Moduli Dynamics in Warped Compactification

1. Moduli Dynamics in Warped Compactification YITP Hideo Kodama Based on the work in collaboration with Kunihito Uzawa

2. Introduction Higher-Dimensional Unification • Dark matter, flat inflaton potential ⇒ SUSY • The repitition of generations with the same structurecan be naturally explained in higher-dimensional unification. • Superstring/M theory is the only (perturbatively) consistent higher-dimensional unfied theory at present. Compactification • Moduli stabilisation • No-Go theorem against accelerating expansion • The KKLT construction and the KKLMMT brane inflation model based on the flux compactification may provide a higher-dimensional cosmological model consistent with osbservations. • In these models, however, the warped structure of the geometry is not fully taken into accout, and the 10D structure of the model is not clear.

3. IIB SUGRA Bosonic field contents • 10D metric • Axion, dilaton: • 3-form flux: • 5-form flux: Action

4. Calabi-Yau Compactification Product-type compactification with no flux • N=2 SUSY vacuum on Moduli • Shape moduli • Dilaton-axion moduli τ: 1 • Complex structure moduli of CY: • Size moduli • Kaehler moduli of CY: this includes the volume modulus of CY

5. 4D Effective Theory KK ansatz 4D effective action (tree) where and

6. Moduli Stabilisation Flux compactification • Discrete set of vacua • Stabilisation of all shape moduli[Giddings, Kachru & Polchinski(2002)] This privides constraints.

7. Difficulties • The size modulus is not stabilised 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 Kaeler moduli are stabilsed by the instanton effets, 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 statilisation 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

8. Warped Compactification Einstein equations If is y-dependent. Hence, the 4D metric depends not only on x but also on the internal coordinate y. Factorisation ansatz In order to preserve some supersymmetry, we often assume that the metrictakes the form

9. Conifold compactification • When Y is Ricci flat , 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

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

11. No-Go Theorem For any (warped) compactification with a compact closed internal space, if the strong energy condition holds in the full theory and all moludi are stabilized, no stationary accelerating expansion of the four-dimensional spacetime is allowed. • Proof For the geometry from the relation for any time-like unit vector V on X, we obtain Hence, if Y is a compact manifold without boundary, is a smooth function on Y, and the strong energy condition is satsified in the (n+4)-dimensional theory, then the strong energy condition is satisfied on X.

12. Moduli Instability: No Flux Case Assumptions • The 10D metric has the form (Einstein frame) • All moduli except for the size modulus are stabilised. • No flux Effective action Since the superpotential potential vanishes, the 4D effective action is given by

13. General solution s.t. h=h(t) and X is a flat Friedmann • There exist two types of general solutions due to the invariance of the action under the trasformation • 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.

14. Moduli Instability: Warped Case Assumptions • Metric • All moduli except for the size modulus are stabilised • Form fields General solution[Kodama & Uzawa hep-th/0504193] If and the metric has no null Killing where

15. Implications • For and no null Killing, it is required that the 4D spacetime X is flat. Hence, in order to allow for a solution whose 4D metric is as in the KKLT construction by the instanton effect, the field equation for and the Einstein equations must be largely modified by the quantum effect. • For , the above solution is not a general solution. • In order to get a solution with compact Y, we have to introduce some singular sources, such as the orientifold O3with negative charge, to cancel the negative term in the right-hand side of the equation in accordance to the No-Go theorem. Such a source gives rise to a singular negative contribution to the warp factor h in general.

16. When is time-like, h can be written . • Therefore, 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. • In the region where the y-dependence of h is small, the dynamical behavior of h is the same as that of the decompactifying solution in the no warp case. • However, in a strongly warped region where is large as at the KS throat, the volume modulus can be effectively stabilised for a long time. • The instability is associated with full SUSY breaking of the order • 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.

17. Summary We have studied the moduli dynamics in the warped compactification of IIB SUGRA and have found • Warp produced by flux has significant effects on the moduli dynamics as well as on the 4D geometry. • In particular, in a region with strong warp, the volume modulus is effectively stabilised for a long time even in the classical framework. Further, its destabilisation can produce a Big-Rip type singularity and SUSY breaking. • These features may be used to construct a realistic cosmological model by combining it with the brane-world scenario, in which accelerated expansion is realised only in a brane. • In order to explore this possibility, we have to investigate various problems: the structure of KK-modes and the 10D description of quantum effects in the warped compactification, brane-world models in 10D framework, extensions of the analysis to more general cases and other string/M theories and so on.