flux compactiﬁcations, generalized geometry and...

Introduction Supersymmetry Coset manifolds Non-susy solutions dS solutions Conclusions

Flux Compactifications, Generalized Geometry andApplications to Coset Manifolds

Uppsala, 15 December 2009

Paul Koerber

Postdoctoral Fellow FWO, ITF KULeuven

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Flux Compactifications, Generalized Geometry and Applications to Coset Manifolds (Paul Koerber)


Introduction I

URL: http://itf.fys.kuleuven.be/~koerber/talks.html

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

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http://itf.fys.kuleuven.be/~koerber/talks.html


Introduction II

Compactification: 10D → 4D

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Introduction II


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

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Introduction II


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

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Introduction III

Supersymmetry

Easier to find solutionsNice link with generalized geometry (SU(3)×SU(3)-structure)

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Introduction III

Supersymmetry


AdS4

Easy where dS4 is difficultinteresting for AdS4/CFT3 dualityAharony, Bergman, Jafferis, Maldacena

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Introduction III

Supersymmetry


AdS4


dS4

Interesting phenomenologically

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Introduction III

Supersymmetry


AdS4


dS4

Interesting phenomenologicallyMore difficult:

Break supersymmetryGoal: find classical solutionVarious no-go theorems Maldacena, Nunez; Hertzberg, Kachru,

Taylor, Tegmark; Gomez-Reino, Louis, Scrucca

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Compactification ansatz

We consider type IIA/IIB supergravity

Metric:

ds2 = e2A(y)g(4)µν(x)dxµdxν + gmn(y)dy

mdyn ,

with g(4) flat Minkowski, AdS4 metric or dS4 metric, A warp factor

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Compactification ansatz

We consider type IIA/IIB supergravity

Metric:

ds2 = e2A(y)g(4)µν(x)dxµdxν + gmn(y)dy

mdyn ,

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 =∑

l

F(l) = F + e4Avol4 ∧ Fel , (Fel = ⋆6σ(F ))

with l even/odd in type IIA/IIB

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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

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ǫ1 = ζ1+ ⊗ η+ + ζ1− ⊗ η− ,

ǫ2 = ζ2+ ⊗ η∓ + ζ2− ⊗ η± ,


Define (poly)forms

Ψ+ = −i

||η||2

∑

l even

(−1)l/21

l!η†+γi1...ilη+dx

i1 ∧ . . . ∧ dxil

Ψ− =i

||η||21

3!η†−γi1...i3η+dx

i1 ∧ dxi2 ∧ dxi3

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ǫ1 = ζ1+ ⊗ η+ + ζ1− ⊗ η− ,

ǫ2 = ζ2+ ⊗ η∓ + ζ2− ⊗ η± ,


Define (poly)forms

Ψ+ = c eiJ

Ψ− = iΩ

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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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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

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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

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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/21

l!η(2)†+ γi1...ilη

(1)+ dxi1 ∧ . . . ∧ dxil

Ψ− = −i

||η||2

∑

l odd

(−1)(l−1)/2 1

l!η(2)†− γi1...ilη

(1)+ dxi1 ∧ dxi2 ∧ dxil

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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)

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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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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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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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Susy equations in polyform notation:

dH(

e4A−ΦReΨ1

)

= (3/R) e3A−ΦRe(eiθΨ2) + e4AFel ,

dH(

e3A−ΦΨ2

)

= (2/R)i e2A−Φe−iθImΨ1 ,


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

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Generalized calibrations

hep-th/0506154 PK, hep-th/0507099 Smyth, Martucci

Nice interpretation susy conditions in terms of generalizedcalibrations

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Mathematical tool to construct D-branes with minimal energy

Corresponds to supersymmetric D-branes in the probe approximation

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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

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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?

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Supersymmetry implies sugra eom

Solve the susy conditions, do we actually have solution equations ofmotion?

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IIA: Lust,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram

Under mild conditions (subtleties time direction): susy and Bianchi& eom form fields =⇒ all others eom

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Let’s add sources: dF = j and specialize to compactificationPK, Tsimpis 0706.1244

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Let’s add sources: dF = j and specialize to compactificationPK, Tsimpis 0706.1244Under mild conditions:

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Bulk supersymmetry conditionsBianchi identities form-fields with sourceSupersymmetry conditions source = generalized calibrationconditions

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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

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What about dS solutions?

dS solutions are non-supersymmetric: solve full equations of motion

No-go theorem Maldacena, Nunez: Minkowski & dScompactifications → negative-tension sources

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Negative-tension sources in string theory: orientifolds

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Difficult to construct solutions with localized sources

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Difficult to construct solutions with localized sources→ smeared orientifoldsFrom a string perspective this is unsatisfactory, since theinterpretation is problematic

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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

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SU(3)-structure AdS4 solutions

Only possible in type IIA

Ψ− = iΩ, Ψ+ = ceiJ

Susy conditions reduce to conditions of Lust,Tsimpis:

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Constant warp factor

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Constant warp factorGeometric flux i.e. non-zero torsion classes:

dJ =3

2Im(W1Ω

∗) +W4 ∧ J +W3

dΩ = W1J ∧ J +W2 ∧ J +W∗

5 ∧ Ω

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dJ =3

2Im(W1Ω

∗)

dΩ = W1J ∧ J +W2 ∧ J

withW1 = −

4i

9eΦf

W2 = −ieΦF

′

2

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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

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Bianchi identities

Susy not enough, we must add the Bianchi identities form fields

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Bianchi identities


Automatically satisfied except for

dF2 +Hm = −j6

Keep possibility of adding a source j6

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Bianchi identities



dF2 +Hm = −j6


Source j6 (O6/D6) must be calibrated (here SLAG):

j6 ∧ J = 0 j6 ∧ ReΩ = 0 ⇒ j6 = −2

5e−ΦµReΩ + w3

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Bianchi identities



dF2 +Hm = −j6



j6 ∧ J = 0 j6 ∧ ReΩ = 0 ⇒ j6 = −2

5e−ΦµReΩ + w3

w3 simple (1,2)+(2,1)

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Bianchi identities



dF2 +Hm = −j6



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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Bianchi identities



dF2 +Hm = −j6



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)

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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

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Summarizing



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

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Summarizing



dW2 ∧ J = 0e2Φm2 = 5

16

(

3|W1|2 − 2|W2|

2)

≥ 0


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Summarizing



dW2 ∧ J = 0e2Φm2 = 5

16

(

3|W1|2 − 2|W2|

2)

≥ 0


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

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Coset manifolds

Homogeneous manifolds G/H , where G acts on the left and theisotropy H on the right

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Coset manifolds


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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Coset manifolds


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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Left-invariant forms

Definition:

Constant coefficients in ei basisf j

a[i1φi2...ip]j = 0

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Definition:


a[i1φi2...ip]j = 0

Globally defined

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Definition:


a[i1φi2...ip]j = 0

Globally defined

Expand all structures and forms in left-invariant forms

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Definition:


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

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Definition:


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

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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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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

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Effective theory

Parameters: not massless moduli, since they change flux quanta

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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

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Relation with ABJM: CP3

CP3 : Sp(2)

S(U(2)×U(1))

σ

25

1: NK 2

b bb

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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

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CP3 : Sp(2)

S(U(2)×U(1))

σ

25

1: NK 2

b bb


CP3 = SU(4)



SO(7)with N = 8

σ = 2/5: m = 0

CFT dual proposed Ooguri, Park

M-theory lift: squashed S7

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CP3 : Sp(2)

S(U(2)×U(1))

σ

25

1: NK 2

b bb


CP3 = SU(4)



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

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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)

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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


M-theory lift for m = 0: Aloff-Wallach spaces

Np,q,r =SU(3)× U(1)

U(1)× U(1)

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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



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

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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



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

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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 ,


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dH(

e3A−ΦΨ2

)

= (2/R)i e2A−Φe−iθImΨ1

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dH(

e3A−ΦΨ2

)


Type IIA (Ψ1,Ψ2) of type (1, 0)

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dH(

e3A−ΦΨ2

)



=⇒ d(e3A−ΦΨ2|0) 6= 0

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dH(

e3A−ΦΨ2

)



=⇒ d(e3A−ΦΨ2|0) 6= 0

Go beyond the coset ansatz =⇒ differential eqs.

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dH(

e3A−ΦΨ2

)



=⇒ d(e3A−ΦΨ2|0) 6= 0


CFT duals of such geometries on CP3 with N = 2, 3 (codim 1,3)

postulated Gaiotto, Tomasiello

& constructed to first order in Romans mass

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dH(

e3A−ΦΨ2

)



=⇒ d(e3A−ΦΨ2|0) 6= 0





Non-trivial superpotential for D2-branes ∝ Ψ2|0

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dH(

e3A−ΦΨ2

)



=⇒ d(e3A−ΦΨ2|0) 6= 0





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

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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-PoorKahler-Einstein point: Lust, Tsimpis; other Danielsson, Hague,Shiu, Van Riet

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Non-susy solution with KE geometry: CFT dual Gaiotto, Tomasiello

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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

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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 .

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Non-supersymmetric solutions III

Results ( Sp(2)S(U(2)×U(1))):

Green: susy, red: unstable left-invariant fluctuations

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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

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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

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Above models have geometric fluxes

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0812.3551 Caviezel, PK, Kors, Lust, Wrase, Zagermann:

Coset models: do not allow dS vacua nor small ǫSU(2)×SU(2): found classical dS vacuum (and small ǫ), but thishas tachyonic directionNo-go theorem Gomez-Reino, Louis, Scrucca: tachyonic directions

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Related work: Flauger, Paban, Robbins, Wrase;Haque, Shiu, Underwood, Van Riet

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Finding dS solutions/inflation seems very difficult!

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Finding dS solutions/inflation seems very difficult!

Work in progress with Uppsala group

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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!

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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

.

. .

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flux compactiﬁcations, generalized geometry and...

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