Cosmological Backgrounds of String Theory, Solvable Algebras and Oxidation A fully algebraic machinery to generate, classify and interpret Supergravity solutions CAPRI COFIN MEETING 2003 Pietro FRE’

NOMIZU OPERATOR SOLVABLE ALGEBRA E8E8 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 E8E8 E 8 maps D=10 backgrounds into D=10 backgrounds Solutions are classified by abstract subalgebras D=3 sigma model Field eq.s reduce to Geodesic equations on D=3 sigma model D=10 SUGRA dimensional oxidation Not unique: classified by different embeddings THE MAIN IDEA COMPENSATOR METHOD TO INTEGRATE 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 (Torino, Italy)  M. Trigiante (Utrecht, The Nederlands)  V. Gili (Pavia, Italy)  A. Sorin (Dubna, Russian Federation)

The Algebraic Basis: a brief summary

Differential Geometry = Algebra

Maximal Susy implies E r+1 series Scalar fields are associated with positive roots or Cartan generators

The relevant Theorem

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

String interpretation of scalar fields

...in the sequential toroidal compactification The sequential toroidal compactification has an algebraic counterpart in the embedding of subalgebras

Sequential Embeddings of Subalgebras and Superstrings

The type IIA chain of subalgebras W is a nilpotent algebra including no Cartan ST algebra

Type IIA versus Type IIB decomposition of the Dynkin diagram Dilaton Ramond scalars The dilaton

The Type IIB chain of subalgebras U duality in D=10

Roots and Fields, Duality and Dynkin diagrams

If we compactify down to D=3 we have E 8(8) Indeed the bosonic Lagrangian of both Type IIA and Type IIB reduces to the gravity coupled sigma model With target manifold

Painting the Dynkin diagram = constructing a suitable basis of simple roots Spinor weight + Type II B painting

A second painting possibility - Type IIA painting

- SO(7,7) Dynkin diagram Neveu Schwarz sector Spinor weight = Ramond Ramond sector Surgery on Dynkin diagram

String Theory understanding of the algebraic decomposition Parametrizes both metrics G ij and B-fields B ij on the Torus Metric moduli space Internal dilaton B-field

Dilaton and radii are in the CSA The extra dimensions are compactified on circles of various radii

The Maximal Abelian Ideal From Number of vector fields in SUGRA in D+1 dimensions

THE FIELD EQUATIONS OF 10d SUPERGRAVITY AS GEODESIC EQUATIONS ON

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!!!

The orbits of geodesics are parametrized by as many parameters as the rank of U Orthogonal decompositionNon orthogonal decomposition 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 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 time dependent backgrounds of supergravity/superstrings we dimensionally reduce to D=3 If we are interested in time dependent backgrounds of supergravity/superstrings we dimensionally reduce to D=3 In D=3 gravity can be decoupled and we just study a sigma model on U/H In D=3 gravity can be decoupled and we just study a sigma model on U/H Field equations of the sigma model reduce to geodesics equations. The Manifold of orbits is parametrized by the dual of the CSA. Field equations of the sigma model reduce to geodesics equations. The Manifold of orbits is parametrized by the dual of the CSA. Geodesic equations are solved in two steps. Geodesic equations are solved in two steps.  First one solves equations for the tangent vectors. They are defined by the Nomizu connection.  Secondly one finds the coset representative Finally we oxide the sigma model solution to D=10, namely we embed the effective Lie algebra used to find the solution into E 8. Note that, in general there are several ways to oxide, since there are several, non equivalent embeddings. Finally we oxide the sigma model solution to D=10, namely we embed the effective Lie algebra used to find the solution into E 8. Note that, in general there are several ways to oxide, since there are several, non 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 A 2 system these equations are

Explicit Integration of the compensator equations for the A 2 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 vectors and 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 A 2 Lie algebra into the E 8 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 Simons term

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 E 8 Where the physical interpretation of the subalgebras and the correspondence with roots is PROBLEM:

8 physically inequivalent embeddings Solv(A 2 ) Solv(E 8 )

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 An euclidean D1-brane in directions 89 or D5 in directions An euclidean D3-brane in directions 3489

If we oxide our particular solution... Note that B 34 = 0 ; C 89 = 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. In this context, the two parameters of the A2 generating solution of the following interpretation:

The effective field equations for this oxidation 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

Explicit Oxidation: The Metric and the Ricci tensor Non vanishing Components of Ricci

Plots of the Radii for the case with We observe the phenomenon of cosmological billiard of Damour, Nicolai, Henneaux

Energy density and equations of state P in 12P in 34 P in 567P in 89

Plots of the Radii for the case with this is a pure D3 brane case

Energy density and equations of state P in 12P in 34 P in 567P in 89

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 Namely a non—trivial Ricci flat meric in d=4 plus the metric of a six dimensional torus Embedding of roots Embedding 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 the maximal number compatible with a non Friedman Universe

We can rewrite metric in group theory language

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

LIGHT LIKE GEODESICS Reddish lines are outgoing null-geodesics while Blueish lines are incoming null-geodesics

A view of outgoing geodesics

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