The eﬀective theory of type IIA AdS4 compactiﬁcations on...

Motivation Type IIA susy vacua Low energy effective theory Inflation Conclusions

The effective theory of type IIA AdS4 compactifications onnilmanifolds and cosets

Based on: 0804.0614 (PK, Tsimpis, Lust), 0806.3458 (Caviezel, PK, Kors, Lust, Tsimpis, Zagermann)

Paul Koerber

Max-Planck-Institut fur Physik, Munich

Zurich, 11 August 2008

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The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (Paul Koerber)

Motivation

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

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Motivation

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

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Motivation

Type IIA compactifications with AdS4 space-time

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Motivation

Early type IIA models on torus orientifolds:Derendinger et al., DeWolfe et al., Camara et al.

All moduli can be stabilized at tree level

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Motivation

All moduli can be stabilized at tree levelUplift susy vacuum to dS, models of inflation: problems

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Motivation

All moduli can be stabilized at tree levelUplift susy vacuum to dS, models of inflation: problemsNo-go theorem modular inflation: fluxes, D6/O6 Hertzberg,

Kachru, Taylor, Tegmark

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Motivation

Way-out: geometric fluxes, NS5-branes, non-geometric fluxes

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Motivation

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Motivation

Geometric fluxes: deviation from Calabi-Yau

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Motivation

Geometric fluxes: deviation from Calabi-Yau

Other application: type IIA on AdS4 × CP3

AdS/CFT Aharony, Bergman, Jafferis, Maldacena M2-branes

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Type IIA susy vacua with AdS4 space-time

Lust, Tsimpis

RR-forms: Romans mass F0 = m, F2, F4,NSNS 3-form: H , dilaton: Φ

SU(3)-structure: J , Ω

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Lust, Tsimpis

Solution susy equations:

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Lust, Tsimpis

Constant warp factor

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Lust, Tsimpis

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

2Im(W1Ω

∗) + W4 ∧ J + W3

dΩ = W1J ∧ J + W2 ∧ J + W∗

5 ∧ Ω

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Lust, Tsimpis

2Im(W1Ω

dΩ = W1J ∧ J + W2 ∧ Jwith

W1 = −4i

W2 = −ieΦF ′

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Lust, Tsimpis

2Im(W1Ω

dΩ = W1J ∧ J + W2 ∧ Jwith

W1 = −4i

W2 = −ieΦF ′

Form-fluxes: AdS4 superpotential W :

5eΦReΩ

9J + F ′

F4 = fvol4 +3m

10J ∧ J

∇µζ− =1

2Wγµζ+ definition

Weiθ = −1

5eΦm +

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Bianchi identities

Automatically satisfied except for

dF2 + HF0 = −j6

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Bianchi identities

dF2 + HF0 = −j6

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

j6 ∧ J = 0 j6 ∧ ReΩ = 0

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Bianchi identities

dF2 + HF0 = −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 + HF0 = −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 + HF0 = −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

(3|W1|2 − |W2|2

)≥ 0

w3 = −ie−ΦdW2

∣∣∣(2,1)+(1,2)

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Smeared orientifolds

Warp factor constant ⇒ only smeared orientifolds

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Solutions without orientifolds:

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Solutions without orientifolds: possible!

No-go theorem Maldacena,Nunez only for Minkowski, not AdS4

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We will introduce orientifolds to obtain N = 1

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Smearing:

j6 = TO6 δ(x4, x5, x6) dx4 ∧ dx5 ∧ dx6 → TOp c dx4 ∧ dx5 ∧ dx6

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Smearing:

We will still associate orientifold involution:

O6 : dx4 → −dx4 , dx5 → −dx5 , dx6 → −dx6

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Smearing:

O6 : dx4 → −dx4 , dx5 → −dx5 , dx6 → −dx6

Write j6 as sum of decomposable forms

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Smearing:

O6 : dx4 → −dx4 , dx5 → −dx5 , dx6 → −dx6

Write j6 as sum of decomposable formsB, H,F2, ReΩ odd, F0, F4, ImΩ even

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Calabi-Yau solution

Calabi-Yau solution Acharya, Benini, Valandro

j6 = −2

5e−ΦµReΩ + w3

e2Φm2 = µ +5

(3|W1|2 − |W2|2

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Calabi-Yau solution

Put W1 = 0,W2 = 0 ⇒ w3 = 0

j6 = −2

5e−ΦµReΩ + w3

e2Φm2 = µ +5

(3|W1|2 − |W2|2

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Calabi-Yau solution

Put W1 = 0,W2 = 0 ⇒ w3 = 0

j6 = −2

5e−ΦµReΩ

e2Φm2 = µ

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Calabi-Yau solution

Put W1 = 0,W2 = 0 ⇒ w3 = 0

j6 = −2

5e−ΦµReΩ

e2Φm2 = µ

Torus orientifolds

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What about equations of motion?

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

With sources: PK, Tsimpis 0706.1244Under mild conditions (subtleties time direction):

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Bulk supersymmetry conditions

Bianchi identities form-fields with source

Supersymmetry conditions source = generalized calibrationconditions PK; Martucci, Smyth

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Bulk supersymmetry conditions

Bianchi identities form-fields with source

Supersymmetry conditions source = generalized calibrationconditions PK; Martucci, Smyth

Einstein equations with source

Dilaton equation of motion with source

Form field equations of motion

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Some older solutions

Nearly-Kahler: only W1 6= 0 Behrndt, Cvetic

SU(2)×SU(2) and the coset spaces G2

SU(3) ,Sp(2)

S(U(2)×U(1)) ,SU(3)

U(1)×U(1)

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Some older solutions

Nearly-Kahler: only W1 6= 0 Behrndt, Cvetic

SU(2)×SU(2) and the coset spaces G2

SU(3) ,Sp(2)

S(U(2)×U(1)) ,SU(3)

U(1)×U(1)

Iwasawa manifold Lust, Tsimpis:

A certain nilmanifold or twisted torus= a group manifold associated to a nilpotent algebraLeft-invariant forms ei obeying Maurer-Cartan relation:

dei = −1

jkej ∧ ek

Singular limit of T2 bundle over K3

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Group manifolds

SU(3)-structures with J,Ω constant in terms of left-invariant formson a group manifold: fertile source of examples

dei = −1

jkej ∧ ek

⇒ algebraic relations

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Group manifolds

SU(3)-structures with J,Ω constant in terms of left-invariant formson a group manifold: fertile source of examples

dei = −1

jkej ∧ ek

⇒ algebraic relations

SU(2)×SU(2)

Nilmanifolds (divide discrete group: compact)

Only Iwasawa & torusNeeds smeared orientifolds

Solvmanifolds (compact?): no solution

Extend to cosets

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Coset manifolds

Tomasiello: W2 6= 0 on two examples Sp(2)S(U(2)×U(1)) ,

SU(3)U(1)×U(1)

PK, Lust, Tsimpis: classification

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Coset manifolds

SU(3)U(1)×U(1)

Maurer-Cartan more complicated:

dei = −1

jkej ∧ ek − f iajω

a ∧ ej

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Coset manifolds

SU(3)U(1)×U(1)

Maurer-Cartan more complicated:

dei = −1

jkej ∧ ek − f iajω

a ∧ ej

Condition left-invariance p-form φ: constant and components

f ja[i1φi2...ip]j = 0

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Type IIA AdS4 susy vacua on coset manifolds

PK, Lust, Tsimpis 0804.0614

SU(2)×SU(2) SU(3)U(1)×U(1)

Sp(2)S(U(2)×U(1))

G2SU(3)

SU(3)×U(1)SU(2)

# of parameters 2 4 4 3 2 4W2 6= 0 No Yes Yes Yes No Yes

j6 ∝ ReΩ Yes No Yes Yes Yes No

Note: geometric flux: W1 6= 0,W2 6= 0

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Type IIA AdS4 susy vacua on coset manifolds

PK, Lust, Tsimpis 0804.0614

SU(2)×SU(2) SU(3)U(1)×U(1)

Sp(2)S(U(2)×U(1))

G2SU(3)

SU(3)×U(1)SU(2)

# of parameters 2 4 4 3 2 4W2 6= 0 No Yes Yes Yes No Yes

j6 ∝ ReΩ Yes No Yes Yes Yes No

Note: geometric flux: W1 6= 0,W2 6= 0

Parameters:

Geometric: scale and shape

J = ae12 + be34 + ce56

Scale a and shape ρ = b/a, σ = c/a:

Orientifold charge µ

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Generalization

SU(3)×SU(3) susy ansatz: 10d → 4d

ǫ1 =ζ+ ⊗ η(1)+ + (c.c.)

ǫ2 =ζ+ ⊗ η(2)∓ + (c.c.)

for IIA/IIB

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Generalization

ǫ1 =ζ+ ⊗ η(1)+ + (c.c.)

ǫ2 =ζ+ ⊗ η(2)∓ + (c.c.)

for IIA/IIB

Strict SU(3)-structure: η(2)+ = eiθη

(1)+ Type: (0,3)

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Generalization

ǫ1 =ζ+ ⊗ η(1)+ + (c.c.)

ǫ2 =ζ+ ⊗ η(2)∓ + (c.c.)

for IIA/IIB

(1)+ Type: (0,3)

Static SU(2)-structure: η(1), η(2) orthogonal everywhereType: (2,1)

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Generalization

ǫ1 =ζ+ ⊗ η(1)+ + (c.c.)

ǫ2 =ζ+ ⊗ η(2)∓ + (c.c.)

for IIA/IIB

(1)+ Type: (0,3)

Static SU(2)-structure: η(1), η(2) orthogonal everywhereType: (2,1)intermediate SU(2)-structure: η(1), η(2) fixed angle,but neither a zero angle nor a right angle Type: (0,1)

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Generalization

ǫ1 =ζ+ ⊗ η(1)+ + (c.c.)

ǫ2 =ζ+ ⊗ η(2)∓ + (c.c.)

for IIA/IIB

(1)+ Type: (0,3)

Static SU(2)-structure: η(1), η(2) orthogonal everywhereType: (2,1)intermediate SU(2)-structure: η(1), η(2) fixed angle,but neither a zero angle nor a right angle Type: (0,1)dynamic SU(3)×SU(3)-structure: angle changes, type may change

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SU(3)×SU(3)-structure susy equations

Grana, Minasian, Petrini, Tomasiello

(e4A−ΦImΨ1

)= 3e3A−ΦIm(W ∗Ψ2) + e4AF

[e3A−ΦRe(W ∗Ψ2)

]= 2|W |2e2A−ΦReΨ1

[e3A−ΦIm(W ∗Ψ2)

(e2A−ΦReΨ1

)= 0 .

dH = d + H∧Ψ1 = Ψ∓ , Ψ2 = Ψ±

/Ψ+ =8

|η(1)||η(2)|η(1)+ ⊗ η

(2)†+ , /Ψ− =

|η(1)||η(2)|η(1)+ ⊗ η

(2)†− .

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(e4A−ΦImΨ1

]= 2|W |2e2A−ΦReΨ1

(e2A−ΦReΨ1

)= 0 .

Example: strict SU(3)-structure in IIA

Ψ1 = Ψ− = −Ω , Ψ2 = Ψ+ = e−iθeiJ

leads to susy equations Lust, Tsimpis

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(e4A−ΦImΨ1

]= 2|W |2e2A−ΦReΨ1

(e2A−ΦReΨ1

)= 0 .

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Generalizations: no-go theorems

]= 2|W |2e2A−ΦReΨ1

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]= 2|W |2e2A−ΦReΨ1

Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)

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]= 2|W |2e2A−ΦReΨ1

Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0

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]= 2|W |2e2A−ΦReΨ1

Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0But we also need 〈Ψ1, Ψ1〉 = −8ivol 6= 0with the Mukai pairing 〈φ1, φ2〉 = φ1 ∧ α(φ2)|top

and α reverses the indices of a form

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]= 2|W |2e2A−ΦReΨ1

Strict SU(3)-structure in IIB: (Ψ1, Ψ2) = (Ψ+, Ψ−)Type: (0,3) Ψ2 3-form ⇒ ReΨ1|0 = ReΨ1|2 = 0But we also need 〈Ψ1, Ψ1〉 = −8ivol 6= 0⇒ impossible

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]= 2|W |2e2A−ΦReΨ1

Intermediate SU(2)-structure:

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]= 2|W |2e2A−ΦReΨ1

Intermediate SU(2)-structure:Type: (0,1) ⇒ general form e2A−ΦΨ1 = ceiω+b

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]= 2|W |2e2A−ΦReΨ1

⇒ Rec = 0 , cω exact

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]= 2|W |2e2A−ΦReΨ1

⇒ Rec = 0 , cω exactImc〈e2A−ΦΨ1, e

2A−Φ, Ψ1〉 = 83! (cω)3

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]= 2|W |2e2A−ΦReΨ1

⇒ Rec = 0 , cω exactImc〈e2A−ΦΨ1, e

2A−Φ, Ψ1〉 = 83! (cω)3

⇒ c not everywhere non-vanishing

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Generalizations: conclusion of no-go theorems

Type IIB: only static SU(2) or dynamic SU(3)×SU(3) that changestype to static SU(2) at least somewhere is possible

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Example: type IIB static SU(2) on nilmanifold 5.1 (table of Grana,

Minasian, Petrini, Tomasiello)

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Example: type IIB static SU(2) on nilmanifold 5.1 (table of Grana,

Minasian, Petrini, Tomasiello)

Type IIA: only strict SU(3) or dynamic SU(3)×SU(3) for whichd

(e2A−Φη(2)†η(1)

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Low energy effective theory: nilmanifolds

Torus (IIA), nilmanifold 5.1 (IIB), Iwasawa (IIA): T-dual to eachother

Calculation mass spectrum

Using effective 4D sugra techniquesFor Iwasawa & torus: direct KK reduction of equations of motionPerfect agreement

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Low energy effective theory: nilmanifolds

Torus (IIA), nilmanifold 5.1 (IIB), Iwasawa (IIA): T-dual to eachother

Calculation mass spectrum

Using effective 4D sugra techniquesFor Iwasawa & torus: direct KK reduction of equations of motionPerfect agreement

Result for M2/|W |2

Complex structure −2, −2, −2Kahler & dilaton 70, 18, 18, 18

Three axions of δC3 0, 0, 0δB & one more axion 88, 10, 10, 10

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Issue: decoupling KK tower

Consistency approximation: gs ≪ 1, ls/Lint ≪ 1

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Consistency approximation: gs ≪ 1, ls/Lint ≪ 1√

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Decoupling KK modes: |W |2L2int =

e2Φm2L2

25 +e2Φf2L2

9 ≪ 1

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e2Φm2L2

25 +e2Φf2L2

9 ≪ 1

Can be tuned with orientifold charge µ:

e2Φm2 = µ +5

(3|W1|

2 − |W2|2) ≈ 0

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e2Φm2L2

25 +e2Φf2L2

9 ≪ 1

eΦfLint ∝ W1Lint ≪ 1:

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e2Φm2L2

25 +e2Φf2L2

9 ≪ 1

eΦfLint ∝ W1Lint ≪ 1:nearly Calabi-Yau

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e2Φm2L2

25 +e2Φf2L2

9 ≪ 1

eΦfLint ∝ W1Lint ≪ 1:nearly Calabi-YauCan be arranged for torus, Iwasawa, nilmanifold 5.1

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e2Φm2L2

25 +e2Φf2L2

9 ≪ 1

eΦfLint ∝ W1Lint ≪ 1:nearly Calabi-YauCan be arranged for torus, Iwasawa, nilmanifold 5.1Harder for coset examples

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Low energy theory: superpotential and Kahlerpotential

We use the superpotential of Grana, Louis, Waldram;

Benmachiche, Grimm; PK, Martucci

Generalizes Gukov, Vafa, Witten

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WE =−i

4κ210

〈Ψ2eδB

︸︷︷︸

, F + i dH

(eδBe−ΦImΨ1 − iδC︸︷︷︸

Z,T holomorphic coordinates,

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WE =−i

4κ210

〈Ψ2eδB

︸︷︷︸

, F + i dH

(eδBe−ΦImΨ1 − iδC︸︷︷︸

Z,T holomorphic coordinates, and the Kahler potential

K = − ln i

〈Z, Z〉 − 2 ln i

〈e−ΦΨ1, e−ΦΨ1〉 + 3 ln(8κ2

10M2P )

where e−ΦeδBΨ1: function of ReT = e−ΦeδBImΨ1 Hitchin.

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Superpotential and Kahler potential: SU(3)

Specialized to SU(3): Ψ1 = −Ω , Ψ2 = e−iθeiJ

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Superpotential:

WE =−ie−iθ

4κ210

〈ei(J−iδB), F − idH

(e−ΦImΩ + iδC3

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Superpotential:

WE =−ie−iθ

4κ210

(e−ΦImΩ + iδC3

Kahler potential:

K = − ln

3J3−2 ln

2 e−ΦImΩ∧e−ΦReΩ+3 ln(8κ210M

2P ) ,

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Superpotential:

WE =−ie−iθ

4κ210

(e−ΦImΩ + iδC3

Kahler potential:

K = − ln

3J3−2 ln

2 e−ΦImΩ∧e−ΦReΩ+3 ln(8κ210M

2P ) ,

Expansion:

Jc = J − iδB = (ki − ibi)Y(2−)i = tiY

(2−)i

e−ΦImΩ + iδC3 = (ui + ici)e−ΦY(3+)i = zie−ΦY

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Comments

We used the same superpotential and Kahler potential for staticSU(2)

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Comments

F-flatness conditions:

DiWE = ∂iWE + ∂iKWE = 0

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Comments

PK, Martucci:

If W = M−1P 〈WE〉 6= 0 i.e. AdS4 compactification

F-flatness conditions ⇔ 10D susy conditions vacuum

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Comments

PK, Martucci:

F-flatness conditions ⇔ 10D susy conditions vacuumIf W = 0 i.e. Minkowski compactificationF-flatness & D-flatness conditions ⇔ 10D susy conditions vacuum

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Comments

PK, Martucci:

The scalar potential is

V = M−2P eK

(KiDiWEDW∗

E − 3|WE|2)

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Comments

PK, Martucci:

The scalar potential is

V = M−2P eK

(KiDiWEDW∗

E − 3|WE|2)

Mass spectrum from quadratic terms

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Low energy theory: cosets

For all cosets (but not for SU(2)×SU(2)): all moduli stabilized attree level

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Example Sp(2)S(U(2)×U(1))

2 4 6 8 10

M2/|W |2

(a) σ = 1

2 4 6 8 10 12

M2/|W |2

(b) σ = 25

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Example Sp(2)S(U(2)×U(1))

2 4 6 8 10

M2/|W |2

(a) σ = 1

2 4 6 8 10 12

M2/|W |2

(b) σ = 25

µ big enough: all mass-squared positive

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Nearly-Calabi Yau limit

Important decoupling KK-modes

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Example Sp(2)S(U(2)×U(1)) ⇒ analytic continuation to negative σ = −2

Twistor bundle description more appropriate Xu

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10 20 30 40 50

M2/|W |2

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10 20 30 40 50

M2/|W |2

Light modes: M2/|W |2 = (−38/49, 130/49)

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Inflation

No-go theorem modular inflation IIAHertzberg, Kachru, Taylor, Tegmark

Dependence on volume-modulus ρ and dilaton-modulus τ

Ingredients: form-fluxes, D6-branes & O6-planes

ǫ > 27/13

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Inflation

ǫ > 27/13

Way out: geometric flux, NS5-branes, other branes, non-geometricflux

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Inflation

ǫ > 27/13

potential geometric flux Vf > 0 ⇔ R < 0

23 / 24

Inflation

ǫ > 27/13

potential geometric flux Vf > 0 ⇔ R < 0not possible for G2

SU(3), possible for all other cosets and SU(2)×SU(2)

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Conclusions

Tractable models with geometric flux: nilmanifold & coset models

Other models: e.g. twistor bundles with negative σ

Uplift: uplifting term or look for dS minimum e.g. Silverstein

Study inflation

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Conclusions

Tractable models with geometric flux: nilmanifold & coset models

Other models: e.g. twistor bundles with negative σ

Uplift: uplifting term or look for dS minimum e.g. Silverstein

Study inflation

end. ..T

he end. . .The end

24 / 24

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