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

Talk by Pietro Frè

at

Corfu 2005”

Let me begin bypresenting....

THE MAIN IDEAfrom a D=3 viewpoint

P. F. ,Trigiante, Rulik, Gargiulo and Sorin

2003,2004, 2005various 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

HUtargetM

NOMIZU OPERATORSOLVABLE 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

UG

D=3 sigma model

HUM /Field eq.s reduce to Geodesic equations on

D=3 sigma model

D>3 SUGRA D>3 SUGRA

dimensional oxidation

Not unique: classified by different embeddings

UG

Time dep. backgrounds

Nomizu connection = LAX PAIRRepresentation. 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 # 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.....!

The cosmic ballString 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.

h1

h2

hD-1

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

Now let us introduce also the roots......

h1

h2

h9

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!!

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 Er+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 wi 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 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 1, 2, 3, 1 2 3and 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:

12

34

183

2 1 31

83

2 1 3

32

34

1382 13

820 1

2143

2 1 31

43

2 1 3

0 0 1322 13

22

its image is 4

0 0 0 10 1 0 00 0 1 01 0 0 0

Limit t

6 0 0 00 2 0 00 0 1 00 0 0 3

Limit t

3 0 0 00 2 0 00 0 1 00 0 0 6

Plot of H.1-3 -2 -1 1 2 3

-5

-2.5

2.5

5

7.5

10

Plot of H.2

-2 -1 1 2

5.5

6

6.5

7

7.5

8

Plot of H.3

-3 -2 -1 1 2 3

44

46

48

50

52

54

Plots of the (integrated) Cartan Fields along

the simple roots

1 2 3

1

2

3

2+3

1+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 r1 is associated a unique subalgebra UTS ½ U which is maximally split.

UTS has rank r2 < r1

The Cartan subalgebra CTS ½ UTS is the true billiard table

Walls in CTS now appear painted as a memory of the parent algebra U

root system

of rank r1

ProjectionSeveral 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 algebrasand 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=2 Symplectic 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

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 ! GÆ

D=3