Presentation on theme: "Cosmic Billiards are fully integrable: Tits Satake projections and Kac Moody extensions Talk by Pietro Frè at Corfu 2005”"— Presentation transcript:
Cosmic Billiards are fully integrable: Tits Satake projections and Kac Moody extensions Talk by Pietro Frè at Corfu 2005”
Let me begin by presenting.... THE MAIN IDEA from a D=3 viewpoint P. F.,Trigiante, Rulik, Gargiulo and Sorin 2003,2004, 2005 various papers
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
NOMIZU OPERATOR 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 Nomizu connection = LAX PAIR Representation. INTEGRATION!
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 # Q SUSY < 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.....!
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: 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
ANSWER: The Cartan subalgebra of a rank D-1 Lie algebra. h1h1 h2h2 h D-1 What is this rank D-1 Lie algebra? It is U D=2 the Duality algebra in D=2 dimensions. This latter is the affine extension of U D=3
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 as many of them as in U D=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!!
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
Differential Geometry = Algebra
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
The Nomizu Operator
Maximal Susy implies E r+1 series Scalar fields are associated with positive roots or Cartan generators
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!
Explicit Form of the Nomizu connection for the maximally split case The metric on the algebra The components of the connection
Let us briefly survey The use of the solvable parametrization as a machinary to obtain solutions, in the split case
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
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 w i are the weights of the representation to which the Lax operator is assigned.
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:
A plotted example with SL(4,R)/O(4) The U Lie algebra is A 3 The rank is r = 3. The Weyl group is S 4 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
Example ( 1 =1, 2 =2, 3 =3) Indeed we have:
Plots of the (integrated) Cartan Fields along the simple roots 11 22 33 2+32+3 1+21+2 1 + 2 + 3 This solution has four bounces
Let us consider The first point: Less SUSY and non split algebras
Scalar Manifolds in Non Maximal SUGRAS and Tits Satake submanifolds WHAT are these new manifolds (split!) associated with the known non split ones....???
The Billiard Relies on Tits Satake Theory To each non maximally non-compact real form U (non split) of a Lie algebra of rank r 1 is associated a unique subalgebra U TS ½ U which is maximally split. U TS has rank r 2 < r 1 The Cartan subalgebra C TS ½ U TS is the true billiard table Walls in C TS now appear painted as a memory of the parent algebra U
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
Two type of roots 1 2 3
The Paint Group 1 2 3 4
Why is it exciting? Since the Nomizu connection depends only on the structure constants of the Solvable Lie algebra
And now let us go the next main point.. Kac Moody Extensions
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)
Structure of the Duality Algebra in D=3 (P.F. Trigiante, Rulik and Gargiulo 2005) Universal, comes from Gravity Comes from vectors in D=4 Symplectic metric in d=2Symplectic metric in 2n dim
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?
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
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 +
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
General Matzner&Misner reduction (P.F. Trigiante, Rulik e Gargiulo 2005) D=4 D=2
The reduction is governed by the embedding
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