Full Integrability of Supergravity Billiards: the arrow of time, asymptotic states and trapped surfaces in the cosmic evolution
Lecture by Pietro Frè and
Alexander S. Sorin
In connection with the nomination for the
JINR prize 2008

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Standard cosmology is based on the cosmological principle.
Homogeneity
Isotropy

From 2001 we know that the Universe is spatially flat (k=0) and that it is dominated by dark energy.
Most probably there has been inflation

The scalar fields drive inflation while rolling down from a maximum to a minimum
• Exponential expansion during slow rolling
• Fast rolling and exit from inflation
•
Oscillations and reheating of the Universe

The milliKelvin angular variations of CMB temperature are the inflation blown up image of Quantum fluctuations of the gravitational potential and the seeds of large scale cosmological structures

Accelerating Universe dominated by Dark
Energy
Equation of State

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This is what happens if there is isotropy !
Relaxing isotropy an entire new world of phenomena opens up
In a multidimensional world, as string theory predicts, there is no isotropy among all dimensions!

A challenging phenomenon, was proposed, at the beginning of this millenium , by a number of authors under the name of cosmic billiards . This proposal was a development of the pioneering ideas of Belinskij, Lifshits and Khalatnikov , based on the Kasner solution of Einstein equations. The Kasner solution corresponds to a regime, where the scale factors of a D -dimensional universe have an exponential behaviour . Einstein equations are simply solved by imposing quadratic algebraic constraints on the coefficients . An inspiring mechanical analogy is at the root of the name billiards .

String Theory implies D=10 space-time dimensions.
Hence 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: are the coordinates of a ball moving linearly with constant velocity

h
2
The Cartan subalgebra of a rank 9 Lie algebra.
h
9
What is this rank 9 Lie algebra?
It is E
9
, namely an affine extension of the Lie algebra E
8 h
1

by the properties of simple roots. For instance for A
3 such that...........


2
2
+

3

3

1
+


2
1
+

3
+

2 we have

1
,

2
,

It suffices to specify the scalar
For instance for A
3

1

2

3 
1
And all the roots are given
There is a simple way of representing these scalar products:Dynkin diagrams

exist for any

1

2

 
1


Algebras of the type exist only for
E
8





6

5

3
 
E series (exceptional)
In an euclidean space we

1 2 1
D=2 ?
is the duality group of String cannot fit more than 8 linear independent vectors with
The group E r angles of 120 degrees !!
D = 10 – r + 1
Theory in dimension

9
How come? More than 8 vectors cannot be fitted in an euclidean space at the prescribed angles !
Yes! Euclidean!! Yet in a non euclidean space we can do that !!
Do you remember the condition on the exponent p i
the little ball)
= (velocity of where
If we diagonalize the matrix
K ij we find the eigenvalues
Here is the non-euclidean signature in the Cartan subalgebra of E
9
. It is an infinite dimensional algebra ( = infinite number of roots!!)

h
2
There are infinitely many, but the time-like ones are in finite number. There are 120 of them as in E
8
. All the others are light-like h
9
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!!
h
1

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 )
Or, in frontal view


Switching a root
 we raise a wall on which the cosmic ball bounces


Before 2003: Rigid Billiards
Asymptotically any time
—dependent solution defines a zigzag in ln a i space
The Supergravity billiard is completely determined by U-duality group h-space walls
CSA of the U algebra hyperplanes orthogonal to positive roots

(h i
) bounces Weyl reflections
Damour, Henneaux,
Nicolai 2002 -billiard region Weyl chamber
Smooth billiards :
Exact cosmological solutions can be constructed using
U-duality ( in fact billiards are exactly integrable) bounces Smooth Weyl reflections Frè, Sorin, and collaborators ,
2003-2008 series of papers walls Dynamical hyperplanes

• The number of effective dimensions varies dynamically in time!
• Some dimensions are suppressed for some cosmic time and then enflate, while others contract.
• The walls are also dynamical. First they do not exist then they raise for a certain time and finally decay once again.
• The walls are euclidean p-branes! (Space-branes)
• When there is the brane its parallel dimensions are big and dominant, while the transverse ones contract.
• When the brane decays the opposite occurs

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Results established by P.Frè and A.Sorin
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The billiard phenomenon is the generic feature of all exact solutions of supergravity restricted to time dependence.
We know all solutions where two scale factors are equal . In this case onedimensional  model on the coset U/H. We proved complete integrability.
We established an integration algorithm which provides the general integral.
We discovered new properties of the moduli space of the general integral. This is the compact coset H/G the Weyl group
W
TS paint
, further modded by the relevant Weyl group . This is of the Tits Satake subalgebra U
TS
½ U .
There exist both trapped and (super)critical surfaces . Asymptotic states of the universe are in one-to-one correspondence with elements of W
TS
.
Classification of integrable supergravity billiards into a short list of universality classes .
Arrow of time . The time flow is in the direction of increasing the disorder:
Disorder is measured by the number of elementary transpositions in a Weyl group element.
Glimpses of a new cosmological entropy to be possibly interpreted in terms of superstring microstates , as it happens for the Bekenstein-Hawking entropy of black holes.

Definition
Statement
Because t-dependent supergravity field equations are equivalent to the geodesic equations for a manifold
U/H
Because U/H is always metrically equivalent to a solvable group manifold exp[Solv(U/H)] and this defines a canonical embedding


It is a theory of maps from one manifold to another one :
World manifold W: coordinates
 
Target manifold M: coordinates

I

Starting from D=3 (
D=2 and D=1, also
) all the (bosonic) degrees of freedom are scalars

5

8

1

2

3

4

6

7
The bosonic Lagrangian of both Type IIA and Type IIB reduces, upon toroidal dimensional reduction from D=10 to
D=3, to the gravity coupled sigma model
With the target manifold being the maximally noncompact coset space
M
 target
E
8 ( 8 )
SO ( 16 )

The 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

Is the discrete group generated by all reflections with respect to all roots
Weyl(L) is a discrete subgroup of the orthogonal group O(r) where r is the rank of L.

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Dimensional reduction to D=3 realizes the identification SUGRA =  -model on U/H
The mechanism of Kac Moody extensions
The solvable parametrization of noncompact U/H
The Tits Satake projection
The Lax representation of geodesic equations and the Toda flow integration algorithm

Lax pair representation and the integration algorithm

Solvable coset representative
Lax operator (symm.)
Connection (antisymm.)
Lax Equation

From initial data we obtain the time flow (complete integral)
Initial data are specified by a pair: an element of the non-compact Cartan
Subalgebra and an element of maximal compact group:

The flow is isospectral
The asymptotic values of the Lax operator are diagonal (Kasner epochs)

Parameter space
Proposition
Trapped submanifolds
ARROW OF
TIME

Available flows on 3-dimensional critical surfaces
Available flows on edges, i.e. 1-dimensional critical surfaces

Plot of

1
¢ h
Plot of

1
¢ h
Future
Trajectory of the cosmic ball
Zoom on this region
PAST
An example of flow on a critical surface for SO(2,4).

2
, i.e. O
2,1
= 0
Future infinity is

8
(the highest Weyl group element), but at past infinity we have

1
(not the highest) = criticality

Plot of

1
¢ h
Future
PAST
Plot of

1
¢ h
O
2,1
' 0.01
(Perturbation of critical surface)
There is an extra primordial bounce and we have the lowest
Weyl group element

5 at t = -1

In case of request of request…..there is encore …..

More details on the underlying
Mathematical Structure

The algebraic structure of duality algebras in D<4 dimensions

Universal, comes from Gravity
Symplectic metric in d=2 Symplectic metric in 2n dim
Comes from vectors in D=4

( Julia, Henneaux, Nicolai, Damour )
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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

Duality algebras for diverse N(Q) from D=4 to D=3
N=8 E
7(7)
E
8(8)
N=6
SO*(12) E
7(-5)
N=5 SU(1,5) E
6(-
14)
N=4
SL(2,R) £SO(6,n)
SO(8,n+2)
N=3 SU(3,n) £ U(1)
Z
SU(4,n+1)

What happens for D<3?
Exceptional E
11- D series for N=8 give a hint

5
2

0
3

8
4

1
5

2
6

3
7

4
8

6

7
9
SO(5,5) SL(5)
SL(2)
+ SL(3)
GL(2,R)
Julia 1981
E
9
= E
8
Æ E
8
E
7
E
6
E
5
E
4
E
3
D
U
D

This extensions is affine!
The new affine triplet: ( L MM
0
, L MM
+
, L MM
-
)
The new triplet is connected to the vector root with a single line , since the SL(2)
MM commutes with U
D=4
U
D=4

0

W
2 exceptions: pure D=4 gravity and N=3 SUGRA
U
D=4

W1

W2

0

1 
0

N=8 E
7(7)
N=6
SO*(12)
D=4 D=3
E
8(8)
E
7(-5)
E
9(9)
E

7
N=5 SU(1,5) E
6(-
14)
E

6
N=4
SL(2,R) £SO(6,n)
SO(8,n+2)
SO(8,n+2)

N=3 SU(3,n) £ U(1)
Z
SU(4,n+1)
D=2

A fondamental ingredient to single out the universality classes and the relevant Weyl group

Several roots of the higher system have the same root
Projection of rank r

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 SUGRA we have E
7(-5)
Compact simple roots define a sugalgebra
H paint

The subalgebra of external automorphisms: is compact and it is the Lie algebra of the paint group

Paint group in diverse dimensions
The paint group survives under dimensional reduction, that adds only non-compact directions to the scalar manifold
D=4 D=3
It means that the Tits Satake projection commutes with the dimensional reduction

Crucial for integrability is the remarkable relation between non-compact cosets U/H and
Solvable Group Manifolds exp[Solv]

i.e.
• 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] , D k [A] = [D k-1 [A] , D k-1 [A] ]
•
D n [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
Solv
N
½ sl(N,R)

There 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
Splitting the Lie algebra U into the maximal compact subalgebra H plus the orthogonal complement K
•There are precise rules to construct Solv(U/H)
•Essentially Solv(U/H) is made by
•the non-compact Cartan generators H i
2 CSA  K and
•those positive root step operators E
 which are not orthogonal to the non compact Cartan subalgebra CSA  K

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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 Ddimensions the E
11-D series of Lie algebras
For lower supersymmetry we always have non-maximally split algebras U