conclusions
Solutions without sources: CP3
CP3 : Sp(2)
S(U(2)×U(1))
σ
25
2
bb
σ = 2: m = 0 Einstein:
CP3 = SU(4)
S(U(3)×U(1))with standard Fubini-Study metric
Supersymmetry enhances to N = 6Global symmetry enhances to SU(4)Geometry of original ABJM in type IIA limitCareful: standard closed Kahler form is not J of N = 1 susyM-theory lift S7 = SO(8)
SO(7)
σ = 2/5: m = 0
CFT dual proposed Ooguri, Park
M-theory lift: squashed S7
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Motivation Overview of general results Application to coset manifolds Conclusions
Solutions without sources: CP3
CP3 : Sp(2)
S(U(2)×U(1))
σ
25
2
bb
σ = 2: m = 0 Einstein:
CP3 = SU(4)
S(U(3)×U(1))with standard Fubini-Study metric
Supersymmetry enhances to N = 6Global symmetry enhances to SU(4)Geometry of original ABJM in type IIA limitCareful: standard closed Kahler form is not J of N = 1 susyM-theory lift S7 = SO(8)
SO(7)
σ = 2/5: m = 0
CFT dual proposed Ooguri, Park
M-theory lift: squashed S7
2/5 < σ < 2: m 6= 0
CFT dual proposed Gaiotto,Tomasiello
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Motivation Overview of general results Application to coset manifolds Conclusions
Massive CFT dual
Giaotto,Tomasiello
They found a CFT dual for the massive case m 6= 0
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 89: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/89.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Massive CFT dual
Giaotto,Tomasiello
They found a CFT dual for the massive case m 6= 0
In fact, several CFT duals withsusy/global symmetry: N = 0: SO(6), N = 1: Sp(2), N = 2:SO(4)×SO(2)R, N = 3: SO(3)×SO(3)R
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Massive CFT dual
Giaotto,Tomasiello
They found a CFT dual for the massive case m 6= 0
In fact, several CFT duals withsusy/global symmetry: N = 0: SO(6), N = 1: Sp(2), N = 2:SO(4)×SO(2)R, N = 3: SO(3)×SO(3)R
N = 0 corresponds to non-susy solution with Einstein-KahlerFubini-Study metric
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 91: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/91.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Massive CFT dual
Giaotto,Tomasiello
They found a CFT dual for the massive case m 6= 0
In fact, several CFT duals withsusy/global symmetry: N = 0: SO(6), N = 1: Sp(2), N = 2:SO(4)×SO(2)R, N = 3: SO(3)×SO(3)R
N = 0 corresponds to non-susy solution with Einstein-KahlerFubini-Study metric
N = 1 corresponds to the above discussed coset solutions
21 / 23
Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 92: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/92.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Massive CFT dual
Giaotto,Tomasiello
They found a CFT dual for the massive case m 6= 0
In fact, several CFT duals withsusy/global symmetry: N = 0: SO(6), N = 1: Sp(2), N = 2:SO(4)×SO(2)R, N = 3: SO(3)×SO(3)R
N = 0 corresponds to non-susy solution with Einstein-KahlerFubini-Study metric
N = 1 corresponds to the above discussed coset solutions
N = 2/N = 3 the dual geometry is unknown
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 93: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/93.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Massive CFT dual
Giaotto,Tomasiello
They found a CFT dual for the massive case m 6= 0
In fact, several CFT duals withsusy/global symmetry: N = 0: SO(6), N = 1: Sp(2), N = 2:SO(4)×SO(2)R, N = 3: SO(3)×SO(3)R
N = 0 corresponds to non-susy solution with Einstein-KahlerFubini-Study metric
N = 1 corresponds to the above discussed coset solutions
N = 2/N = 3 the dual geometry is unknown
Non-homogeneousOnly one susy can be strict SU(3)-structure, the others genuineSU(3)×SU(3)-structure
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
SU(3)U(1)×U(1)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ = σ
σ = 1
ρ = 1
σ
ρ
Figure: Plot m = 0 curve configuration space shape parameters (ρ, σ)
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 95: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/95.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
SU(3)U(1)×U(1)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ = σ
σ = 1
ρ = 1
σ
ρ
Figure: Plot m = 0 curve configuration space shape parameters (ρ, σ)
M-theory lift for m = 0: Aloff-Wallach spaces
Np,q,r =SU(3) × U(1)
U(1) × U(1)
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 96: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/96.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
SU(3)U(1)×U(1)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ = σ
σ = 1
ρ = 1
σ
ρ
Figure: Plot m = 0 curve configuration space shape parameters (ρ, σ)
M-theory lift for m = 0: Aloff-Wallach spaces
Np,q,r =SU(3) × U(1)
U(1) × U(1)
Red Points (1, 2), (2, 1), (1/2, 1/2)M-theory lift: N = 3 susy, IIA reduction: only N = 1
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 97: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/97.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
SU(3)U(1)×U(1)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ = σ
σ = 1
ρ = 1
σ
ρ
Figure: Plot m = 0 curve configuration space shape parameters (ρ, σ)
M-theory lift for m = 0: Aloff-Wallach spaces
Np,q,r =SU(3) × U(1)
U(1) × U(1)
Red Points (1, 2), (2, 1), (1/2, 1/2)M-theory lift: N = 3 susy, IIA reduction: only N = 1
CFT dual unknown as far as I knowrelated work Jafferis, Tomasiello
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
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Motivation Overview of general results Application to coset manifolds Conclusions
Conclusions
General properties of N = 1 SU(3)×SU(3)-structure type II sugracompactifications
Susy conditions backgroundRelation generalized calibrated and thus susy D-branesSusy and Bianchis imply all eoms
Hard to find examples
AdS4 SU(3)-structure compactifications on coset manifolds
Susy solutions, non-susy AdS solutionsEffective theoryAttempts to construct dS vacuaMore natural in AdS4/CFT correspondence
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)
![Page 99: Flux Compactifications on Coset Manifolds and Applicationskoerber/utrecht2009.pdf · 2009. 9. 30. · Paul Koerber Max-Planck-Institut fu¨r Physik, Munich Utrecht, 19 March 2009](https://reader035.vdocument.in/reader035/viewer/2022071415/611049ff2ea3693c52420d21/html5/thumbnails/99.jpg)
Motivation Overview of general results Application to coset manifolds Conclusions
Conclusions
General properties of N = 1 SU(3)×SU(3)-structure type II sugracompactifications
Susy conditions backgroundRelation generalized calibrated and thus susy D-branesSusy and Bianchis imply all eoms
Hard to find examples
AdS4 SU(3)-structure compactifications on coset manifolds
Susy solutions, non-susy AdS solutionsEffective theoryAttempts to construct dS vacuaMore natural in AdS4/CFT correspondence
The
end. ..T
he end. . .The end
.
. .
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Flux Compactifications on Coset Manifolds and Applications (Paul Koerber)