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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’
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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
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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)
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The Algebraic Basis:
a brief summary
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Differential Geometry = Algebra
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Maximal Susy implies Er+1 series
Scalar fields are associated with positive roots or Cartan generators
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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
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The Nomizu Operator
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Explicit Form of the Nomizu connection
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Let us recall the definition of the cocycle N
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The type IIA chain of subalgebras
W is a nilpotent algebra including no Cartan
ST algebra
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The Type IIB chain of subalgebras
U duality in D=10
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Roots and Fields, Duality and Dynkin
diagrams
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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
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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
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A second painting possibility
8
5
1 2 3 4 6 7 -
)( 7654321821
7
322
211
188
656
655
544
433
Type IIA painting
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-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
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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
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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
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THE FIELD EQUATIONS OF 10d SUPERGRAVITY
AS GEODESIC EQUATIONS
ON
)16(8
SO
E
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Decoupling of 3D gravity
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Decoupling 3D gravity continues...
K is a constant by means of the field equations of scalar fields.
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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
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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:
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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
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Let us consider the geodesics equation explicitly
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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
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The Main Differential system:
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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.
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The paradigma of the A2 Lie Algebra
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The A2 differential system
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The H compensator method
THIS A SYSTEM OF DIFFERENTIAL EQUATIONS FOR THE H-PARAMETERS
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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
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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 !!
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AS AN EXAMPLE WE DISCUSS
THE SIMPLEST SOLUTION (One rotation only )
and SOME OF ITS OXIDATIONS
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Explicit solution for the tangent vectorsand the scalar fields after one rotation
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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
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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.
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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
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The type IIB field equations
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OXIDATION =EMBEDDING
SUBALGEBRAS
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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
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8 physically inequivalent embeddingsSolv(A2) Solv(E8)
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Embedding description continued
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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
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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:
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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
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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
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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
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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
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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
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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
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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
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Inserting the sigma model solution with just one root switched on
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A Ricci flat metric in d=4 is our result
4 Killings is themaximal number compatible with a non FriedmanUniverse
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We can rewrite metric in group theory language
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This metric falls into Bianchi classification of Homogeneous Cosmological metrics in d=4
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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
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X
T
r
t
A view of outgoing geodesics
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Test for TeXPoint 2