Compactification and 4D Effective Theory

1. Compactification and 4D Effective Theory YITP Hideo Kodama HK and Kunihito Uzawa, JHEP0507:061(2005) HK and Kunihito Uzawa, JHEP0603:035(2006)

2. Three Major Requirements on HUNT • 4-dimensional universe at present • compactification of extra dimensions • moduli stabilisation • Primordial inflation and dark energy • violation of strong energy condition • Realistic low energy physics • gauge group, gauge/susy symmetry breaking, chiral fermions, generation repetition, , …

3. Moduli Stabilisation Compactification without flux In this case, D=11/10 sugra can be reduced 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 ⇒,the complex structureJ of CY: h 2,1 +1 • Size moduli ⇒ Kahler moduli of CY: h1,1 4D effective action (tree) [Gukov, Vafa, Witten 2000;Taylor, Vafa 2000;Giddings, Kachru, Polchinski 2002]

5. Discrete set of vacua For  =const, Hence, W depends only on the cohomology classes of 3-form fluxes: • Stabilisation of all shape moduli This privides h2,1+1 constraints for the h2,1+1 complex moduli. [Giddings, Kachru & Polchinski(2002)] • No scale model Because W is independent of ρ and DW=dW + dK W , we have The size L = e u of the extra-dimension is not stabilised.

6. 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

7. 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

8. 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.

9. Warped Compactification

10. Flux Produces Warp Einstein Equations If G30 and F5 0 , they have non-trivial y-dependence. Hence, R is y-dependent, and as a consequence, the 4D metric g 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

11. 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 AdS5£ S5 with full SUSY: • Note that when D3 branes and anti-D3 branes coexist, naked singularities appear,because

12. Klebanov-Strassler solution • Constant , 5-form flux and ISD 3-form flux; *YG3= iG3 • CY=deformed conifold regular at r=0 Then we have • The orientifold/F-theory compactification of this solution was discussed by Giddings, Kachru and Polchinski (2002)

13. Size Moduli Dynamics Assumptions • Metric • All moduli except for the size modulus are stabilised • Form fields General solution[Kodama & Uzawa, JHEP0507:061(2005)] If G30 and the metric has no null Killing where proof

14. 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 h1(y) 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)].

15. 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, it appears that in order to realise the moduli stabilisation and the inflation, we have to allow for the existence of naked singularities in the classical limit. Is this unavoidable? Are these singularities harmless? Cf. IIA moduli stabilisation [DeWolfe, Giryavets, Kachru & Taylor 2005]

16. Four-Dimensional Effective Theories

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

18. Characteristic Features • This effective 4-dim theory is equivalent to the original 10-dim theory under the above ansatz. • For =0 , the effective action is invariant under the trasformation in the Einstein frame: h!h-1 • The solution in which h increases with the cosmic expansion: • The solution in which h decreases with the cosmic expansion. • We will see that only the first decompactifying solution survives in the warped compactification.

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

20. Effective Action The effective action for 4-dim fields h0(x) and g(X) 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 H!k/H in the “Einstein frame” that is not respected in the original 10-dim theory:

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

22. 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

23. Summary and Discussions

24. 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 h= h0(x) + h1(y). In particular, the instability is effectively suppressed in a strongly warped region. • This instability can be globally stabilised only by quantum effects or other corrections to sugra in higher dimensions.

25. 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 theory under the ansatz used to obtain the 4-dim effective theory. Cf. Arroja and Koyama [hep-th/0602068] : criticised our metric ansatz in the 5-dim HW theory.