Cosmic Billiards are fully integrable: Tits Satake projections and Kac Moody extensions

1. Cosmic Billiards are fully integrable:Tits Satake projections and Kac Moody extensions Talk by Pietro Frè at Corfu 2005”

2. Let me begin bypresenting.... THE MAIN IDEA from a D=3 viewpoint P. F. ,Trigiante, Rulik, Gargiulo and Sorin 2003,2004, 2005 various papers

3. Starting from D=3(D=2 and D=1, also)all the (bosonic) degrees of freedom are scalars The bosonic Lagrangian of any Supergravity, can be reduced in D=3, to a gravity coupled sigma model

4. U D>3 SUGRA D>3 SUGRA dimensional oxidation UmapsD>3 backgrounds into D>3 backgrounds dimensional reduction Not unique: classified by different embeddings U D=3 sigma model D=3 sigma model Field eq.s reduce to Geodesic equations on Solutions are classified by abstract subalgebras Since all fields are chosen to depend only on one coordinate, t = time, then we can just reduce everything to D=3, D=2 or D=1. In these dimensions every degree of freedom (bosonic) is a scalar Time dep. backgrounds NOMIZU OPERATOR SOLVABLE ALGEBRA Nomizu connection = LAX PAIR Representation. INTEGRATION!

5. With this machinery..... • We can obtain exact solutions for time dependent backgrounds • We can see the bouncing phenomena (=billiard) We have to extend the idea to lower supersymmetry # QSUSY < 32 and... • We do not have to stop at D=3. For time dependent backgrounds we can start from D=2 or D=1 • In D=2 and D=1 we have affine and hyperbolic Kac Moody algebras, respectively.....!

6. The cosmic ball String Theory implies D=10 space-time dimensions. In general in dimension D 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

7. hD-1 h2 h1 ANSWER: The Cartan subalgebra of a rankD-1Lie algebra. What is this rank D-1 Lie algebra? It is UD=2 the Duality algebra in D=2 dimensions. This latter is the affine extension of UD=3

8. 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 as many of them as in UD=3. 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!!

9. Cosmic Billiards in Corfu

10.    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

11. Differential Geometry = Algebra

12. How to build the solvable algebra Given the Real form of the algebra U, for each positive root there is an appropriate step operator belonging to such a real form

13. The Nomizu Operator

14. Maximal Susy implies Er+1 series Scalar fields are associated with positive roots or Cartan generators

15. From the algebraic view point..... • Maximal SUSY corresponds to... • MAXIMALLY non-compact real forms: • i.e. SPLIT ALGEBRAS. • This means: • All Cartan generators are non compact • Step operators E2 Solv , 82+ • The representation is completely real • The billiard table is the Cartan subalgebra of the isometry group!

16. The metric on the algebra Explicit Form of the Nomizu connection for the maximally split case The components of the connection

17. Let us briefly survey The use of the solvable parametrization as a machinary to obtain solutions, in the split case

18. The general integration formula • Initial data at t=0 are • A) , namely an element of the Cartan subalgebra determining the eigenvalues of the LAX operator • B) , namely an element of the maximal compact subgroup Then the solution algorithm generates a uniquely defined time dependent LAX operator

19. Properties of the solution • For each element of the Weyl group • The limits of the LAX operator at t=§1 are diagonal • At any instant of time the eigenvalues of the LAX operator are constant1, ...,n • where wiare the weights of the representation to which the Lax operator is assigned.

20. Disconnected classes of solutions Property (2) and property (3) combined together imply that the two asymptotic values L§1 of the Lax operator are necessarily related to each other by some element of the Weyl group which represents a sort of topological charge of the solution: The solution algorithm induces a map:

21. A plotted example with SL(4,R)/O(4) • The U Lie algebra is A3 • The rank is r = 3. • The Weyl group is S4 with 4! elements • The compact subgroup H = SO(4) The integration formula can be easily encoded into a computer programme and for any choice of the eigenvalues and for any choice of the group element2 O(4) The programme CONSTRUCTS the solution

22. Example (1=1, 2=2 , 3=3) Indeed we have:

23. 2+3 1+2 +3 3 2 1+2 1 Plots of the (integrated) Cartan Fields along the simple roots This solution has four bounces

24. Let us consider The first point: Less SUSY and non split algebras

25. Scalar Manifolds in Non Maximal SUGRAS and Tits Satake submanifolds WHAT are these new manifolds (split!) associated with the known non split ones....???

26. The Billiard Relies on Tits Satake Theory • To each non maximally non-compact real form U (non split) of a Lie algebra of rank r1 is associated a unique subalgebra UTS½ U which is maximally split. • UTS has rank r2 < r1 • The Cartan subalgebra CTS½ UTS is the true billiard table • Walls in CTS now appear painted as a memory of the parent algebra U

27. 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

28. Two type of roots 1 2 3

29. The Paint Group 1 2 3 4

30. Why is it exciting? • Since the Nomizu connection depends only on the structure constants of the Solvable Lie algebra

31. And now let us go the next main point.. Kac Moody Extensions

32. 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

33. 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(P.F. Trigiante, Rulik and Gargiulo 2005)

34. The Kac Moody extension of the D=3 Duality algebra In D=2 the duality algebra becomes the Kac Moody extension of the algebra in D=3. Why is that so?

35. 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

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

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

38. D=4 D=2 General Matzner&Misner reduction (P.F. Trigiante, Rulik e Gargiulo 2005)

39. The reduction is governed by the embedding

40. Symmetries of MM Lagrangian • GD=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 GD=3 of the Ehlers reduction these symmetries generate the affine extension of GD=3! GÆD=3