cosmic billiards are fully integrable: tits satake projections and special geometries
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Cosmic Billiards are fully integrable: Tits Satake projections and Special Geometries
Lectures by Pietro Frè
at Dubna JINR
July 2007”В Дубне 2007:
Как всегда я очень рад быть здесь и хотел бы сказать всем моим друзям огромное спосибо за приглажение.
Introduction to cosmic billiards
I begin by introducing, somewhat heuristically, the idea of cosmic billiards
Then I will illustrate the profound relation between this pictorial description of cosmic evolution and the fundamental duality symmetries of string theory
-The Universe is expanding, in the presence of matter the Universe cannot be static,
- all directions of a FRW universe expand in the same way, (We introduce only one scale factor)
- going back in time we turn to the moment when the Universe was very small and matter was concentrated in infinetely small region of space, matter density was infinite: Big Bang,
- the character of the expansion depends on the equation of state: P = w
FRW : The Observed universe is homogeneous and isotropic
g
-1a2 (t)
a2 (t)
a2 (t)0
0
T$ homogeneous isotropic medium with pressure P and density
Standard Cosmology
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
-1a1
2 (t)
a22 (t)
a32 (t)0
0
Useful pictorial representation:A light-like trajectory of a ball in the lorentzian space of
hi(t)= log[ai(t)]
h1
h2
h3
These equations are the Einstein equations
Let us now consider, the coupling of a vector field to diagonal gravity
If Fij = const this term adds a potential to the ball’s hamiltonian
Free motion
Inaccessible region
Wall position or bounce condition
Asymptoticaly
Introducing Billiard Walls
Billiard: a paradigm for multidimensional cosmology
Scale factor logarithms hi(t) describe a trajectory of a ball in
D-1 dimensional space with Minkovsky signature.
If there are no matter fields or off-diagonal metric components,this trajectory is a straight line – Kasner solution with momenta pi
In the presence of matter, radiation and non-diagonal metric components (spatial curvature) the motion of the ball is bounded by exponential potential walls
Damour,Henneaux,Nicolaihep-th/0212256
We sawan example
If Fij = const this term adds a potential to the ball’s hamiltonian
Free motion (Kasner epoch)
Inaccessible region
Wall position or bounce conditionCij
2>0 ! the potential is repulsive!
BKL1970’: in the vicinity of spacelike singularity space points decouple,cosmological evolution is a series of Kasner epochs,mixmaster behaviour
t ! 0
1
2
3
The Rigid billiard h
h
a wallω(h) = 0
ball trajectoryWhen the ball reaches the wall it bounces against it: geometric reflectionIt 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-space CSA of the U algebra
walls hyperplanes orthogonalto positive roots (hi)
bounces Weyl reflections
billiard region Weyl chamber
The Supergravity billiard is completely determined by U-duality group
Smooth billiards:
Asymptotically any time—dependent solution defines a zigzag in ln ai space
Damour, Henneaux, Nicolai 2002 --
Exact cosmological solutions can be constructed using U-duality
bounces Smooth Weyl reflections
walls Dynamical hyperplanes
Dualities: their action on the bosonic fields
The noncompact global invariance is realized nonlinearly on the scalar fields
local invariance
The scalars typically parametrizea coset manifold
The p-forms are in linear representations of UD
In D=2n duality is a symmetry of the equations of motionfor (n-1)-forms
UD
Supergravities in different dimensions are connected by
dimensional reduction
a , gMN , … M = 0, .., D=1
a ( g , g i , gi,j )
SL(N)/SO(N)
i gMN = 0
More scalars and more global symmetries !
= 0,.. , di = i,..,N
Smooth supergravity and superstring
billiards....
THE MAIN IDEAfrom a D=3 viewpointP. F. ,Trigiante, Rulik, Gargiulo, Sorin
Van Proeyen and Roossel
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.....!
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!
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 now introduce more structure of
SUGRA/STRING THEORY
The first point:
Less SUSY (NQ < 32)
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
To say it in a more detailed way:Non split algebras arise as duality algebras in non maximal supergravities N< 8
r – split rank
Under the involutive automorphism that defines the non split real section
compact roots
non compact roots
root pairs
Non split real algebras are represented by Satake diagrams
For example, for N=6 SUGRAwe have E7(-5)
Compact simple rootsdefine a sugalgebraHpaint
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
Paint group in diverse dimensions
The paint group survives under dimensional reduction, that adds only non-compact directions to the scalar manifold
D=3D=4
It means that the Tits Satake projection commutes with the dimensional reduction
More about the Tits Satake projection
It is a projection....
Supergravity Models fall into Universality classes
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=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
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
Let me remind you…
about the relation betweenabout the relation between the Cartan presentation and the Chevalley Serre
presentation of a Lie algebra
Cartan presentation
Cartan subalgebra (CSA)
Root generators
Chevalley-Serre presentation
Simple roots i
Dynkin diagram
Cij
Cartan matrix
TA (L0 L+ L-) (W+W-)
The new Chevalley-Serre triplet: we take the highest weight of thesymplectic representation h
new symmetries
UD=4
W
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
Let us review
The algebraic mechanism of this extension
N=8 E8(8)
N=6 E7(-5)
N=5 E6(-14)
N=4
SO(8,n+2)
N=3
SU(4,n+1)
Duality algebras for diverse N(Q) from D=4 to D=3
E7(7)
SO*(12)
SU(1,5)
SL(2,R)£SO(6,n)
SU(3,n) £ U(1)Z
What happens for D<3?
7
9
5
6
8
4
7
3
6
2
5
1
4
8
3 D
Exceptional E11- D series for N=8 give a hint
E8
Julia 1981
2
E9 = E8Æ
UD
GL(2,R)SL(2) + SL(3)
SL(5)
E4E3
SO(5,5)
E5E6E7
0
This extensions is affine!
UD=4
W0
The new triplet is connected to the vector root with a single line,since the SL(2)MM commutes with UD=4
2 exceptions: pure D=4 gravity and N=3 SUGRA
UD=4
W1
0
W2
1
The new affine triplet: (LMM0, LMM
+, LMM-)
N=8 E8(8)
N=6 E7(-5)
N=5 E6(-14)
N=4
SO(8,n+2)
N=3
SU(4,n+1)
D=4
E7(7)
SO*(12)
SU(1,5)
SL(2,R)£SO(6,n)
SU(3,n) £ U(1)Z
E9(9)
E7
E6
SO(8,n+2)
D=3 D=2
Conclusions
The algebraic structure of U duality algebras governs many aspects of String Theory
In particular it is responsible for the cosmic billiard paradigma of multidimensional cosmologies
The integrability of the maximal split case has to be extended to the non maximal split cases taking advantage of TS projection (work in progress)
The exact integrability in D=3 has to be extended to the affine and hyperbolic cases
This is for ungauged supergravities……
Gauged Supergravities
Gauging is equivalent to introducing fluxes In gauged supergravities we have -models
with potentials… The integrability of these cases is an entirely
new chapter…. May be for next year seminar….!
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