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Cosmic Billiards are fully integrable: Tits Satake projections and Kac Moody extensions
Talk by Pietro Frè
at
Corfu 2005”
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Let me begin bypresenting....
THE MAIN IDEAfrom a D=3 viewpoint
P. F. ,Trigiante, Rulik, Gargiulo and Sorin
2003,2004, 2005various papers
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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
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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!
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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.....!
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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
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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
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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!!
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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
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Differential Geometry = Algebra
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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
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The Nomizu Operator
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Maximal Susy implies Er+1
series
Scalar fields are associated with positive roots or Cartan generators
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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!
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Explicit Form of the Nomizu connection for the maximally split case
The metric on the algebra
The components
of the connection
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Let us briefly survey
The use of the solvable parametrization as a machinary
to obtain solutions,
in the split case
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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
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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.
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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:
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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
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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
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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
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Let us consider
The first point:
Less SUSY
and
non split algebras
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Scalar Manifolds in Non Maximal SUGRAS and Tits Satake submanifolds
WHAT are these new manifolds (split!) associated with the known non split ones....???
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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
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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
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Two type of roots
1
2
3
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The Paint Group
1
2
3
4
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Why is it exciting?
Since the Nomizu connection depends only on the structure constants of the Solvable Lie algebra
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And now let us go the next main point..
Kac Moody Extensions
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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)
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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
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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?
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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
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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
+
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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
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General Matzner&Misner reduction (P.F. Trigiante, Rulik e Gargiulo 2005)
D=4
D=2
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The reduction is governed by the embedding
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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