flux compactifications, generalized geometry and...
TRANSCRIPT
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Flux Compactifications, Generalized Geometry andApplications to Coset Manifolds
Based on: 0707.1038, 0706.1244, 0710.5530, 0804.0614, 0806.3458, 0812.3551, 0904.0012
Collaborators: Caviezel, Kors, Lust, Martucci, Tsimpis, Wrase, Zagermann
Work in progress with S. Kors; Uppsala/Madison group
http://itf.fys.kuleuven.be/~koerber/talks.html
Paul Koerber
Postdoctoral fellow FWOInstituut voor Theoretische Fysica, K.U. Leuven
Heidelberg, 10 December 2009
1 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Introduction I
Compactification: 10D → 4D
2 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Introduction I
Compactification: 10D → 4D
external 4D space: AdS4, Minkowski, dS4
add RR, NSNS fluxes (lifting moduli)geometric fluxes (away from Calabi-Yau)possibly add sources (D-branes, orientifolds, NS5-branes,...)supersymmetric or non-supersymmetric
2 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Introduction I
Compactification: 10D → 4D
external 4D space: AdS4, Minkowski, dS4
add RR, NSNS fluxes (lifting moduli)geometric fluxes (away from Calabi-Yau)possibly add sources (D-branes, orientifolds, NS5-branes,...)supersymmetric or non-supersymmetric
Goals
General picture of how compactifications look like,general propertiesFind low-energy theory in the presence of fluxes(geometric, RR, NSNS, ...)Concrete simple examples e.g. coset manifolds
2 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Introduction II
Supersymmetry
Easier to find solutionsNice link with generalized geometry (SU(3)×SU(3)-structure)
3 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Introduction II
Supersymmetry
Easier to find solutionsNice link with generalized geometry (SU(3)×SU(3)-structure)
AdS4
Easy where dS4 is difficultinteresting for AdS4/CFT3 dualityAharony, Bergman, Jafferis, Maldacena
3 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Introduction II
Supersymmetry
Easier to find solutionsNice link with generalized geometry (SU(3)×SU(3)-structure)
AdS4
Easy where dS4 is difficultinteresting for AdS4/CFT3 dualityAharony, Bergman, Jafferis, Maldacena
dS4
Interesting phenomenologically
3 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Introduction II
Supersymmetry
Easier to find solutionsNice link with generalized geometry (SU(3)×SU(3)-structure)
AdS4
Easy where dS4 is difficultinteresting for AdS4/CFT3 dualityAharony, Bergman, Jafferis, Maldacena
dS4
Interesting phenomenologicallyMore difficult:
Break supersymmetryGoal: find classical solutionVarious no-go theorems Maldacena, Nunez; Hertzberg, Kachru,
Taylor, Tegmark; Gomez-Reino, Louis, Scrucca
3 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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, AdS4 metric or dS4 metric, A warp factor
4 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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, AdS4 metric or dS4 metric, A warp factor
NSNS-flux H
RR-fluxes:
Democratic formalism: double fields (e.g. in IIA: F(0), F(2), F(4),F(6), F(8), F(10)), impose duality conditionCombine forms into one polyform
Ftot =X
l
F(l) = F + e4Avol4 ∧ Fel , (Fel = ⋆6σ(F ))
with l even/odd in type IIA/IIB
4 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry ansatz: SU(3)
Ansatz for the two Majorana-Weyl susy generatorsSU(3)-structure ansatz:
ǫ1 = ζ1+ ⊗ η+ + ζ1
− ⊗ η− ,
ǫ2 = ζ2+ ⊗ η∓ + ζ2
− ⊗ η± ,
ζ1,2: 4d spinor characterizes preserved susy in 4d (N = 2)η: fixed 6d-spinor, property background, defines SU(3)-structure
5 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry ansatz: SU(3)
Ansatz for the two Majorana-Weyl susy generatorsSU(3)-structure ansatz:
ǫ1 = ζ1+ ⊗ η+ + ζ1
− ⊗ η− ,
ǫ2 = ζ2+ ⊗ η∓ + ζ2
− ⊗ η± ,
ζ1,2: 4d spinor characterizes preserved susy in 4d (N = 2)η: fixed 6d-spinor, property background, defines SU(3)-structure
Define (poly)forms
Ψ+ = −i
||η||2
∑
l even
(−1)l/2 1
l!η†+γi1...il
η+dxi1 ∧ . . . ∧ dxil
Ψ− =i
||η||21
3!η†−γi1...i3η+dx
i1 ∧ dxi2 ∧ dxi3
5 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry ansatz: SU(3)
Ansatz for the two Majorana-Weyl susy generatorsSU(3)-structure ansatz:
ǫ1 = ζ1+ ⊗ η+ + ζ1
− ⊗ η− ,
ǫ2 = ζ2+ ⊗ η∓ + ζ2
− ⊗ η± ,
ζ1,2: 4d spinor characterizes preserved susy in 4d (N = 2)η: fixed 6d-spinor, property background, defines SU(3)-structure
Define (poly)forms
Ψ+ = c eiJ
Ψ− = iΩ
5 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry ansatz: SU(3)
In the absence of fluxes, susy conditions: dJ = 0, dΩ = 0=⇒ SU(3)-holonomy i.e. CY=⇒ J Kahler-form, Ω holomorphic three-form
6 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry ansatz: SU(3)
In the absence of fluxes, susy conditions: dJ = 0, dΩ = 0=⇒ SU(3)-holonomy i.e. CY=⇒ J Kahler-form, Ω holomorphic three-form
With fluxes:
Generically dJ 6= 0, dΩ 6= 0For a susy solution we must put ζ1 = ζ2
=⇒ theory: N = 2, solution: N = 1
6 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry ansatz: SU(3)×SU(3)
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
7 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry ansatz: SU(3)×SU(3)
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
7 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry ansatz: SU(3)×SU(3)
These polyforms Ψ± can be considered as spinors of TM ⊕ T ⋆M
Not every polyform is related to a spinor bilinear: only pure spinors
Type of the pure spinor: lowest dimension of the polyforme.g. SU(3)-structure: (ceiJ , iΩ) −→ (0, 3)Generic case: (0, 1)
8 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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
9 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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 . . .
9 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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
9 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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
9 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Generalized calibrations
hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci
Nice interpretation susy conditions in terms of generalizedcalibrations
10 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Generalized calibrations
hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci
Nice interpretation susy conditions in terms of generalizedcalibrations
Mathematical tool to construct D-branes with minimal energy
Corresponds to supersymmetric D-branes in the probe approximation
10 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Generalized calibrations
hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci
Nice interpretation susy conditions in terms of generalizedcalibrations
Mathematical tool to construct D-branes with minimal energy
Corresponds to supersymmetric D-branes in the probe approximation
Calibration forms are exactly the polyforms of the background susyconditions:
ωsf = e4A−ΦReΨ1 ,
ωDWφ = e3A−ΦRe(eiφΨ2) ,
ωstring = e2A−ΦImΨ1 .
More subtle for AdS4 compactifications0710.5530 PK, Martucci
10 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Generalized calibrations
hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci
Nice interpretation susy conditions in terms of generalizedcalibrations
Mathematical tool to construct D-branes with minimal energy
Corresponds to supersymmetric D-branes in the probe approximation
Calibration forms are exactly the polyforms of the background susyconditions:
ωsf = e4A−ΦReΨ1 ,
ωDWφ = e3A−ΦRe(eiφΨ2) ,
ωstring = e2A−ΦImΨ1 .
More subtle for AdS4 compactifications0710.5530 PK, Martucci
What about back-reacting sources?
10 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry implies sugra eom
Solve the susy conditions, do we actually have solution equations ofmotion?
11 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry implies sugra eom
Solve the susy conditions, do we actually have solution equations ofmotion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
Under mild conditions (subtleties time direction): susy and Bianchi& eom form fields =⇒ all others eom
11 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry implies sugra eom
Solve the susy conditions, do we actually have solution equations ofmotion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
Under mild conditions (subtleties time direction): susy and Bianchi& eom form fields =⇒ all others eom
Let’s add sources: dF = j and specialize to compactificationPK, Tsimpis 0706.1244
11 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry implies sugra eom
Solve the susy conditions, do we actually have solution equations ofmotion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
Under mild conditions (subtleties time direction): susy and Bianchi& eom form fields =⇒ all others eom
Let’s add sources: dF = j and specialize to compactificationPK, Tsimpis 0706.1244Under mild conditions:
11 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry implies sugra eom
Solve the susy conditions, do we actually have solution equations ofmotion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
Under mild conditions (subtleties time direction): susy and Bianchi& eom form fields =⇒ all others eom
Let’s add sources: dF = j and specialize to compactificationPK, Tsimpis 0706.1244Under mild conditions:
Bulk supersymmetry conditionsBianchi identities form-fields with sourceSupersymmetry conditions source = generalized calibrationconditions
11 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Supersymmetry implies sugra eom
Solve the susy conditions, do we actually have solution equations ofmotion?
IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram
Under mild conditions (subtleties time direction): susy and Bianchi& eom form fields =⇒ all others eom
Let’s add sources: dF = j and specialize to compactificationPK, Tsimpis 0706.1244Under mild conditions:
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
11 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about dS solutions?
dS solutions are non-supersymmetric: solve full equations of motion
No-go theorem Maldacena, Nunez: Minkowski & dScompactifications → negative-tension sources
12 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about dS solutions?
dS solutions are non-supersymmetric: solve full equations of motion
No-go theorem Maldacena, Nunez: Minkowski & dScompactifications → negative-tension sources
Negative-tension sources in string theory: orientifolds
12 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about dS solutions?
dS solutions are non-supersymmetric: solve full equations of motion
No-go theorem Maldacena, Nunez: Minkowski & dScompactifications → negative-tension sources
Negative-tension sources in string theory: orientifolds
Difficult to construct solutions with localized sources
12 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about dS solutions?
dS solutions are non-supersymmetric: solve full equations of motion
No-go theorem Maldacena, Nunez: Minkowski & dScompactifications → negative-tension sources
Negative-tension sources in string theory: orientifolds
Difficult to construct solutions with localized sources→ smeared orientifoldsFrom a string perspective this is unsatisfactory, since theinterpretation is problematic
12 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about dS solutions?
dS solutions are non-supersymmetric: solve full equations of motion
No-go theorem Maldacena, Nunez: Minkowski & dScompactifications → negative-tension sources
Negative-tension sources in string theory: orientifolds
Difficult to construct solutions with localized sources→ smeared orientifoldsFrom a string perspective this is unsatisfactory, since theinterpretation is problematic
AdS4 compactifications can avoid this no-go theorem!I.e. we can find flux vacua without introducing source terms
12 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
SU(3)-structure AdS4 solutions
Only possible in type IIA
Ψ− = iΩ, Ψ+ = ceiJ
Susy conditions reduce to conditions of Lust,Tsimpis:
13 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
SU(3)-structure AdS4 solutions
Only possible in type IIA
Ψ− = iΩ, Ψ+ = ceiJ
Susy conditions reduce to conditions of Lust,Tsimpis:
Constant warp factor
13 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
SU(3)-structure AdS4 solutions
Only possible in type IIA
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
13 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
SU(3)-structure AdS4 solutions
Only possible in type IIA
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 ∧ J
withW1 = −
4i
9eΦf
W2 = −ieΦF
′
2
13 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
SU(3)-structure AdS4 solutions
Only possible in type IIA
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 ∧ J
withW1 = −
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
13 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
14 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
Automatically satisfied except for
dF2 +Hm = −j6
Keep possibility of adding a source j6
14 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
Automatically satisfied except for
dF2 +Hm = −j6
Keep possibility of adding a source j6
Source j6 (O6/D6) must be calibrated (here SLAG):
j6 ∧ J = 0 j6 ∧ ReΩ = 0 ⇒ j6 = −2
5e−ΦµReΩ + w3
14 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
Automatically satisfied except for
dF2 +Hm = −j6
Keep possibility of adding a source 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)
14 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
Automatically satisfied except for
dF2 +Hm = −j6
Keep possibility of adding a source 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
14 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Bianchi identities
Susy not enough, we must add the Bianchi identities form fields
Automatically satisfied except for
dF2 +Hm = −j6
Keep possibility of adding a source 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 − 2|W2|
2)
≥ 0
w3 = −ie−ΦdW2
∣
∣
∣
(2,1)+(1,2)
14 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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 − 2|W2|
2´
≥ 0
15 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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 − 2|W2|
2´
≥ 0
If we want a solution without source term, we put µ = 0,dW2 ∝ ReΩ in the above
15 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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 − 2|W2|
2´
≥ 0
If we want a solution without source term, we put µ = 0,dW2 ∝ ReΩ in the above
15 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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 − 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 examples known are the homogeneous manifolds:
SU(2)×SU(2), G2
SU(3) = S6, Sp(2)S(U(2)×U(1)) = CP
3, SU(3)U(1)×U(1)
So let us look in more detail at coset manifolds
15 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Coset manifolds
Homogeneous manifolds G/H , where G acts on the left and theisotropy H on the right
16 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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
16 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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
16 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Left-invariant forms
Definition:
Constant coefficients in ei basisf j
a[i1φi2...ip]j = 0
17 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Left-invariant forms
Definition:
Constant coefficients in ei basisf j
a[i1φi2...ip]j = 0
Globally defined
17 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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
17 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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
=⇒ N = 2 low-energy theory
17 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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
=⇒ N = 2 low-energy theory
Big advantage: all differential equations reduce to algebraicequations using the Maurer-Cartan relations:
dei = −1
2f i
jkej ∧ ek−f i
ajωa ∧ ej
17 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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 /
18 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions 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, overall scale
Shape
Orientifold charge µ
Last line shows the solutions without source µ = 0
18 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Effective theory
Parameters: not massless moduli, since they change flux quanta
19 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Effective theory
Parameters: not massless moduli, since they change flux quanta
Effective theory studied in 0806.3458
Caviezel, PK, Kors, Lust, Tsimpis, Zagermann
For all cosets (but not for SU(2)×SU(2)): generically all modulistabilized at tree level
19 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Relation with ABJM: CP3
CP3 : Sp(2)
S(U(2)×U(1))
σ
25
1: NK 2
b bb
20 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Relation with ABJM: CP3
CP3 : Sp(2)
S(U(2)×U(1))
σ
25
1: NK 2
b 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)with N = 8
20 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Relation with ABJM: CP3
CP3 : Sp(2)
S(U(2)×U(1))
σ
25
1: NK 2
b 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)with N = 8
σ = 2/5: m = 0
CFT dual proposed Ooguri, Park
M-theory lift: squashed S7
20 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Relation with ABJM: CP3
CP3 : Sp(2)
S(U(2)×U(1))
σ
25
1: NK 2
b 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)with N = 8
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
Romans mass: Chern-Simons levels k1 6= −k2
20 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Relation with ABJM: 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)
21 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Relation with ABJM: 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)
21 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Relation with ABJM: 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 (tri-Sasakian), IIA reduction: only N = 1
21 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Relation with ABJM: 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 (tri-Sasakian), IIA reduction: only N = 1
CFT dual unknown as far as I knowrelated work Jafferis, Tomasiello
21 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about SU(3)×SU(3)-structure?
Susy equations for AdS4 compactifications 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 ,
22 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about SU(3)×SU(3)-structure?
Susy equations for AdS4 compactifications 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 ,
22 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about SU(3)×SU(3)-structure?
dH
(
e3A−ΦΨ2
)
= (2/R)i e2A−Φe−iθImΨ1
23 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about SU(3)×SU(3)-structure?
dH
(
e3A−ΦΨ2
)
= (2/R)i e2A−Φe−iθImΨ1
Type IIA (Ψ1,Ψ2) of type (1, 0)
23 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about SU(3)×SU(3)-structure?
dH
(
e3A−ΦΨ2
)
= (2/R)i e2A−Φe−iθImΨ1
Type IIA (Ψ1,Ψ2) of type (1, 0)
=⇒ d(e3A−ΦΨ2|0) 6= 0
23 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about SU(3)×SU(3)-structure?
dH
(
e3A−ΦΨ2
)
= (2/R)i e2A−Φe−iθImΨ1
Type IIA (Ψ1,Ψ2) of type (1, 0)
=⇒ d(e3A−ΦΨ2|0) 6= 0
Go beyond the coset ansatz =⇒ differential eqs.
23 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about SU(3)×SU(3)-structure?
dH
(
e3A−ΦΨ2
)
= (2/R)i e2A−Φe−iθImΨ1
Type IIA (Ψ1,Ψ2) of type (1, 0)
=⇒ d(e3A−ΦΨ2|0) 6= 0
Go beyond the coset ansatz =⇒ differential eqs.
CFT duals of such geometries on CP3 with N = 2, 3 (codim 1,3)
postulated Gaiotto, Tomasiello
& constructed to first order in Romans mass
23 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about SU(3)×SU(3)-structure?
dH
(
e3A−ΦΨ2
)
= (2/R)i e2A−Φe−iθImΨ1
Type IIA (Ψ1,Ψ2) of type (1, 0)
=⇒ d(e3A−ΦΨ2|0) 6= 0
Go beyond the coset ansatz =⇒ differential eqs.
CFT duals of such geometries on CP3 with N = 2, 3 (codim 1,3)
postulated Gaiotto, Tomasiello
& constructed to first order in Romans mass
Non-trivial superpotential for D2-branes ∝ Ψ2|0
23 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
What about SU(3)×SU(3)-structure?
dH
(
e3A−ΦΨ2
)
= (2/R)i e2A−Φe−iθImΨ1
Type IIA (Ψ1,Ψ2) of type (1, 0)
=⇒ d(e3A−ΦΨ2|0) 6= 0
Go beyond the coset ansatz =⇒ differential eqs.
CFT duals of such geometries on CP3 with N = 2, 3 (codim 1,3)
postulated Gaiotto, Tomasiello
& constructed to first order in Romans mass
Non-trivial superpotential for D2-branes ∝ Ψ2|0
Related numeric N = 2 SU(3)×SU(3) solutions on reductions ofM3,2 and Q1,1,1 Petrini, Zaffaroni; Lust, Tsimpis
23 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Non-supersymmetric solutions I
P.K., Kors, work in progress
Non-supersymmetric AdS4 solutions found in particular casesnearly Kahler point:Lust,Marchesano,Martucci,Tsimpis;Cassani,Kashani-Poor
Kahler-Einstein point: Lust, Tsimpis; other Danielsson, Hague,
Shiu, Van Riet
24 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Non-supersymmetric solutions I
P.K., Kors, work in progress
Non-supersymmetric AdS4 solutions found in particular casesnearly Kahler point:Lust,Marchesano,Martucci,Tsimpis;Cassani,Kashani-Poor
Kahler-Einstein point: Lust, Tsimpis; other Danielsson, Hague,
Shiu, Van Riet
Non-susy solution with KE geometry: CFT dual Gaiotto, Tomasiello
24 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Non-supersymmetric solutions I
P.K., Kors, work in progress
Non-supersymmetric AdS4 solutions found in particular casesnearly Kahler point:Lust,Marchesano,Martucci,Tsimpis;Cassani,Kashani-Poor
Kahler-Einstein point: Lust, Tsimpis; other Danielsson, Hague,
Shiu, Van Riet
Non-susy solution with KE geometry: CFT dual Gaiotto, Tomasiello
For every susy solution in M-theory with F4 = 0non-susy solution on same geometry with F4 6= 0 Englert
24 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Non-supersymmetric solutions II
Let us scan for non-susy solution with same geometry susy solution,different NSNS-, RR-fluxes
Ansatz for RR-fluxes:
eΦF0 = f1 ,
eΦF2 = f2 J + f3W2 ,
eΦF4 = f4 J ∧ J + f5W2 ∧ J ,
eΦF6 = f6 vol6 ,
H = f7 ReΩ .
25 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Non-supersymmetric solutions III
Results ( Sp(2)S(U(2)×U(1))):
Green: susy, red: unstable left-invariant fluctuations
26 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
dS vacua and inflation
On top of generic no-go theorem Maldacena, Nunez, requiringsourcesMore specific no-go theorem modular inflation: fluxes, D6/O6Hertzberg, Kachru, Taylor, Tegmark
=⇒ no dS vacua nor small ǫ inflation
27 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
dS vacua and inflation
On top of generic no-go theorem Maldacena, Nunez, requiringsourcesMore specific no-go theorem modular inflation: fluxes, D6/O6Hertzberg, Kachru, Taylor, Tegmark
=⇒ no dS vacua nor small ǫ inflation
Way-out: geometric fluxes, NS5-branes, KK-monopoles,non-geometric fluxese.g. Silverstein
27 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
dS vacua and inflation
Above models have geometric fluxes
28 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
dS vacua and inflation
Above models have geometric fluxes
0812.3551 Caviezel, PK, Kors, Lust, Wrase, Zagermann:
Coset models: do not allow dS vacua nor small ǫ
SU(2)×SU(2): found classical dS vacuum (and small ǫ), but thishas tachyonic directionNo-go theorem Gomez-Reino, Louis, Scrucca: tachyonic directions
28 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
dS vacua and inflation
Above models have geometric fluxes
0812.3551 Caviezel, PK, Kors, Lust, Wrase, Zagermann:
Coset models: do not allow dS vacua nor small ǫ
SU(2)×SU(2): found classical dS vacuum (and small ǫ), but thishas tachyonic directionNo-go theorem Gomez-Reino, Louis, Scrucca: tachyonic directions
Related work: Flauger, Paban, Robbins, Wrase;Haque, Shiu, Underwood, Van Riet
28 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
dS vacua and inflation
Above models have geometric fluxes
0812.3551 Caviezel, PK, Kors, Lust, Wrase, Zagermann:
Coset models: do not allow dS vacua nor small ǫ
SU(2)×SU(2): found classical dS vacuum (and small ǫ), but thishas tachyonic directionNo-go theorem Gomez-Reino, Louis, Scrucca: tachyonic directions
Related work: Flauger, Paban, Robbins, Wrase;Haque, Shiu, Underwood, Van Riet
Finding dS solutions/inflation seems very difficult!
28 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
dS vacua and inflation
Above models have geometric fluxes
0812.3551 Caviezel, PK, Kors, Lust, Wrase, Zagermann:
Coset models: do not allow dS vacua nor small ǫ
SU(2)×SU(2): found classical dS vacuum (and small ǫ), but thishas tachyonic directionNo-go theorem Gomez-Reino, Louis, Scrucca: tachyonic directions
Related work: Flauger, Paban, Robbins, Wrase;Haque, Shiu, Underwood, Van Riet
Finding dS solutions/inflation seems very difficult!
Work in progress with Uppsala group
28 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Conclusions
Supersymmetry conditions of type II supergravity are naturallydescribed in generalized geometry formalism
Coset manifolds: simple and rich examples of both susy andnon-susy compactifications to AdS4
Applications to AdS4/CFT3
Go beyond coset ansatz
Classical dS-solutions: difficult because of several no-go theorems!
29 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)
Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions
Conclusions
Supersymmetry conditions of type II supergravity are naturallydescribed in generalized geometry formalism
Coset manifolds: simple and rich examples of both susy andnon-susy compactifications to AdS4
Applications to AdS4/CFT3
Go beyond coset ansatz
Classical dS-solutions: difficult because of several no-go theorems!
The
end. ..T
he end. . .The end
.
. .
29 / 29
Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)