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The arrow of time and the Weyl group:all supergravity billiards are integrable
Talk by Pietro Frè
at SISSA Oct. 2007”based on work of P.F. with A.S.Sorin
FUTURE
PAST
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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
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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 (Kasner Epoch)
Inaccessible region
Wall position or bounce condition
Asymptoticaly
Introducing Billiard Walls
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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
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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 (in fact billiards are exactly integrable)
bounces Smooth Weyl reflections
walls Dynamical hyperplanes
Frè, Rulik, Sorin,Frè, Rulik, Sorin,
TrigianteTrigiante
2003-2007
series of papers
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Main PointsDefinition
Statement
Because t-dependentsupergravity field equationsare equivalent to thegeodesic equations fora manifold
U/H
Because U/H is alwaysmetrically equivalent toa solvable group manifold exp[Solv(U/H)]and this defines acanonical embedding
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The discovered PrincipleThe relevant
Weyl group is that
of the Tits Satake
projection. It is
a property of a universality class
of theories.
There is an interesting topology of parameter space for the LAX EQUATION
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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
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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
INGREDIENT 1
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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
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
LAX PAIRINTEGRATION!
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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] = 0 for some n > 0 then 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
SolvN ½ sl(N,R)
INGREDIENT 2
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The solvable parametrizationThere 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
•There are precise rules to construct Solv(U/H)
•Essentially Solv(U/H) is made by
•the non-compact Cartan generators Hi 2 CSA K and
•those positive root step operators E which are not orthogonal to the non compact Cartan subalgebra CSA K
Splitting the Lie algebra U into the maximal compact subalgebra H plus the orthogonal complement K
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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
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Tits Satake Projection: an example
The D3 » A3 root system contains 12 roots:
Complex Lie algebra SO(6,C)
We consider the real section SO(2,4)
The Dynkin diagram is
Let us distinguish the roots that have
a non-zero
z-component,
from those that have
a vanishing
z-component
INGREDIENT 3
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Tits Satake Projection: an example
The D3 » A3 root system contains 12 roots:
Complex Lie algebra SO(6,C)
We consider the real section SO(2,4)
The Dynkin diagram is
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
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Tits Satake Projection: an example
The D3 » A3 root system contains 12 roots:
Complex Lie algebra SO(6,C)
We consider the real section SO(2,4)
The Dynkin diagram is
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
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Tits Satake Projection: an example
The D3 » A3 root system contains 12 roots:
Complex Lie algebra SO(6,C)
We consider the real section SO(2,4)
The Dynkin diagram is
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
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Tits Satake Projection: an example
This system of
vectors is actually
a new root system in rank r = 2.
1
2
1 2
2 1 2
It is the root system B2 » C2 of the Lie Algebra Sp(4,R) » SO(2,3)
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Tits Satake Projection: an example
1
2
1 2
2 1 2The 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.
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Scalar Manifolds in Non Maximal SUGRAS and Tits Satake submanifolds
An overview of the Tits Satake projections.....and affine extensions
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Universality Classes
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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
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The paint group
The subalgebra of external automorphisms:
is compact and it is the Lie algebra of the paint group
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Lax Representation andIntegration Algorithm
INGREDIENT 4
Solvable coset
representative
Lax operator (symm.)
Connection (antisymm.)
Lax Equation
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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 CartanSubalgebra and an element of maximal compact group:
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Properties of the flowsThe flow is isospectral
The asymptotic values of the Lax operator are diagonal (Kasner epochs)
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Proposition
Trapped
submanifolds
ARROW OF
TIME
Parameter space
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Example. The Weyl group of Sp(4)» SO(2,3)
Available flows
on 3-dimensional
critical surfaces
Available flows on edges, i.e. 1-dimensional critical surfaces
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An example of flow on a critical surface for SO(2,4).
2 , i.e. O2,1 = 0
Future
PAST
Plot of 1 ¢ h
Plot of 1 ¢ h
Future infinity is 8 (the highest Weyl group element), but at past infinitywe have 1 (not the highest) = criticality
Zoom on this region
Trajectory of the Trajectory of the cosmic ballcosmic ball
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Future
PAST
Plot of 1 ¢ h
Plot of 1 ¢ h
O2,1 ' 0.01 (Perturbation of
critical surface)
There is an extra primordial
bounce and we have the lowest Weyl group element 5 at
t = -1
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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.... ......