Is the four-dimensional effective theory effective?

1. Is the four-dimensional effective theory effective? YITP Hideo Kodama HK and Kunihito Uzawa, JHEP0507:061(2005) HK and Kunihito Uzawa, hep-th/0512104

2. Three Major Requirements on HUT • 4-dimensional universe at present • compactification of the extra dimension • moduli stabilisation

3. Moduli Stabilisation Compactification without flux In this case, D=11/10 sugra reduces to a theory for gravity coupled with dilaton. Assume Then In the cosmological context, the second equation leads to

4. Flux compactification Moduli for IIB Sugra • Shape moduli ⇒ τ,the complex structureJ of CY: • Size moduli ⇒ Kahler moduli of CY: 4D effective action (tree)[Gukov, Vafa & Witten (2000)]

5. Discrete set of vacua For τ=const, Hence, W depends only on the cohomology classes of 3-form fluxes: • Stabilisation of all shape moduli This privides constraints. [Giddings, Kachru & Polchinski(2002)] • No scale model Because W is independent of ρ and DW=dW + dK W , we have The size modulus ρ is not stabilised.

6. Three Major Requirements on HUT • 4-dimensional universe at present • compactification of the extra dimension • moduli stabilisation • Realistic low energy physics • gauge/generation structure, symmetry breaking, … • Primordial inflation and dark energy • violation of strong energy condition

7. Obstructions against inflation • No de Sitter SUSY vacuum • No-Go theorem against accelerated cosmic expansion • Assume • The spacetime metric is of the warped product type • The internal space is static and compact without boundary. • The warp factor is regular and bounded everywhere. • The strong energy condition is satisfied in the full theory. • Then, no accelerated expansion is allowed for the four-dimensional spacetime Proof

8. KKLT Construction (2003) Stabilisation of the volume modulus • When there exists a special 4-cycle Z on Y • Euclidean D3-brane wrapped on Z ⇒ • D7 brane wrapped on Z ⇒ gaugino condensate ⇒ • Combining with the no scale model ⇒Stable supersymmetric AdS vacuum with all moduli fixed Uplifting the vacuum • Adding N anti-D3 branes at the KS throat • Adjusting the number of anti-D3 branes ⇒(Metastable) dS vacuum

9. Difficulties • Non-genericity of the size modulus stabilisation KKLT construction: Instanton effects ⇒ Kaehler moduli stabilisation • The volume modulus is not stabilised in a model with a single Kaeher modulus [Denef, Douglas and Florea 2004] • There exist models in which all Kaehler moduli are stabilsed by the instanton effects, but they are not generic [Denef et al 2005, Aspinwall & Kallosh 2005] • Fluxes produce warped geometry Warped structure is not taken into account in the volume modulus stabilisation argument. • The orbifold singularity with negative charge • In order to evade the No-Go theorem, singular objects or branes/orientifold planes with negative tension are required. • These singular objects produce naked singularities in classical supergravity, which may not be resolved in quantum theory.

10. Warped Compactification

11. Flux Produces Warp Einstein Equations If is y-dependent. Hence, the 4D metric depends not only on x but also on the internal coordinate y. Warped Geometry In order to preserve some supersymmetry, we assume that the metrictakes the warped product form

12. Conifold compactification • When Y is Ricci flat, the 3-form flux vanishes and the 5-form flux takes the form the field equations are reduced to • For example, for the 5-form flux produced by D3 branes with flat X, we obtain a regular spacetime with full SUSY: • Note that when D3 branes and anti-D3 branes coexist, naked singularities appear,because

13. Klebanov-Strassler solution • Constant τ, 5-form flux and ISD 3-form flux • CY=deformed conifold regular at r=0 Then we have • Compactification of this solution was discussed by Giddings, Kachru and Polchinski (2002)

14. Size Moduli Dynamics Assumptions • Metric • All moduli except for the size modulus are stabilised • Form fields General solution[Kodama & Uzawa, JHEP0507:061(2005)] If and the metric has no null Killing where proof

15. Implications • The size modulus is still unstable even if warp is taken into account and is associated with full SUSY breaking of the order • However, when the volume modulus is x-dependent, the x-dependent part does not factor out from the warp factor as is assumed in most effective four-dimensional theories. • The concept of the Einstein frame loses meaning,. • In a strongly warped region where is large as at the KS throat,the volume modulus can be effectively stabilised for a long time. • Hence, 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 without quantum stabilisation effects. • When the instability grows, it produces a Big-Rip singularity if a<0. [cf. Gibbons, Lu & Pope(2005)].

16. Cosmology Orders Naked Singularities • In order to get a solution with compact and closed Y, we have to introduce some singular sources, such as the orientifold O3with negative charge, in order to cancel the negative term in the right-hand side of the equation in accordance to the No-Go theorem (the tadpole condition). • Such a source gives rise to a singular negative contribution to the warp factor h in general . • Hence, in order to realise the moduli stabilisation and the inflation, we have to allow for the existence of naked singularities in the classical limit. Cf. IIA moduli stabilisation [DeWolfe, Giryavets, Kachru & Taylor 2005]

17. Four-Dimensional Effective Theories

18. No Flux Case Assumptions • The 10D metric has the form • All moduli except for the size modulus are stabilised • No flux Effective action

19. Characteristic Features • This effetive 4-dim theory is equivalent to the original 10-dim theory under the above ansatz. • For , the effective action is invariant under the trasformation in the Einstein frame: • The solution in which h increases with the cosmic expandion: • The solution in which h decreases with the cosmic expansion. . • We will see that only the first decompactifying solution survives in the warped compactification.

20. IIB Supergravity with Flux Ansatz on 10-dim IIB Fields • Metric: • Dilaton : • Gauge fields: • Warp factor:

21. Effective Action The effective action for 4-dim fields and is given by where Characteristic Features • The 4-dim effective action has the same form as that in the no-flux case. • In particular, the effecitve 4-dim theory has the modular invariance in the “Einstein frame” that is not respected in the original 10-dim theory:

22. Heterotic M Theory 11-dim  5-dim Ansatz Field equations 5-dim effective action

23. 5-dim  4-dim CCGLP-type solution[Chen et al (2005), Kodama & Uzawa(2005)] Under the ansatz the general solution is given by From this it follows that 4-dim effective action

24. Summary and Discussions

25. Size Modulus Instability For warped compactifications in the type IIB sugra and the heterotic M theory • In the classical sugra framework, the size modulus is unstable even for the warped compactification with flux. • The instability mode produces an x-dependent additive correction to the warp factor as . In particular, the instability is effectively suppressed in a strongly warped region. • This instability can be stabilised only by quantum effects or other corrections to sugra in higher dimensions.

26. 4-dim effective theory • We can construct an effective 4-dim theory for the metric and the size modulus that reproduces the instability in the original theory. • The effective theories obtained in this way for warped compactification are identical to that in the no warp case. • In particular,the effective theory allows a wider class of solutions than the original higher-dimensional theoryunder the ansatz used to obtain the 4-dim effective theory.

27. Cf. Arroja and Koyama [hep-th/0602068] argued that this conclusion is not correct at least in the 5-dim HW theory, by showing that any solution to the effective 4-dim theory can be uplifted 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 include a slightly wider class. Answer to this critique is • They have not shown that the uplifting is possible in the full order. • Our argument in the IIB case is much more general, and it does not seem possible to construct uplifting of 4-dim solutions by simple modification of the ansatz. • If we do not impose any ansatz on the higher-dimensional configurations, it is always possible to find some uplifting of 4-dim solutions, but uplifting is far from unique. The essential point is to find a good ansatz on higher-dimensional solutions.