Dynamics of Compactification in HUNT

1. Dynamics of Compactification in HUNT YITP Kyoto University Hideo Kodama 細谷暁夫教授還暦記念研究会 2006年6月23日－２５日 箱根湯元

2. 1960 ADM formalism 1964 Dirac Qz of gravity 1967 Wheeler-DeWitt equation 1968-70 Bianchi cosmology 1982 Creation of Universe from Nothing 1983 No boundary proposal ・・・ Hawking – Vilenkin controversy 1988 Baby universe model 1989 Non-predictivity of EPI QG [Halliwell, Louko; Polchinski] 1926 Kaluza-Klein theory ・ ・ ・ ・ 1978 D=11 SUGRA model 1981Inflation model 1982New inflation model Galaxies from quantum fluctuations 1984Superstring theories 1984 Finite temp. QFT [Hosaya, Sakagami, ..] 1985 Chaos in KK model [Furusawa, Hosoya] 1990 (2+1)-dim QG [Hosoya, Nakao] Brief History of Quantum Cosmology

3. 1998 Acceleration of the Universe (SN IIb obs.) 1998 TeV QG 1999 BOOMERANG 1999 RS braneworld model 2003 WMAP(1st year) 2003 KKLMMT model 2005 BOOMERANG03 2006 WMAP(3rd year) 1995 Unification of SSTs by Duality M theory 1996D branes F theory, Horava-Witten theory 1997 AdS/CFTConjecture 2000 Landscapeproblem 2002 Moduli stabilisation by flux 2003 KKLT construction 2004 Racetrack model 2006 Better racetrack model Toward New Framework

4. Initial Condition of Universe and Low Energy Physics Monotheism vs Polytheism • 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 freezes the low energy physics in the observed region. • 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  All

5. 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. Compactification Transition 1993 Flop transition [Witten E; Aspinwall, Greene, Morrison 1994] 1995 Conifold transition [Greene, Morrison, Strominger] 1997 Semi-classical stability of sugra vacua [Taylor-Robinson MM] 2003 Cosmological flop transition in M-theory [Brandle M, Lukas A] 2005 Cosmological conifold transition in IIB sugra [Lukas, Palti, Saffin] 2006 Toroical compactification-decompactification transition in M-theory [Kodama H, Ohta N ]

6. M Theory Field Contents Action for the bosonic sector Field Equations

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

8. PP-wave Solutions Solutions[ Hull CM (1984)] where [3]is a harmonic function on R9, and H is determined from [3] as Supersymmetry This class of solutions has 16 Killing spinors in general. Further, when [3] is a constant 3-form, and H is quadratic with respect to x, the solution can have 18, 20, 22, 24, 26 and 32 Killing spinors for special choices of [3] and H. These solutions can be transformed into the x-independent light-like form in terms of the coordinate transformation

9. Kowalski-Glikman Solution The Kowalski-Glikman solution is expressed as This solution can be transformed into the x-independent form by the coordinate transformation

10. Locally Flat Solution In the diagonal case the curvature can be written The apparently non-trivial solution for vanishing flux can be transformed into the explicitly flat form in terms of the coordinate transformation

11. Isometries The SCI light-like solution is invariant under the transformation where a=(ai), b=(bi) and c are constants, and G=(Gij) is defined by This type of transformations form the maximal common isometry group Gm. The multiplication of two elements in this group is expressed as In particular, from Gm is a nilpotent group.

12. Toroidal Compactification Metric Ansatz We consider a class of light-like solutions that can be written as the product of 4-dim Minkowski spacetime (x0,x) ( x =(x10,y) =(x10,x1,x2) ) and a 6-dim internal space z=(x3,,x9): Abelian Discrete Isometry Group The elements of Gm commute with each other and generate an Abelian group Z7 :

13. General Behavior of the Solution Apparent singularity of g(u) Let us diagonalise g(u) by an orthogonal matrix O(u) and define the anti-symmetric matrix  as Then R-- can be written Further, for=(det g)1/14, we obtain where the last equality holds only when the spacetime is locally flat. Hence, if the solution is asymptotically flat at u!1 and u! -1 , det g, hence some Dp should vanish at a finite value of u.

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

15. Inclusion of boosts This kind of singularity can be avoided if we modify the lattice to include boosts as In this case, from the transformation g2 can be written at u»0 as for in the coordinates

16. 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.

17. 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.

18. 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.