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

Cosmic Billiards are fully integrable: Tits Satake projections and Kac Moody extensions. Seminar by Pietro Frè at Milano I April 19th 2007”. Introduction to cosmic billiards. I begin by introducing, somewhat heuristically, the idea of cosmic billiards

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

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  1. Cosmic Billiards are fully integrable:Tits Satake projections and Kac Moody extensions Seminar by Pietro Frè at Milano I April 19th 2007”

  2. Introduction to cosmic billiards • I begin by introducing, somewhat heuristically, the idea of cosmic billiards • Then I will illustrate the profound relation between this pictorial description of cosmic evolution and the fundamental duality symmetries of string theory

  3. Standard Cosmology g= -1 0 a2 (t) a2 (t) 0 a2 (t) FRW : TheObserved universe is homogeneous and isotropic Tmn $ homogeneous isotropic medium with pressure P and density r • The Universe is expanding, in the presence of matter the Universe cannot be static, • all directions of a FRW universe expand in the same way, (We introduce only one scale factor) • going back in time we turn to the moment when the Universe was very small and matter was concentrated in infinetely small region of space, matter density was infinite: Big Bang, • the character of the expansion depends on the equation of state: P = w 

  4. g= 0 -1 a12 (t) a22 (t) 0 a32 (t) h2 Useful pictorial representation: A light-like trajectory of a ball in the lorentzian space of h3 hi(t)= log[ai(t)] h1 Standard FRW cosmology is concerned with studying the evolution of specific general relativity solutions, but we want to ask what more general type of evolution is conceivable just under GR rules. What if we abandon isotropy? The Kasner universe: an empty, homogeneous, but non-isotropic universe Some of the scale factors expand, but some other have to contract: an anisotropic universe is not static even in the absence of matter! These equations are the Einstein equations

  5. Let us now consider, the coupling of a vector field to diagonal gravity Introducing Billiard Walls If Fij = const this term adds a potential to the ball’s hamiltonian Free motion Asymptoticaly Inaccessible region Wall position or bounce condition

  6. in the vicinity of spacelike singularity space points decouple, cosmological evolution is a series of Kasner epochs, mixmaster behaviour BKL1970’: Scale factor logarithms hi(t) describe a trajectory of a ball in D-1 dimensional space with Minkovsky signature. Damour, Henneaux, Nicolai hep-th/0212256 If there are no matter fields or off-diagonal metric components, this trajectory is a straight line – Kasner solution with momenta pi In the presence of matter, radiation and non-diagonal metric components (spatial curvature) the motion of the ball is bounded by exponential potential walls We saw an example If Fij = const this term adds a potential to the ball’s hamiltonian Free motion (Kasner epoch) Inaccessible region Cij2>0 ! the potential is repulsive! Wall position or bounce condition Billiard: a paradigm for multidimensional cosmology

  7. h1 a wall ω(h) = 0 h2 H1 t ! 0 H3 H2 The Rigid billiard When the ball reaches the wall it bounces against it: geometric reflection ball trajectory It means that the space directions transverse to the wallchange their behaviour: they begin to expand if they were contracting and vice versa Billiard table: the configuration of the walls -- the full evolution of such a universe is a sequence of Kasner epochs with bounces between them -- the number of large (visible) dimensions can vary in time dynamically -- the number of bounces and the positions of the walls depend on the field content of the theory: microscopical input

  8. Damour, Henneaux, Nicolai 2002 -- Smooth Billiards and dualities 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 Smooth billiards: bounces Smooth Weyl reflections walls Dynamical hyperplanes

  9. The noncompact global invariance is realized nonlinearly on the scalar fields local invariance are in linear representations of UD The p-forms Dualities: their action on the bosonic fields The scalars typically parametrize a coset manifold In D=2n duality is a symmetry of the equations of motion for (n-1)-forms UD

  10. i gMN = 0 SL(N)/SO(N) Supergravities in different dimensions are connected by dimensional reduction a , gMN , … M = 0, .., D=1 a ( g , g i , gi,j ) = 0,.. , d i = i,..,N More scalars and more global symmetries !

  11. Smooth supergravity and superstring billiards.... THE MAIN IDEA from a D=3 viewpoint P. F. ,Trigiante, Rulik, Gargiulo, Sorin Van Proeyen and Roossel 2003,2004, 2005 various papers

  12. 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

  13. 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!

  14. 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.....!

  15. Differential Geometry = Algebra

  16. 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

  17. The Nomizu Operator

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

  19. 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!

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

  21. 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

  22. 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.

  23. 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:

  24. 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

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

  26. 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

  27. Let us now introduce more structure of SUGRA/STRINGTHEORY The first point: Less SUSY (NQ < 32) and non split algebras

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

  29. 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

  30. 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

  31. Two type of roots 1 2 3

  32. 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)

  33. The Paint Group 1 2 3 4

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

  35. 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

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

  37. 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

  38. 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)

  39. Let me remind you… • about the relation between • the Cartan presentation • and the Chevalley Serre presentation of a Lie algebra

  40. Cartan subalgebra (CSA) Root generators Simple roots i Cij Dynkin diagram Cartan matrix Chevalley-Serre presentation Cartan presentation

  41. UD=4 W TA (L0 L+ L-) (W+ W-) new symmetries The new Chevalley-Serre triplet: we take the highest weight of the symplectic representation h

  42. 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?

  43. 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

  44. Let us review The algebraic mechanism of this extension

  45. 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)

  46. 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

  47. 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

  48. 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)

  49. Conclusions • The algebraic structure of U duality algebras governs many aspects of String Theory • In particular it is responsible for the cosmic billiard paradigma of multidimensional cosmologies • The integrability of the maximal split case has to be extended to the non maximal split cases taking advantage of TS projection (work in progress) • The exact integrability in D=3 has to be extended to the affine and hyperbolic cases • This is for ungauged supergravities……

  50. Gauged Supergravities • Gauging is equivalent to introducing fluxes • In gauged supergravities we have -models with potentials… • The integrability of these cases is an entirely new chapter…. • May be for next year seminar….!

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