Is the four-dimensional effective theory

effective?YITP Hideo Kodama

HK and Kunihito Uzawa, JHEP0507:061(2005)

HK and Kunihito Uzawa, hep-th/0512104

Three Major Requirements on HUTThree Major Requirements on HUT

4-dimensional universe at present4-dimensional universe at presentcompactification of the extra dimensioncompactification of the extra dimensionmoduli stabilisationmoduli stabilisation

Moduli StabilisationModuli Stabilisation

Compactification without fluxCompactification without fluxIn this case, D=11/10 sugra reduces to a theory for gravity cIn this case, D=11/10 sugra reduces to a theory for gravity coupled with dilaton.oupled with dilaton.

AssumeAssume

ThenThen

In the cosmological context, the second equation leads toIn the cosmological context, the second equation leads to

Flux compactificationFlux compactification

Moduli for IIB SugraModuli for IIB Sugra

Shape moduli Shape moduli ⇒ ⇒ ττ,, the complex structurethe complex structure J J of CY: of CY: Size moduli Size moduli ⇒ ⇒ Kahler moduli of CY:Kahler moduli of CY:

4D effective action (tree)4D effective action (tree) [Gukov, Vafa & Witten (2000)][Gukov, Vafa & Witten (2000)]

Discrete set of vacuaDiscrete set of vacuaFor τ=const, For τ=const,

Hence,Hence, W W depends only on the cohomology classes of 3-form fluxes: depends only on the cohomology classes of 3-form fluxes:

StabilisationStabilisation of all shape moduli of all shape moduli

This privides constraints. This privides constraints. [Giddings, Kachru & Polchinski(2002)][Giddings, Kachru & Polchinski(2002)]

No scale modelNo scale modelBecause Because WW is independent of ρ and is independent of ρ and DW=dW + dK W , DW=dW + dK W , we havewe have

The size modulus ρ is not stabilised.The size modulus ρ is not stabilised.

Three Major Requirements on HUTThree Major Requirements on HUT

4-dimensional universe at present4-dimensional universe at presentcompactification of the extra dimensioncompactification of the extra dimensionmoduli stabilisationmoduli stabilisation

Realistic low energy physicsRealistic low energy physicsgauge/generation structure, symmetry breaking, gauge/generation structure, symmetry breaking,

……Primordial inflation and dark energyPrimordial inflation and dark energy violation of strong energy conditionviolation of strong energy condition

Obstructions against inflationObstructions against inflation

No de Sitter SUSY vacuumNo de Sitter SUSY vacuumNo-Go theorem against accelerated cosmic expansionNo-Go theorem against accelerated cosmic expansion

Assume Assume • The spacetime metric is of the warped product typeThe spacetime metric is of the warped product type

• The internal space is static and compact without boundary.The internal space is static and compact without boundary.• The warp factor is regular and bounded everywhere.The warp factor is regular and bounded everywhere.• The strong energy condition is satisfied in the full theory.The strong energy condition is satisfied in the full theory. Then, no accelerated expansion is allowed for the four-dimensional Then, no accelerated expansion is allowed for the four-dimensional spacetimespacetime

Proof

KKLT Construction KKLT Construction (2003)(2003)

Stabilisation of the volume modulusStabilisation of the volume modulusWhen there exists a special 4-cycle Z on YWhen there exists a special 4-cycle Z on Y

• Euclidean D3-brane wrapped on Z Euclidean D3-brane wrapped on Z ⇒ ⇒ • D7 brane wrapped on Z D7 brane wrapped on Z ⇒⇒ gaugino condensate gaugino condensate ⇒⇒

Combining with the no scale model Combining with the no scale model ⇒ ⇒ Stable supersymmetric AdS vacuum with all moduli fixedStable supersymmetric AdS vacuum with all moduli fixed

Uplifting the vacuumUplifting the vacuumAdding N anti-D3 branes at the KS throatAdding N anti-D3 branes at the KS throat

Adjusting the number of anti-D3 branesAdjusting the number of anti-D3 branes ⇒⇒ (Metastable) dS vacuum(Metastable) dS vacuum

DifficultiesDifficultiesNon-genericity of the size modulus stabilisationNon-genericity of the size modulus stabilisationKKLT construction: Instanton effects KKLT construction: Instanton effects ⇒⇒ Kaehler moduli stabilisation Kaehler moduli stabilisation

The volume modulus is not stabilised in a model with a single The volume modulus is not stabilised in a model with a single Kaeher modulus Kaeher modulus [Denef, Douglas and Florea 2004][Denef, Douglas and Florea 2004]There exist models in which all Kaehler moduli are stabilsed There exist models in which all Kaehler moduli are stabilsed by the instanton effects, but they are not genericby the instanton effects, but they are not generic

[Denef et al 2005, Aspinwall & [Denef et al 2005, Aspinwall & KalloshKallosh 2005] 2005]

Fluxes produce warped geometryFluxes produce warped geometryWarped structure is not taken into account in the volume moduluWarped structure is not taken into account in the volume modulus stabilisation argument.s stabilisation argument.

The orbifold singularity with negative chargeThe orbifold singularity with negative charge In order to evade the No-Go theorem, singular objects or branesIn order to evade the No-Go theorem, singular objects or branes/orientifold planes with negative tension are required./orientifold planes with negative tension are required.These singular objects produce naked singularities in classical These singular objects produce naked singularities in classical supergravity, which may not be resolved in quantum theory.supergravity, which may not be resolved in quantum theory.

Warped Compactification

Flux Produces WarpFlux Produces WarpEinstein EquationsEinstein Equations

If is y-dependent. Hence, the 4D metricIf is y-dependent. Hence, the 4D metric depends not only on x but also on the internal coordinate y.depends not only on x but also on the internal coordinate y.

Warped GeometryWarped Geometry In order to preserve some supersymmetry, we assume that the In order to preserve some supersymmetry, we assume that the

metricmetric takes the warped product formtakes the warped product form

Conifold compactificationConifold compactificationWhen When YY is Ricci flat, the 3-form flux vanishes and the 5-form flux is Ricci flat, the 3-form flux vanishes and the 5-form flux takes the formtakes the form

the field equations are reduced tothe field equations are reduced to

For example, for the 5-form flux produced by D3 branes with flat For example, for the 5-form flux produced by D3 branes with flat X, we obtain a regular spacetime with full SUSY:X, we obtain a regular spacetime with full SUSY:

Note that Note that when D3 branes and anti-D3 branes coexist, naked siwhen D3 branes and anti-D3 branes coexist, naked singularities appear,ngularities appear, becausebecause

Klebanov-Strassler solutionKlebanov-Strassler solutionConstant τ, 5-form flux and ISD 3-form fluxConstant τ, 5-form flux and ISD 3-form flux

CY=deformed conifold regular at r=0CY=deformed conifold regular at r=0

Then we haveThen we have

Compactification of this solution was discussed by Giddings, KCompactification of this solution was discussed by Giddings, Kachru and Polchinski (2002)achru and Polchinski (2002)

Size Moduli DynamicsSize Moduli DynamicsAssumptionsAssumptions

MetricMetricAll moduli except for the size modulus are stabilisedAll moduli except for the size modulus are stabilisedForm fieldsForm fields

General solutionGeneral solution [Kodama & Uzawa, JHEP0507:061(2005)][Kodama & Uzawa, JHEP0507:061(2005)]

If and the metric has no null KillingIf and the metric has no null Killing

wherewhere

proof

ImplicationsImplicationsThe size modulus is still unstable even if warp is taken into The size modulus is still unstable even if warp is taken into account and is associated with full SUSY breaking of the orderaccount and is associated with full SUSY breaking of the order

However, when the volume modulus is However, when the volume modulus is xx-dependent, -dependent, the the xx--dependent part does not factor out from the warp factordependent part does not factor out from the warp factor as is as is assumed in most effective four-dimensional theories.assumed in most effective four-dimensional theories.

The concept of the Einstein frame loses meaning,.The concept of the Einstein frame loses meaning,.In a strongly warped region where is large as at the KS In a strongly warped region where is large as at the KS throat,throat, the volume modulus can be effectively stabilised for a long the volume modulus can be effectively stabilised for a long time.time.Hence, Hence, if we live in a brane at a deep KS throat, all moduli can be if we live in a brane at a deep KS throat, all moduli can be stabilised and SUSY is preserved for a sufficiently long time, even stabilised and SUSY is preserved for a sufficiently long time, even without quantum stabilisation effects.without quantum stabilisation effects.When the instability grows, it produces a Big-Rip singularity if a<0.When the instability grows, it produces a Big-Rip singularity if a<0.[cf. Gibbons, Lu & Pope(2005)].[cf. Gibbons, Lu & Pope(2005)].

Cosmology Orders Naked SingularitiesCosmology Orders Naked SingularitiesIn order to get a solution with compact and closed Y, we have to In order to get a solution with compact and closed Y, we have to introduce some singular sources, such as the orientifold O3introduce some singular sources, such as the orientifold O3 with with negative charge, in order to cancel the negative term in the right-negative charge, in order to cancel the negative term in the right-hand side of the equationhand side of the equation

in accordance to the No-Go theorem (in accordance to the No-Go theorem (the tadpole conditionthe tadpole condition).). Such a source gives rise to a singular negative contribution to the Such a source gives rise to a singular negative contribution to the warp factor warp factor hh in general . in general .Hence, in order to realise the moduli stabilisation and the inflation, Hence, in order to realise the moduli stabilisation and the inflation, we have to allow for the existence of naked singularities in the we have to allow for the existence of naked singularities in the classical limit.classical limit.Cf. IIA moduli stabilisation Cf. IIA moduli stabilisation [DeWolfe, Giryavets, Kachru & Taylor 2005][DeWolfe, Giryavets, Kachru & Taylor 2005]

Four-Dimensional Effective Theories

No Flux CaseNo Flux CaseAssumptionsAssumptions

The 10D metric has the formThe 10D metric has the form

All moduli except for the size modulus are stabilisedAll moduli except for the size modulus are stabilised

No fluxNo flux

Effective actionEffective action

Characteristic FeaturesCharacteristic FeaturesThis effetive 4-dim theory is This effetive 4-dim theory is equivalentequivalent to the original 10-dim to the original 10-dim theory under the above ansatz.theory under the above ansatz.

For For , the effective action is invariant under the , the effective action is invariant under the trasformation trasformation 　　　　 in the Einstein frame:in the Einstein frame:

The solution in which h increases with the cosmic expandion: The solution in which h increases with the cosmic expandion:

The solution in which h decreases with the cosmic expansion. The solution in which h decreases with the cosmic expansion. . .

We will see that We will see that only the first decompactifying solution survives in only the first decompactifying solution survives in the warped compactificationthe warped compactification..

IIB Supergravity with FluxIIB Supergravity with FluxAnsatz on 10-dim IIB FieldsAnsatz on 10-dim IIB Fields

Metric:Metric:

Dilaton :Dilaton :Gauge fields:Gauge fields:

Warp factor:Warp factor:

Effective ActionEffective ActionThe effective action for 4-dim fields and is The effective action for 4-dim fields and is given bygiven by

wherewhere

Characteristic FeaturesCharacteristic FeaturesThe 4-dim effective action has the same form as that in the nThe 4-dim effective action has the same form as that in the no-flux case.o-flux case.In particular, the effecitve 4-dim theory has the modular invarIn particular, the effecitve 4-dim theory has the modular invariance in the “Einstein frame” that is not respeciance in the “Einstein frame” that is not respected in the original 10-dim theory:ted in the original 10-dim theory:

Heterotic M TheoryHeterotic M Theory11-dim 11-dim 5-dim 5-dim

AnsatzAnsatz

Field equationsField equations

5-dim effective action5-dim effective action

5-dim 5-dim 4-dim 4-dim CCGLP-type solutionCCGLP-type solution [Chen et al (2005), Kodama & Uzawa(2005)][Chen et al (2005), Kodama & Uzawa(2005)]

Under the ansatzUnder the ansatz

the general solution is given bythe general solution is given by

From this it follows thatFrom this it follows that

4-dim effective action4-dim effective action

Summary and

Discussions

Size Modulus Instability Size Modulus Instability

FFor warped compactifications in the type IIB sugra anor warped compactifications in the type IIB sugra and the heterotic M theoryd the heterotic M theory

In the classical sugra framework, the size modulus In the classical sugra framework, the size modulus is unstable even for the warped compactification wis unstable even for the warped compactification with flux.ith flux. The instability mode produces an x-dependent adThe instability mode produces an x-dependent additive correction to the warp factor as ditive correction to the warp factor as . In particular, the instability is effectively sup . In particular, the instability is effectively suppressed in a strongly warped region.pressed in a strongly warped region.This instability can be stabilised only by quantum eThis instability can be stabilised only by quantum effects or other corrections to sugra in higher dimenffects or other corrections to sugra in higher dimensions.sions.

4-dim effective theory4-dim effective theory

We can construct an effective 4-dim theory for the We can construct an effective 4-dim theory for the metric and the size modulus that reproduces the inmetric and the size modulus that reproduces the instability in the original theory. stability in the original theory. The effective theories obtained in this way for warpThe effective theories obtained in this way for warped compactification are identical to that in the no ed compactification are identical to that in the no warp case. warp case. In particular,In particular, the effective theory allows a wider clathe effective theory allows a wider class of solutions than the original higher-dimensional ss of solutions than the original higher-dimensional theorytheory under the ansatz used to obtain the 4-dim efunder the ansatz used to obtain the 4-dim effective theory.fective theory.

Cf. Cf. Arroja and Koyama [hep-th/0602068] argued that this conclArroja and Koyama [hep-th/0602068] argued that this conclusion is not correct at least in the 5-dim HW theory, by showiusion is not correct at least in the 5-dim HW theory, by showing that any solution to the effective 4-dim theory can be uplifng that any solution to the effective 4-dim theory can be uplifted to a 5-dim solution by the gradient expansion method up ted to a 5-dim solution by the gradient expansion method up to the first order, if the ansatz on the 5-dim metric is modified to the first order, if the ansatz on the 5-dim metric is modified to include a slightly wider class.to include a slightly wider class.

Answer to this critique isAnswer to this critique is They have not shown that the uplifting is possible in the fThey have not shown that the uplifting is possible in the full order.ull order. Our argument in the IIB case is much more general, anOur argument in the IIB case is much more general, and it does not seem possible to construct uplifting of 4-dim d it does not seem possible to construct uplifting of 4-dim solutions by simple modification of the ansatz.solutions by simple modification of the ansatz. If we do not impose any ansatz on the higher-dimensionIf we do not impose any ansatz on the higher-dimensional configurations, it is always possible to find some upliftial configurations, it is always possible to find some uplifting of 4-dim solutions, but uplifting is far from unique. The ng of 4-dim solutions, but uplifting is far from unique. The essential point is to find a good ansatz on higher-dimensiessential point is to find a good ansatz on higher-dimensional solutions.onal solutions.