Lecture by Pietro Frè and Alexander S. Sorin

1. Full Integrability of Supergravity Billiards: the arrow of time, asymptotic states and trapped surfaces in the cosmic evolution Lecture by Pietro Frè and Alexander S. Sorin In connection with the nomination for the JINR prize 2008

2. Standard Cosmology • Standard cosmology is based on the cosmological principle. • Homogeneity • Isotropy

3. Evolution of the scale factorwithout cosmological constant

4. From 2001 we know that the Universe is spatially flat (k=0) and that it is dominated by dark energy. Most probably there has been inflation

5. The scalar fields drive inflation while rolling down from a maximum to a minimum • Exponential expansion during slow rolling • Fast rolling and exit from inflation • Oscillations and reheating of the Universe

6. The isotropy and homogeneity are proved by the CMB spectrum

7. WMAP measured anisotropies of CMB The milliKelvin angular variations of CMB temperature are the inflation blown up image of Quantum fluctuations of the gravitational potential and the seeds of large scale cosmological structures

8. Equation of State Accelerating Universe dominated by Dark Energy

9. Non isotropic Universes • This is what happens if there is isotropy ! • Relaxing isotropy an entire new world of phenomena opens up • In a multidimensional world, as string theory predicts, there is no isotropy among all dimensions!

10. Cosmic Billiards before 2003 A challenging phenomenon, was proposed, at the beginning of this millenium, by a number of authors under the name of cosmic billiards. This proposal was a development of the pioneering ideas of Belinskij, Lifshits and Khalatnikov, based on the Kasner solution of Einstein equations. The Kasner solution corresponds to a regime, where the scale factors of a D-dimensional universe have an exponential behaviour . Einstein equations are simply solved by imposing quadratic algebraic constraints on the coefficients . An inspiring mechanical analogy is at the root of the name billiards.

11. Some general considerations on roots and gravity....... String Theory implies D=10 space-time dimensions. Hence a generalization of the standard cosmological metric is of the type: In the absence of matter the conditions for this metric to be Einstein are: are the coordinates of a ball moving linearly with constant velocity Now comes an idea at first sight extravagant.... Let us imagine that What is the space where this fictitious ball moves

12. h9 h2 h1 ANSWER: The Cartan subalgebra of a rank 9 Lie algebra. What is this rank 9 Lie algebra? It is E9, namely an affine extension of the Lie algebra E8

13. Lie algebras and root systems

14. 2+3 1+2 +3 3 2 1+2 1 Lie algebras are classified....... by the properties of simple roots. For instance for A3 we have 1, 2 , 3 such that........... It suffices to specify the scalar products of simple roots For instance for A3 And all the roots are given There is a simple way of representing these scalar products:Dynkin diagrams

15. exist for any exist only for E series (exceptional) In an euclidean space we cannot fit more than 8 linear independent vectors with angles of 120 degrees !! Algebras of the type Algebras of the type In D=3 we haveE8 Then what do we have for D=2 ? The groupEris the duality group of String Theory in dimensionD = 10 – r + 1

16. where We have E9 ! How come? More than 8 vectors cannot be fitted in an euclideanspaceat the prescribed angles ! Yes! Euclidean!! Yet in a non euclidean space we can do that !! Do you remember the condition on the exponent pi = (velocity of the little ball) If we diagonalize the matrix Kij we find the eigenvalues Here is the non-euclidean signature in the Cartan subalgebra of E9. It is an infinite dimensional algebra ( = infinite number of roots!!)

17. h9 h2 h1 Now let us introduce also the roots...... There are infinitely many, but the time-like ones are in finite number. There are 120 of them as in E8. All the others are light-like Time like roots, correspond to the light fields of Superstring Theory different from the diagonal metric: off-diagonal components of the metric and p-form fields When we switch on the roots, the fictitious cosmic ball no longer goes on straight lines. It bounces!!

18.    The cosmic Billiard Or, in frontal view The Lie algebra roots correspond to off-diagonal elements of the metric, or to matter fields (the p+1 forms which couple to p-branes) Switching a root we raise a wall on which the cosmic ball bounces

19. Damour, Henneaux, Nicolai 2002 -- Before 2003: Rigid Billiards Asymptotically any time—dependent solution defines a zigzag in ln ai space The Supergravity billiard is completely determined by U-duality group h-space CSA of the U algebra hyperplanes orthogonal to positive roots(hi) walls bounces Weyl reflections billiard region Weyl chamber Exact cosmological solutions can be constructed using U-duality (in fact billiards are exactly integrable) Smooth billiards: bounces Smooth Weyl reflections Frè, Sorin, and collaborators, 2003-2008 series of papers walls Dynamical hyperplanes

20. What is the meaning of the smooth cosmic billiard ? • The number of effective dimensions varies dynamically in time! • Some dimensions are suppressed for some cosmic time and then enflate, while others contract. • The walls are also dynamical. First they do not exist then they raise for a certain time and finally decay once again. • The walls are euclidean p-branes! (Space-branes) • When there is the brane its parallel dimensions are big and dominant, while the transverse ones contract. • When the brane decays the opposite occurs

21. Cosmic Billiards in 2008 Results established by P.Frè and A.Sorin • The billiard phenomenon is the generic feature of all exact solutions of supergravity restricted to time dependence. • We know all solutions where two scale factors are equal. In this case one-dimensional -model on the coset U/H. We proved complete integrability. • We established an integration algorithm which provides the general integral. • We discovered new properties of the moduli spaceof the general integral. This is the compact coset H/Gpaint , further modded by the relevant Weyl group. This is the Weyl group WTS of the Tits Satake subalgebra UTS½ U. • There exist both trapped and (super)critical surfaces. Asymptotic states of the universe are in one-to-one correspondence with elements of WTS. • Classification of integrable supergravity billiards into a short list of universality classes. • Arrow of time. The time flow is in the direction of increasing the disorder: • Disorder is measured by the number of elementary transpositions in a Weyl group element. • Glimpses of a new cosmological entropy to be possibly interpreted in terms of superstring microstates, as it happens for the Bekenstein-Hawking entropy of black holes.

22. Definition Statement Main Points Because t-dependent supergravity field equations are equivalent to the geodesic equations for a manifold U/H Because U/H is always metrically equivalent to a solvable group manifold exp[Solv(U/H)] and this defines a canonical embedding

23. World manifold W: coordinates  Target manifold M: coordinates I What is a  - model ? It is a theory of maps from one manifold to another one:

24. With the target manifold being the maximally non-compact coset space Starting from D=3 (D=2 and D=1, also) all the (bosonic) degrees of freedom are scalars The bosonic Lagrangian of both Type IIA and Type IIB reduces, upon toroidal dimensional reduction from D=10 to D=3, to the gravity coupled sigma model

25. The discovered Principle The relevant Weyl group is that of the Tits Satake projection. It is a property of auniversality class of theories. There is an interesting topology of parameter space for the LAX EQUATION

26. The Weyl group of a Lie algebra • Is the discrete group generated by all reflections with respect to all roots • Weyl(L) is a discrete subgroup of the orthogonal group O(r) where r is the rank of L.

27. The mathematical ingredients • Dimensional reduction to D=3 realizes the identification SUGRA = -model on U/H • The mechanism of Kac Moody extensions • The solvable parametrization of non-compact U/H • The Tits Satake projection • The Lax representation of geodesic equations and the Toda flow integration algorithm

28. Full Integrability Lax pair representation and the integration algorithm

29. Solvable coset representative Lax operator (symm.) Connection (antisymm.) Lax Equation Lax Representation andIntegration Algorithm

30. Parameters of the time flows From initial data we obtain the time flow (complete integral) Initial data are specified by a pair: an element of the non-compact Cartan Subalgebra and an element of maximal compact group:

31. Properties of the flows The flow is isospectral The asymptotic values of the Lax operator are diagonal (Kasner epochs)

32. Parameter space Proposition Trapped submanifolds ARROW OF TIME

33. Available flows on 3-dimensional critical surfaces Available flows on edges, i.e. 1-dimensional critical surfaces Example. The Weyl group of Sp(4)» SO(2,3)

34. Plot of 1¢ h Plot of 1¢ h Future PAST An example of flow on a critical surface for SO(2,4). 2 , i.e. O2,1 = 0 Zoom on this region Future infinity is 8 (the highest Weyl group element), but at past infinity we have 1 (not the highest) = criticality Trajectory of the cosmic ball

35. Plot of 1¢ h Plot of 1¢ h Future O2,1' 0.01 (Perturbation of critical surface) There is an extra primordial bounce and we have the lowest Weyl group element5 at t = -1 PAST

36. END of this RECITALThank you for your attention In case of request of request…..there is encore…..

37. For subject lovers More details on the underlying Mathematical Structure

38. Duality Algebras The algebraic structure of duality algebras in D<4 dimensions

39. Universal, comes from Gravity Comes from vectors in D=4 Symplectic metric in d=2 Symplectic metric in 2n dim Structure of the Duality Algebra in D=3

40. Affine and Hyperbolic algebrasand the cosmic billiard (Julia, Henneaux, Nicolai, Damour) • We do not have to stop to D=3 if we are just interested in time dependent backgrounds • We can step down to D=2 and also D=1 • In D=2 the duality algebra becomes an affine Kac-Moody algebra • In D=1 the duality algebra becomes an hyperbolic Kac Moody algebra • Affine and hyperbolic symmetries are intrinsic to Einstein gravity

41. Duality algebrasfor diverse N(Q) from D=4 to D=3 N=8 E7(7) E8(8) SO*(12) E7(-5) N=6 N=5 SU(1,5) E6(-14) N=4 SL(2,R)£SO(6,n) SO(8,n+2) N=3 SU(3,n) £ U(1)Z SU(4,n+1)

42. 3 4 5 6 7 What happens for D<3? Exceptional E11- D series for N=8 give a hint 0 2 9 D 8 SL(2) + SL(3) GL(2,R) SO(5,5) SL(5) Julia 1981 UD E8 E3 E9 = E8Æ E7 E6 E5 E4

43. UD=4 W 0 UD=4 W1 W2 0 This extensions is affine! The new affine triplet: (LMM0, LMM+, LMM-) The new triplet is connected to the vector root with a single line, since the SL(2)MM commutes with UD=4 2 exceptions: pure D=4 gravity and N=3 SUGRA 0 1

44. D=4 D=3 D=2 N=8 E7(7) E8(8) E9(9) SO*(12) E7(-5) N=6 E7 N=5 SU(1,5) E6(-14) E6 N=4 SL(2,R)£SO(6,n) SO(8,n+2) SO(8,n+2) N=3 SU(3,n) £ U(1)Z SU(4,n+1)

45. The Tits Satake projection A fondamental ingredient to single out the universality classes and the relevant Weyl group

46. Several roots of the higher system have the same projection. These are painted copies of the same wall. The Billiard dynamics occurs in the rank r2 system Projection root system of rank r1 rank r2 < r1 root lattice

47. Two type of roots 1 2 3

48. r – split rank compact roots non compact roots root pairs Compact simple roots define a sugalgebra Hpaint To say it in a more detailed way: Non split algebras arise as duality algebras in non maximal supergravities N< 8 Under the involutive automorphism  that defines the non split real section Non split real algebras are represented by Satake diagrams For example, for N=6 SUGRA we have E7(-5)

49. The paint group The subalgebra of external automorphisms: is compact and it is the Lie algebra of the paint group

50. D=4 D=3 Paint group in diverse dimensions The paint group survives under dimensional reduction, that adds only non-compact directions to the scalar manifold It means that the Tits Satake projection commutes with the dimensional reduction