Dynamical Compactification by Flux in M-theory

1. Dynamical Compactification by Flux in M-theory Cosmophysics Group IPNS, KEK, Japan Hideo Kodama GRG18, Sydney, 10 July 2007

2. Contents • Introduction • Light-Like Solutions in M-theory • Tilted Toroidal Compactifications • Summary

3. Uniqueness and Diversity • M-theory  (Might be) Unique Theory of Everything But, the theory allows vast spectra of competent compactifications and ground states/low energy physics.  Landscape Problem • No boundary proposal  Unique IC of the Universe But, the quantum nature of the initial state prohibits unique predictions about the low energy physics and may lead to the chaotic inflation scenario. • Chaotic inflation  All possible states are realised. Inflation eventually freezes the low energy physics in the observed region.

4. What determines the low energy physics? Accidental ? Anthropic? Dynamical? In order to see whether the low energy physics can be selected or restricted by dynamics, we have to compare various ground states/compactifications and investigate their stability by connecting them spatially or temporally.

5. How to Compare Different Vacua Classical Stability Analysis by Perturbations Dynamical Solution Spatially Connecting Two Solutions space Semi-Classical Stability Analysis Solution I Solution II time Vacuum Solution I Instanton time Vacuum Solution II

6. Compactification Transition Semi-Classical Transitions in SUGRA 1982 Kaluza-Klein bubble: instability of E3,1£ S1[Witten] 1987 Instability of En,1£ Tm , stability of E3,1£ CY [Mazur;Brill, Horowitz 1991] 1997 Stability due to spin structure [Tayler-Robinson] Singular Transitions/Topology Changes in SUGRA 1993 Flop transition [Witten; Aspinwall, Greene, Morrison 1994] 1995 Conifold transition [Greene, Morrison, Strominger] 2002 Symplectic conifold transitions [Smith I, Thomas, Yau] 2003 Cosmological flop transition in M-theory [Brandle, Lukas] 2005 Cosmological conifold transition in IIB sugra [Lukas, Palti, Saffin] Transitions in String Theories 2004 Dynamical dimension change in supercritical string theories [Hellerman et al 2004-2007] 2006 D-duality [Silverstein et al] 2007 Heterotic Kahler/non-Kahler transition [Becker, Tseng, Yau; Sethi] Non-Singular Transitions in SUGRA 2006 Toroidal compactification-decompactification transition in M-theory [Kodama, Ohta]

7. Light-Like Solutions in M-theory

8. M Theory Bosonic Field Contents Action for the bosonic sector Field Equations

9. SCI Light-Like Solutions Ansatz • Metric: • The spacetime and F[4] are regular. • Supersymmetry General Solution Supersymmetry • 16 Killing spinors at least. • 32 Killings spinors for the locally flat case (F[4]=0) and the Kowalski-Glikman solution( a special pp-wave solution). HK & N Ohta, hep-th/0605179; T Ishino, HK & N Ohta, PLB631:68(2005)

10. Apparent Singularity of the Solution If the solution is asymptotically flat at u!1 or u! -1, det g should vanish at a finite value of u, because we can show that u2 (det g)1/18 <0 unless the spacetime is locally flat. This does not imply that the spacetime is always singular because For example, the apparently non-trivial solution for vanishing flux can be transformed into the explicitly flat form in terms of the coordinate transformation

11. Diagonal Solutions with Two AF Regions Let us consider the diagonal solution where ef in u>0 behaves as Then, this solution represents a spacetime that is asymptotically flat at u»1. Further, it can be extended to the exactly flat spacetime in u<0, because the metric can be put in the form . by the coordinate transformation

12. Tilted Toroidal Compactification

13. Simple Example Standard compactification Let us consider the diagonal solution representing a spacetime that is asymptotically flat at u»1 and exactly flat in u<0, where ef behaves as If we compactify this spacetime by the standard lattice consisting of pure translations in the z-directions Then, the compactified internal space becomes singular at u=0.

14. Inclusion of boosts In order to avoid this type of singularity, let us include boosts in the transformations as where and Then, in the coordinates the above transformation can be written at u»0 for .

15. Degenerate Singularity In the asymptotically flat region u»1, the corresponding transformation can be written where G’pq=(Gp-e’p)pq. Hence, the size of the compactified z-space diverges as u !1. This region, however, cannot be connected smoothly to the static region u· 0 because G’ becomes degenerate at some u=u*>0.

16. Tilted compactification This type of singularity associated with the degeneracy of the lattice action can be avoided if we modify the transformation to have a y-translation component. For example, consider the case in which the internal space is 1-dimensional, and G’ behaves as G’» k(u-u1) . If we identify this spacetime by the translation we obtain a regular spacetime around u» u1 if a  0. Thus, in general, instead of singularity, the compactified directions abruptly rotate every time when G’ becomes degenerate.

17. Compactification-Decompactification Transition Because the u<0 region in the previous example is the product of the 4-dim Minkowski spacetime and a static torus T7, we can construct a solution describing creation of a static compactified region in a decompactified background, by the cut and paste of two copies of the solution.

18. Summary

19. We have constructed a supersymmetric classical solution with flux that spatially connects two susy solutions, E3,1£ T7 and E1,10, in M-theory. • Supersymmetric solutions can be unstable (possibly even if the fermion sector is taken into account). • A chaotic landscape in the early universe might be dynamically modified. • It would be quite interesting if we can find dynamical solutions connecting more realistic vacua or some systematic method to construct them. • Such solutions break supersymmetry in general.