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….!