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

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)

http://wwwth.mppmu.mpg.de/members/koerber/talks.html


Motivation

Compactification: 10D → 4D, add RR, NSNS fluxes

2 / 23



Motivation


Type IIB orientifold with D3/D7-branes on conformal CY:well-studied

2 / 23



Motivation



stabilization Kahler moduli through non-perturbative effectsuplift susy vacuum to dS, models of inflation

2 / 23



Motivation




Generically: fluxes =⇒ not CY

2 / 23



Motivation




Generically: fluxes =⇒ not CY geometric fluxes

2 / 23



Motivation





Models with SU(3)-structureInteresting class: type IIA compactifications with AdS4 space-time

Models with SU(3)×SU(3)-structure generalized geometry

2 / 23



Motivation






Early type IIA models on torus orientifolds (DeWolfe et al.): allmoduli can be stabilized at tree level


2 / 23



Motivation






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


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Motivation






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


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Motivation







New application to AdS4/CFT Aharony, Bergman, Jafferis,

MaldacenaModels with SU(3)×SU(3)-structure generalized geometry

2 / 23



Motivation









Many properties can be proven in general for this class

2 / 23



Motivation









Many properties can be proven in general for this classHard to find examples ’06: Grana, Minasian, Petrini, Tomasiello,’08: Andriot

2 / 23



Motivation









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

2 / 23



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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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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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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ǫ1 = ζ+ ⊗ η+ + ζ− ⊗ η− ,

ǫ2 = ζ+ ⊗ η∓ + ζ− ⊗ η± ,


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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ǫ1 = ζ+ ⊗ η+ + ζ− ⊗ η− ,

ǫ2 = ζ+ ⊗ η∓ + ζ− ⊗ η± ,


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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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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ǫ1 = ζ+ ⊗ η(1)+ + ζ− ⊗ η

(1)− ,

ǫ2 = ζ+ ⊗ η(2)∓ + ζ− ⊗ η

(2)± ,


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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ǫ1 = ζ+ ⊗ η(1)+ + ζ− ⊗ η

(1)− ,

ǫ2 = ζ+ ⊗ η(2)∓ + ζ− ⊗ η

(2)± ,


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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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 ,

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

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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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Calibration form:mathematical tool to construct submanifolds with minimal volume

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Calibration form:mathematical tool to construct submanifolds with minimal volumee.g. CY case:

eiJ ⇒ complex submanifoldsΩ ⇒ special Lagrangian

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Generalized calibration form φ:constructs D-branes with minimal energy

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Calibrated D-brane (Σ,F) with F the world-volume gauge fieldmust satisfy

e−Φ√

g + F|Σ = φ|Σ ∧ eF

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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



Natural generalized calibration forms

Martucci, Smyth

Calibration forms are the polyforms:

ωsf = e4A−ΦReΨ1 ,

ωDWφ = e3A−ΦRe(eiφΨ2) ,

ωstring = e2A−ΦImΨ1 .

8 / 23




Martucci, Smyth





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




Martucci, Smyth





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



Sugra equations of motion

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

9 / 23





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

Under mild conditions: susy and Bianchi & eom form fields =⇒ allothers eom

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Let’s add sources: dF = jPK, Tsimpis 0706.1244

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Let’s add sources: dF = jPK, Tsimpis 0706.1244Under mild conditions (subtleties time direction):

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

9 / 23








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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Problems constructing solutions

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

10 / 23





Negative-tension sources in string theory: orientifolds

10 / 23






Difficult to construct solutions with localized sources

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Difficult to construct solutions with localized sources→ smeared orientifolds

10 / 23







AdS4 compactifications can avoid the no-go theorem!

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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








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



SU(3)-structure AdS4 solutions

Ψ− = iΩ, Ψ+ = ceiJ

Susy conditions reduce to conditions of Lust,Tsimpis:

11 / 23






Constant warp factor

11 / 23






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 ∧ Jwith

W1 = −4i

9eΦf

W2 = −ieΦF ′

2

11 / 23







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



Bianchi identities

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

Automatically satisfied except for

dF2 +Hm = −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

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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)

12 / 23



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 − |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 − |W2|

2)

≥ 0

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Summarizing



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

13 / 23



Summarizing



dW2 ∧ J = 0e2Φm2 = 5

16

(

3|W1|2 − |W2|

2)

≥ 0


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Summarizing



dW2 ∧ J = 0e2Φm2 = 5

16

(

3|W1|2 − |W2|

2)

≥ 0


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)

13 / 23



Coset manifolds

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

14 / 23



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

14 / 23



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

14 / 23



Left-invariant forms

Definition:

Constant coefficients in ei basisf j

a[i1φi2...ip]j = 0

15 / 23




Definition:


a[i1φi2...ip]j = 0

Globally defined

15 / 23




Definition:


a[i1φi2...ip]j = 0

Globally defined

Expand all structures and forms in left-invariant forms

15 / 23




Definition:


a[i1φi2...ip]j = 0

Globally defined

Expand all structures and forms in left-invariant forms=⇒ consistent reduction Cassani, Kashani-Poor

15 / 23




Definition:


a[i1φi2...ip]j = 0

Globally defined


Disadvantage: a genuine SU(3)×SU(3)-structure solution is notpossibleCaviezel, PK, Kors, Lust, Tsimpis, Zagermann

15 / 23




Definition:


a[i1φi2...ip]j = 0

Globally defined


Disadvantage: a genuine SU(3)×SU(3)-structure solution is notpossibleCaviezel, PK, Kors, Lust, Tsimpis, Zagermann

=⇒ strict SU(3)-structure

15 / 23



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 /

16 / 23



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

16 / 23



Effective theory

Parameters: not massless moduli, since they change flux quanta

17 / 23



Effective theory


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



Effective theory


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

17 / 23



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 ).

18 / 23



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



dS vacua and inflation

No-go theorem modular inflation: fluxes, D6/O6 Hertzberg,

Kachru, Taylor, Tegmark

19 / 23






Way-out: geometric fluxes, NS5-branes, KK-monopoles,non-geometric fluxese.g. Silverstein

19 / 23







Above models have geometric fluxes

19 / 23








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








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!

19 / 23



Solutions without sources: CP3

CP3 : Sp(2)

S(U(2)×U(1))

σ

25

2

bb

20 / 23




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)

20 / 23




CP3 : Sp(2)

S(U(2)×U(1))

σ

25

2

bb


CP3 = SU(4)



SO(7)

σ = 2/5: m = 0

CFT dual proposed Ooguri, Park

M-theory lift: squashed S7

20 / 23




CP3 : Sp(2)

S(U(2)×U(1))

σ

25

2

bb


CP3 = SU(4)



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

20 / 23



Massive CFT dual

Giaotto,Tomasiello

They found a CFT dual for the massive case m 6= 0

21 / 23



Massive CFT dual

Giaotto,Tomasiello


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

21 / 23



Massive CFT dual

Giaotto,Tomasiello



N = 0 corresponds to non-susy solution with Einstein-KahlerFubini-Study metric

21 / 23



Massive CFT dual

Giaotto,Tomasiello




N = 1 corresponds to the above discussed coset solutions

21 / 23



Massive CFT dual

Giaotto,Tomasiello





N = 2/N = 3 the dual geometry is unknown

21 / 23



Massive CFT dual

Giaotto,Tomasiello





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

21 / 23



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)

22 / 23



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


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

Np,q,r =SU(3) × U(1)

U(1) × U(1)

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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



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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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



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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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

23 / 23



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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Documents

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