The arrow of time and the Weyl group: all supergravity billiards are integrable Talk by Pietro Frè at SISSA Oct. 2007” based on work of P.F. with A.S.Sorin FUTUR E PAST

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? Some of the scale factors expand, but some other have to contract: an anisotropic universe is not static even in the absence of matter! The Kasner universe : an empty, homogeneous, but non-isotropic universe g  a 1 2 (t) a 2 2 (t) a 3 2 (t) 0 0 Useful pictorial representation: A light-like trajectory of a ball in the lorentzian space of h i (t)= log[a i (t)] h1h1 h2h2 h3h3 These equations are the Einstein equations

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

11 22 33 The Rigid billiard hh hh a wall ω( h) = 0 ball trajectory When the ball reaches the wall it bounces against it: geometric reflection It means that the space directions transverse to the wall change 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

Smooth Billiards and dualities 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 Exact cosmological solutions can be constructed using U-duality (in fact billiards are exactly integrable) bouncesSmooth Weyl reflections walls Dynamical hyperplanes Frè, Rulik, Sorin, Trigiante series of papers

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

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

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

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 INGREDIENT 1

SOLVABLE ALGEBRA U dimensional reduction 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 U U maps D>3 backgrounds into D>3 backgrounds Solutions are classified by abstract subalgebras D=3 sigma model Field eq.s reduce to Geodesic equations on D=3 sigma model D>3 SUGRA dimensional oxidation Not unique: classified by different embeddings Time dep. backgrounds LAX PAIR INTEGRATION!

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) INGREDIENT 2

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

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  There exists, however, the Tits Satake projection

Tits Satake Projection: an example The D 3 » A 3 root system contains 12 roots: Complex Lie algebra SO(6,C) We consider the real section SO(2,4) The Dynkin diagram is Let us distinguish the roots that have a non-zero z-component, from those that have a vanishing z-component INGREDIENT 3

Tits Satake Projection: an example The D 3 » A 3 root system contains 12 roots: Complex Lie algebra SO(6,C) We consider the real section SO(2,4) The Dynkin diagram is Let us distinguish the roots that have a non-zero z-component, from those that have a vanishing z-component Now let us project all the root vectors onto the plane z = 0

Tits Satake Projection: an example The D 3 » A 3 root system contains 12 roots: Complex Lie algebra SO(6,C) We consider the real section SO(2,4) The Dynkin diagram is Let us distinguish the roots that have a non-zero z-component, from those that have a vanishing z-component Now let us project all the root vectors onto the plane z = 0

Tits Satake Projection: an example The D 3 » A 3 root system contains 12 roots: Complex Lie algebra SO(6,C) We consider the real section SO(2,4) The Dynkin diagram is The projection creates new vectors in the plane z = 0 They are images of more than one root in the original system Let us now consider the system of 2-dimensional vectors obtained from the projection

Tits Satake Projection: an example This system of vectors is actually a new root system in rank r = 2. It is the root system B 2 » C 2 of the Lie Algebra Sp(4,R) » SO(2,3)

Tits Satake Projection: an example The root system B 2 » C 2 of the Lie Algebra Sp(4,R) » SO(2,3) so(2,3) is actually a subalgebra of so(2,4). It is called the Tits Satake subalgebra The Tits Satake algebra is maximally split. Its rank is equal to the non compact rank of the original algebra.

Scalar Manifolds in Non Maximal SUGRAS and Tits Satake submanifolds An overview of the Tits Satake projections.....and affine extensions

Universality Classes

Classification of special geometries, namely of the scalar sector of supergravity with 8 supercharges In D=5, D=4 and D=3 D=5D=4D=3

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

Lax Representation and Integration Algorithm INGREDIENT 4 Solvable coset representative Lax operator (symm.) Connection (antisymm.) Lax Equation

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:

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

Proposition Trapped submanifolds ARROW OF TIME Parameter space

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

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

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

Conclusions  Supergravity billiards are an exciting paradigm for string cosmology and are all completely integrable  There is a profound relation between U-duality and the billiards and a notion of entropy associated with the Weyl group of U.  Supergravity flows are organized in universality classes with respect to the TS projection.  We have a phantastic new starting point....A lot has still to be done: Extension to Kac-Moody Inclusion of fluxes Comparison with the laws of BH mechanics