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. Standard cosmology is based on the cosmological principle. Homogeneity Homogeneity Isotropy Isotropy

3. Evolution of the scale factor without 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. Accelerating Universe dominated by Dark Energy Equation of State

9. Non isotropic Universes This is what happens if there is isotropy ! This is what happens if there is isotropy ! Relaxing isotropy an entire new world of phenomena opens up 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! 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: Now comes an idea at first sight extravagant.... Let us imagine that are the coordinates of a ball moving linearly with constant velocity What is the space where this fictitious ball moves

12. ANSWER: The Cartan subalgebra of a rank 9 Lie algebra. h1h1 h2h2 h9h9 What is this rank 9 Lie algebra? It is E 9, namely an affine extension of the Lie algebra E 8

13. Lie algebras and root systems

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

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

16. We have E 9 ! How come? More than 8 vectors cannot be fitted in an euclidean space at the prescribed angles ! Yes! Euclidean!! Yet in a non euclidean space we can do that !! Do you remember the condition on the exponent p i = (velocity of the little ball) where If we diagonalize the matrix K ij we find the eigenvalues Here is the non-euclidean signature in the Cartan subalgebra of E 9. It is an infinite dimensional algebra ( = infinite number of roots!!)

17. Now let us introduce also the roots...... h1h1 h2h2 h9h9 There are infinitely many, but the time-like ones are in finite number. There are 120 of them as in E 8. 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    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 Or, in frontal view

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

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 The billiard phenomenon is the generic feature of all exact solutions of supergravity restricted to time dependence. 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 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 established an integration algorithm which provides the general integral. We discovered new properties of the moduli space of the general integral. This is the compact coset H/G paint, further modded by the relevant Weyl group. This is the Weyl group W TS of the Tits Satake subalgebra U TS ½ U. We discovered new properties of the moduli space of the general integral. This is the compact coset H/G paint, further modded by the relevant Weyl group. This is the Weyl group W TS of the Tits Satake subalgebra U TS ½ U. There exist both trapped and (super)critical surfaces. Asymptotic states of the universe are in one-to-one correspondence with elements of W TS. There exist both trapped and (super)critical surfaces. Asymptotic states of the universe are in one-to-one correspondence with elements of W TS. Classification of integrable supergravity billiards into a short list of universality classes. 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: 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. 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. 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. Results established by P.Frè and A.Sorin

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

24. 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 With the target manifold being the maximally non- compact coset space

25. The discovered Principle The relevant Weyl group is that of the Tits Satake projection. It is a property of a universality 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 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. 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. Lax Representation and Integration Algorithm Solvable coset representative Lax operator (symm.) Connection (antisymm.) Lax Equation

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. Proposition Trapped submanifolds ARROW OF TIME Parameter space

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

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

35. Future PAST Plot of  1 ¢ h O 2,1 ' 0.01 (Perturbation of critical surface) There is an extra primordial bounce and we have the lowest Weyl group element  5 at t = -1

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

40. Affine and Hyperbolic algebras and the cosmic billiard 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 (Julia, Henneaux, Nicolai, Damour)

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

42. What happens for D<3? 9 8 76543D Exceptional E 11- D series for N=8 give a hint E8E8 Julia 1981 2 E 9 = E 8 Æ UDUD GL(2,R) SL(2) + SL(3) SL(5) E4E4 E3E3 SO(5,5) E5E5 E6E6 E7E7 00

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

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

45. The reason is... That there are two ways of stepping down from D=4 to D=2 The Ehlers reduction The Matzner&Misner reduction The two routes give two different lagrangians with two different finite algebra of symmetries There are non local relations between the fields of the two lagrangians The symmetries of one Lagrangian have a non local realization on the other and vice versa Together the two finite symmetry algebras provide a set of Chevalley generators for the Kac Moody algebra

46. Ehlers reduction of pure gravity CONFORMAL GAUGE DUALIZATION OF VECTORS TO SCALARS D=4 D=3 D=2 Liouville field SL(2,R)/O(2)  - model +

47. Matzner&Misner reduction of pure gravity D=4 D=3 D=2 CONFORMAL GAUGE NO DUALIZATION OF VECTORS !! Liouville field SL(2,R)/O(2)  - model DIFFERENT SL(2,R) fields non locally related

48. General Matzner&Misner reduction D=4 D=2

49. The reduction is governed by the embedding

50. Symmetries of MM Lagrangian G D=4 through pseudorthogonal embedding SL(2,R) MM through gravity reduction Local O(2) symmetry acting on the indices A,B etc [ O(2) 2 SL(2,R) MM ] Combined with G D=3 of the Ehlers reduction these symmetries generate the affine extension of G D=3 ! G Æ D=3

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

52. root system of rank r 1 Projection 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

53. Two type of roots 1 2 3

54. To say it in a more detailed way: Non split algebras arise as duality algebras in non maximal supergravities N< 8 r – split rank Under the involutive automorphism  that defines the non split real section compact roots non compact roots root pairs Non split real algebras are represented by Satake diagrams For example, for N=6 SUGRA we have E 7(-5) Compact simple roots define a sugalgebra H paint

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

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

57. The Solvable Parametrization Crucial for integrability is the remarkable relation between non-compact cosets U/H and Solvable Group Manifolds exp[Solv]

58. Solvable Lie Algebras: i.e. triangular matrices What is a solvable Lie algebra A ? It is an algebra where the derivative series ends after some steps i.e. D[A] = [A, A], D k [A] = [D k-1 [A], D k-1 [A] ] D n [A] = 0 for some n > 0 then A = solvable THEOREM: All linear representations of a solvable Lie algebra admit a basis where every element T 2 A is given by an upper triangular matrix For instance the upper triangular matrices of this type form a solvable subalgebra Solv N ½ sl(N,R)

59. The solvable parametrization There is a fascinating theorem which provides an identification of the geometry of moduli spaces with Lie algebras for (almost) all supergravity theories. THEOREM: All non compact (symmetric) coset manifolds are metrically equivalent to a solvable group manifold There are precise rules to construct Solv(U/H) Essentially Solv(U/H) is made by the non-compact Cartan generators H i 2 CSA  K and those positive root step operators E  which are not orthogonal to the non compact Cartan subalgebra CSA  K Splitting the Lie algebra U into the maximal compact subalgebra H plus the orthogonal complement K

60. Maximally split cosets U/H  U/H is maximally split if CSA = CSA  K is completelly non-compact  Maximally split U/H occur if and only if SUSY is maximal # Q =32.  In the case of maximal susy we have (in D- dimensions the E 11-D series of Lie algebras  For lower supersymmetry we always have non-maximally split algebras U