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

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 (Kasner Epoch)

Inaccessible region

Wall position or bounce condition

Asymptoticaly

Introducing Billiard Walls

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 (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

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

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

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

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

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!

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

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

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

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

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

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

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

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)

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.

Scalar Manifolds in Non Maximal SUGRAS and Tits Satake submanifolds

An overview of the Tits Satake projections.....and affine extensions

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

Lax Representation andIntegration Algorithm

INGREDIENT 4

Solvable coset

representative

Lax operator (symm.)

Connection (antisymm.)

Lax Equation

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:

Properties of the flowsThe flow is isospectral

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

Proposition

Trapped

submanifolds

ARROW OF

TIME

Parameter space

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

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

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

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.... ......