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The arrow of time and the Weyl group: all supergravity billiards are integrable

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## The arrow of time and the Weyl group: all supergravity billiards are integrable

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**The arrow of time and the Weyl group:all supergravity**billiards are integrable FUTURE Talk by Pietro Frè at Dubna 2008 PAST**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**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 (Kasner Epoch) Asymptoticaly Inaccessible region Wall position or bounce condition**h1**a wall ω(h) = 0 h2 H1 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**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 (in fact billiards are exactly integrable) Smooth billiards: Frè, Rulik, Sorin, Trigiante 2003-2007 series of papers bounces Smooth Weyl reflections walls Dynamical hyperplanes**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**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**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**D=4 vector fields**model of D=4 scalars Field dependent coupling constants and theta angles Example: from D=4 to D=3 and back The Bosonic action of any supergravity model (ungauged) can be written In the following form: Dimensionally reducing to D=3 we obtain just a larger group sigma model**We obtain the sigma model:**Where the old and new scalars are:**INGREDIENT 2**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] , Dk[A] = [Dk-1[A] , Dk-1[A] ] • Dn[A] = 0for some n > 0then 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 SolvN½ sl(N,R)**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 Splitting the Lie algebra U into the maximal compact subalgebra H plus the orthogonal complementK • There are precise rules to construct Solv(U/H) • Essentially Solv(U/H) is made by • the non-compact Cartan generators Hi2 CSA K and • those positive root step operators E which are not orthogonal to the non compact Cartan subalgebra CSA 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 E11-D series of Lie algebras • For lower supersymmetry we always have non-maximally split algebras U • There exists, however, the Tits Satake projection**The Dynkin diagram is**Tits Satake Projection: an example The D3» A3 root system contains 12 roots: Let us distinguish the roots that have a non-zero z-component, from those that have a vanishing z-component Complex Lie algebra SO(6,C) INGREDIENT 3 We consider the real section SO(2,4)**The D3» A3 root system contains 12 roots:**Complex Lie algebra SO(6,C) The Dynkin diagram is We consider the real section SO(2,4) Tits Satake Projection: an example 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**The D3» A3 root system contains 12 roots:**Complex Lie algebra SO(6,C) The Dynkin diagram is We consider the real section SO(2,4) Tits Satake Projection: an example 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**The D3» A3 root system contains 12 roots:**Complex Lie algebra SO(6,C) The Dynkin diagram is We consider the real section SO(2,4) Tits Satake Projection: an example 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**The root system B2» C2 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.**Tits Satake Projection: an example**This system of vectors is actually a new root system in rank r = 2. It is the root system B2» C2 of the Lie Algebra Sp(4,R) » SO(2,3)**Scalar Manifolds in Non Maximal SUGRAS and Tits Satake**submanifolds An overview of the Tits Satake projections.....and affine extensions**Classification**of special geometries, namely of the scalar sector of supergravity with 8 supercharges In D=5, D=4 and D=3 D=5 D=4 D=3**The paint group**The subalgebra of external automorphisms: is compact and it is the Lie algebra of the paint group**Solvable coset**representative Lax operator (symm.) Connection (antisymm.) Lax Equation Lax Representation andIntegration Algorithm INGREDIENT 4**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)**Parameter space**Proposition Trapped submanifolds ARROW OF TIME**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)**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**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 element5 at t = -1 PAST**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.... • ......