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The arrow of time and the Weyl group: all supergravity billiards are integrable FUTURE Talk by Pietro Frè at Dubna 2008 PAST g = 0 -1 a 1 2 (t) a 2 2 (t) 0 a 3 2 (t) h 2 Useful pictorial representation: A light-like trajectory of a ball in the lorentzian space of h 3

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the arrow of time and the weyl group all supergravity billiards are integrable

The arrow of time and the Weyl group:all supergravity billiards are integrable

FUTURE

Talk by Pietro Frè

at Dubna 2008

PAST

slide2

g=

0

-1

a12 (t)

a22 (t)

0

a32 (t)

h2

Useful pictorial representation:

A light-like trajectory of a ball in

the lorentzian space of

h3

hi(t)= log[ai(t)]

h1

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?

The Kasner universe: an empty, homogeneous, but non-isotropic universe

Some of the scale factors expand, but some other have to contract: an anisotropic universe is not static even in the absence of matter!

These equations are the Einstein equations

introducing billiard walls

Let us now consider, the coupling of a vector field to diagonal gravity

Introducing Billiard Walls

If Fij = const this term adds a potential to the ball’s hamiltonian

Free motion (Kasner Epoch)

Asymptoticaly

Inaccessible region

Wall position or bounce condition

slide4

h1

a wall

ω(h) = 0

h2

H1

H3

H2

The Rigid billiard

When the ball reaches the wall it bounces against it:

geometric reflection

ball trajectory

It means that the space directions transverse to the wallchange 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

Damour, Henneaux,

Nicolai 2002 --

Smooth Billiards and dualities

Asymptotically any time—dependent solution defines a zigzag in ln ai space

The Supergravity billiard is completely determined by U-duality group

h-space

CSA of the U algebra

hyperplanes orthogonal

to positive roots(hi)

walls

bounces

Weyl reflections

billiard region

Weyl chamber

Exact cosmological solutions can be constructed using

U-duality (in fact billiards are exactly integrable)

Smooth billiards:

Frè, Rulik, Sorin,

Trigiante

2003-2007

series of papers

bounces

Smooth Weyl reflections

walls

Dynamical hyperplanes

main points

Definition

Statement

Main Points

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

the discovered principle
The discovered Principle

The relevant

Weyl group is that

of the Tits Satake

projection. It is

a property of auniversality class

of theories.

There is an interesting topology of parameter space for the LAX EQUATION

the mathematical ingredients
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
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

INGREDIENT 1

example from d 4 to d 3 and back

D=4 vector fields

model of D=4 scalars

Field dependent coupling constants and theta angles

Example: from D=4 to D=3 and back

The Bosonic action of any supergravity model (ungauged) can be written

In the following form:

Dimensionally reducing to D=3 we obtain

just a larger group sigma model

we obtain the sigma model
We obtain the sigma model:

Where the old and new scalars are:

solvable lie algebras i e triangular matrices

INGREDIENT 2

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] = 0for some n > 0then 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

SolvN½ sl(N,R)

the solvable parametrization
The solvable parametrization

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 complementK

  • There are precise rules to construct Solv(U/H)
  • Essentially Solv(U/H) is made by
    • the non-compact Cartan generators Hi2 CSA  K and
    • those positive root step operators E which are not orthogonal to the non compact Cartan subalgebra CSA  K
maximally split cosets u h
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 Dynkin diagram is

Tits Satake Projection: an example

The D3» A3 root system contains 12 roots:

Let us distinguish the roots that have

a non-zero

z-component,

from those that have

a vanishing

z-component

Complex Lie algebra SO(6,C)

INGREDIENT 3

We consider the real section SO(2,4)

tits satake projection an example16

The D3» A3 root system contains 12 roots:

Complex Lie algebra SO(6,C)

The Dynkin diagram is

We consider the real section SO(2,4)

Tits Satake Projection: an example

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 example17

The D3» A3 root system contains 12 roots:

Complex Lie algebra SO(6,C)

The Dynkin diagram is

We consider the real section SO(2,4)

Tits Satake Projection: an example

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 example18

The D3» A3 root system contains 12 roots:

Complex Lie algebra SO(6,C)

The Dynkin diagram is

We consider the real section SO(2,4)

Tits Satake Projection: an example

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 example19
Tits Satake Projection: an example

The 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.

tits satake projection an example20
Tits Satake Projection: an example

This system of

vectors is actually

a new root system in rank r = 2.

It is the root system B2» C2 of the Lie Algebra Sp(4,R) » SO(2,3)

scalar manifolds in non maximal sugras and tits satake submanifolds
Scalar Manifolds in Non Maximal SUGRAS and Tits Satake submanifolds

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

slide24

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 paint group

The subalgebra of external automorphisms:

is compact and it is the Lie algebra of the paint group

lax representation and integration algorithm

Solvable coset

representative

Lax operator (symm.)

Connection (antisymm.)

Lax Equation

Lax Representation andIntegration Algorithm

INGREDIENT 4

parameters of the time flows
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 Cartan

Subalgebra and an element of maximal compact group:

properties of the flows
Properties of the flows

The flow is isospectral

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

slide29

Parameter space

Proposition

Trapped

submanifolds

ARROW OF

TIME

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

Example. The Weyl group of Sp(4)» SO(2,3)
slide31

Plot of 1¢ h

Plot of 1¢ h

Future

PAST

An example of flow on a critical surface for SO(2,4).

2 , i.e. O2,1 = 0

Zoom on this region

Future infinity is 8

(the highest Weyl group element),

but at past infinity

we have 1 (not the highest) = criticality

Trajectory of the cosmic ball

slide32

Plot of 1¢ h

Plot of 1¢ h

Future

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

PAST

conclusions
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....
    • ......