Cosmological Backgrounds of String Theory, Solvable Algebras and Oxidation

A fully algebraic machinery to A fully algebraic machinery to generate, classify and interpret generate, classify and interpret

Supergravity solutionsSupergravity solutions

CAPRI COFIN MEETING 2003

Pietro FRE’

NOMIZU OPERATORSOLVABLE ALGEBRA

E8

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

E8

E8 maps D=10 backgrounds

into D=10 backgrounds

Solutions are classified by abstract subalgebras

8EG

D=3 sigma model

)16(/8 SOEM Field eq.s reduce to Geodesic equations on

D=3 sigma model

D=10 SUGRA D=10 SUGRA

dimensional oxidation

Not unique: classified by different embeddings

8EG

THE MAIN IDEA

COMPENSATOR METHOD TOINTEGRATE GEODESIC EQUATIONS

What follows next is a report on work both published and to be published Based on a collaboration by:Based on a collaboration by:

P. F. , F. Gargiulo, K. Rulik (P. F. , F. Gargiulo, K. Rulik (Torino, Torino, ItalyItaly))

M. Trigiante M. Trigiante (Utrecht, The Nederlands)(Utrecht, The Nederlands)V. Gili V. Gili (Pavia, Italy)(Pavia, Italy)A. SorinA. Sorin (Dubna, Russian Federation) (Dubna, Russian Federation)

The Algebraic Basis:

a brief summary

Differential Geometry = Algebra

Maximal Susy implies Er+1 series

Scalar fields are associated with positive roots or Cartan generators

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

Explicit Form of the Nomizu connection

Let us recall the definition of the cocycle N

The type IIA chain of subalgebras

W is a nilpotent algebra including no Cartan

ST algebra

The Type IIB chain of subalgebras

U duality in D=10

Roots and Fields, Duality and Dynkin

diagrams

8

5

1 2 3 4 6 7

If we compactify down to D=3we have E8(8)

Indeed the bosonic Lagrangian of both Type IIA and Type IIB reduces to the gravity coupled sigma model

)16()8(8

SO

E

targetM With target manifold

Painting the Dynkin diagram = constructing a suitable basis of simple roots

)( 7654321821

7

322

211

188

656

655

544

433

Spinor

weight

8

5

1 2 3 4 6 7+

Type II B painting

A second painting possibility

8

5

1 2 3 4 6 7 -

)( 7654321821

7

322

211

188

656

655

544

433

Type IIA painting

-4

5

8 1 2 3 6 7

66

SO(7,7) Dynkin diagram

Neveu Schwarz sector

48 1 2 3 6

5

7

Spinor weight =Ramond Ramond sector

Surgery on Dynkin diagram

String Theory understanding of the algebraic decomposition

)7()7()7,7(

torusT theof space Moduli 7

SOSOSO

Parametrizes both metrics Gij and B-fields Bij on the Torus ijijSOSO

SO BGL exp)7()7()7,7(

B2)1,1(O

)7(

),7(

)7(

)7,7(W

SO

RSLSolv

SO

SOSolv

Metric moduli spaceInternal dilaton

B-field

Dilaton and radii are in the CSA

The extra dimensions are compactified on circles of The extra dimensions are compactified on circles of various radiivarious radii

THE FIELD EQUATIONS OF 10d SUPERGRAVITY

AS GEODESIC EQUATIONS

ON

)16(8

SO

E

Decoupling of 3D gravity

Decoupling 3D gravity continues...

K is a constant by means of the field equations of scalar fields.

The matter field equations are geodesic equations in the target manifold U/H

Geodesics are fixed by initial conditions

The starting point The direction of the initial tangent vector

Since U/H is a homogeneous space all initial points are equivalent

Initial tangent vectors span a representation of H and by means of H transformations can be reduced to normal form.

The orbits of geodesics contain as many parameters as that normal form!!!

02

2

d

d

d

d

d

d JIIJK

I

HUI / 0

K 0 I

The orbits of geodesics are parametrized by as many parameters as the rank of U

)( , 2

1 EEHT iA

)/( HGSolv HKHG

Orthogonal decomposition Non orthogonal decomposition

ofspanK

ofspanH )( 2

1 EEJsitive rootset of pos

Indeed we have the following identification of the

representation K to which the tangent vectors belong:

and since

We can conclude that any tangent vector can be brought to have only CSA components by means of H transformations

CSA , ; 0 , ii HHEE

The cosmological solutions in D=10 are therefore parametrized by 8 essential parameters. They can be obtained from an 8 parameter generating solution of the sigma model by means of SO(16) rotations.

The essential point is to study these solutions and their oxidations

Let us consider the geodesics equation explicitly

and turn them to the anholonomic basis

The strategy to solve the differential equations consists now of two steps:

•First solve the first order differential system for the tangent vectors

•Then solve for the coset representative that reproduces such tangent vectors

The Main Differential system:

Summarizing: If we are interested in If we are interested in time dependent backgroundstime dependent backgrounds of of

supergravity/superstrings we dimensionally reduce to supergravity/superstrings we dimensionally reduce to D=3D=3 In D=3 In D=3 gravity can be decoupledgravity can be decoupled and we just study a and we just study a sigma model on sigma model on

U/HU/H Field equations of the sigma model reduce to Field equations of the sigma model reduce to geodesics equationsgeodesics equations. The . The

Manifold of orbitsManifold of orbits is parametrized by the is parametrized by the dual of the CSAdual of the CSA.. Geodesic equationsGeodesic equations are solved in two steps. are solved in two steps.

First one solves equations for the First one solves equations for the tangent vectorstangent vectors. They are defined . They are defined by the by the Nomizu connectionNomizu connection..

Secondly one finds the Secondly one finds the coset representativecoset representative Finally we Finally we oxideoxide the sigma model solution to D=10, namely we the sigma model solution to D=10, namely we embed embed

the effective Lie algebra used to find the solution into Ethe effective Lie algebra used to find the solution into E8.8. Note that, in Note that, in general there are several ways to oxide, since there are several, non general there are several ways to oxide, since there are several, non equivalent embeddings.equivalent embeddings.

The paradigma of the A2 Lie Algebra

The A2 differential system

The H compensator method

THIS A SYSTEM OF DIFFERENTIAL EQUATIONS FOR THE H-PARAMETERS

The Compensator Equations

Solving the differential system for the compensators is fully equivalent to solving the original system of equations for the tangent vectors

The compensator system however is triangular and can be integrated by quadratures

For instance for the A2 system these equations are

Explicit Integration of the compensator equations for the A2 system

The solution contains three integration constants. Together with the two constanst of the generating solution this makes five.

We had five equations of the first order. Hence we have the general integral !!

AS AN EXAMPLE WE DISCUSS

THE SIMPLEST SOLUTION (One rotation only )

and SOME OF ITS OXIDATIONS

Explicit solution for the tangent vectorsand the scalar fields after one rotation

Next we consider the equations for the scalar fields:

The equations for the scalar fields can always be integrated because they are already reduced to quadratures. The form of the vielbein is obtained by calculating the left invariant 1—form from the coset representative:

The order is crucial: from left to right, decreasing grade. This makes exact comparison with supergravity

This is the final solution for the scalar fields, namely the parameters in the Solvable Lie algebra representation

This solution can be OXIDED in many different ways to a complete solution of D=10 Type IIA or Type IIB supergravity. This depends on the various ways of embedding the A2 Lie algebra into the E8 Lie algebra.

The physical meaning of the various oxidations is very much different, but they are related by HIDDEN SYMMETRY transformations.

Type II B Action and Field equations in D=10

Where the field strengths are:

Note that the Chern Simons term couples the RR fields to the NS fields !!

Chern Simonsterm

The type IIB field equations

OXIDATION =EMBEDDING

SUBALGEBRAS

There are several inequivalent ways, due to the following graded structure of the Solvable Lie algebra of E8

fieldB roots B

form-k RR roots ...C

moduli field-B roots

KK vectors roots

moduli metric roots SL(7)

i8]1[

217[k]

]2[

8]1[

i

kiii

ji

ii

ji

B

AA

SolvΑ

Where the physical interpretation of the subalgebras and the correspondence with roots is

PROBLEM: )( 82 ESolvASolv

8 physically inequivalent embeddingsSolv(A2) Solv(E8)

Embedding description continued

Choosing an example of type 4 embedding

Physically this example corresponds to a superposition of three extended objects:

1. An euclidean NS 1-brane in directions 34 or NS5 in directions 1256789

2. An euclidean D1-brane in directions 89 or D5 in directions 1234567

3. An euclidean D3-brane in directions 3489

If we oxide our particular solution...

Note that B34 = 0 ; C89= 0 since in our particular solution the tangent vector fields associated with the roots are zero. Yet we have also the second Cartan swtiched on and this remembers that the system contains not only the D3 brane but also the 5-branes. This memory occurs through the behaviour of the dilaton field which is not constant rather it has a non trivial evolution.

The rolling of the dilaton introduces a distinction among the directions pertaining to the D3 brane which have now different evolutions.

charge" brane D5 "

charge" brane D3"

In this context, the two parameters of the A2 generating solution of the following interpretation:

The effective field equations for this oxidation

0FF*

F F 150 2

0*

[5][5]

]5[N

[5]M2

1

dd

R

dd

NMMN

For our choice of oxidation the field equations of type IIB supergravity reduce to

and one can easily check that they are explicitly satisfied by use of the A2 model solution with the chosen identifications

5 brane contribution to

the stress energy tensor

D3 brane contribution to

the stress energy tensor

Out[54]= dt2 t2323 Cosh

t 2

t232

23 Cosh t 2dx12 dx2

2

t6 dx32 dx4

2Cosh t 2 Cosh

t 2dx52 dx6

2 dx72

t6 dx82 dx9

2Cosh t 2

2)10( Dds

Explicit Oxidation: The Metricand the Ricci tensor

Ricci11 12882 9 2 2 Cosh2 t Sech

t 22

Ricci22 132

12 t2

3 2 2 Sech t 22

Ricci44 132

16 t62

3 22 Sech

t 23

Ricci66 132

t23 2 2 Sech

t 22

Ricci99 132

16 t62

3 22 Sech

t 23

00Ric

2211 RicRic

776655 RicRicRic

4433 RicRic

9988 RicRic

Non vanishing

Components

of Ricci

Plots of the Radii for the case with

5 10 15 20

1

2

3

4

5

6

7

R12

2 4 6 8 10

0.65

0.7

0.75

0.8

0.85

R34

2.5 5 7.5 10 12.5 15

1.1

1.2

1.3

1.4

R567

5 10 15 20

0.5

0.6

0.7

0.8

0.9

1.1

1.2

R89

1

2

We observe the phenomenon of cosmological billiardof Damour, Nicolai, Henneaux

Energy density and equations of state 1

2

Plot of the total energy density

5 10 15 20

0.01

0.02

0.03

0.04

0.05

0.06

total

Plot of the ratiobrane energytotal energy

2.5 5 7.5 10 12.5 15

0.82

0.84

0.86

0.88

0.9

0.92

brane

Plot of the pressures of the brane

5 10 15 20

-0.04

-0.03

-0.02

-0.01

P12

5 10 15 20

0.01

0.02

0.03

0.04

P345 10 15 20

-0.04

-0.03

-0.02

-0.01

P567

5 10 15 20

0.01

0.02

0.03

0.04

P89

P in 12 P in 34 P in 567 P in 89

Plots of the Radii for the case withthis is a pure D3 brane case 0

2

2.5 5 7.5 10 12.5 15 17.5

1

2

3

4

5

6

7

R122.5 5 7.5 10 12.5 15 17.5

0.6

0.7

0.8

0.9

R34

2.5 5 7.5 10 12.5 15 17.5

1.1

1.2

1.3

1.4

R5672.5 5 7.5 10 12.5 15 17.5

0.6

0.7

0.8

0.9

R89

Energy density and equations of state 0

2

Plot of the pressures of the brane

P in 12 P in 34 P in 567 P in 89

Plot of the total energy density

2.5 5 7.5 10 12.5 15 17.5

0.01

0.02

0.03

0.04

total

Plot of the ratiobrane energytotal energy

2.5 5 7.5 10 12.5 15 17.5

1

1

1

1

brane

2.5 5 7.5 10 12.5 15 17.5

0.01

0.02

0.03

0.04

P89

2.5 5 7.5 10 12.5 15 17.5

0.01

0.02

0.03

0.04

P342.5 5 7.5 10 12.5 15 17.5

-0.04

-0.03

-0.02

-0.01

P122.5 5 7.5 10 12.5 15 17.5

-0.04

-0.03

-0.02

-0.01

P567

A new embedding (TYPE 6)Choosing embedding of Type 6 we obtain a purely gravitational configuration.There are no dilaton or p—forms excited and we get a Ricci flat metric in D=10 which is of the form

224

2)10( 6TDD dsdsds

Namely a non—trivial Ricci flat meric in d=4 plus the metric of a six dimensional torus

Embedding of rootsEmbedding of CSA

Gives diagonal metric scale factors

Inserting the sigma model solution with just one root switched on

A Ricci flat metric in d=4 is our result

4 Killings is themaximal number compatible with a non FriedmanUniverse

We can rewrite metric in group theory language

This metric falls into Bianchi classification of Homogeneous Cosmological metrics in d=4

5 10 15 20r

-10

-8

-6

-4

-2

2

t

LIGHT LIKE GEODESICS

Reddish lines are outgoing null-geodesics

while Blueish lines are incoming null-geodesics

X

T

r

t

A view of outgoing geodesics

Test for TeXPoint 2