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Hidden Symmetries, Solvable Lie Algebras, Reduction and Oxidation in Superstring Theory Pietro Fré Dubna July 2003 An algebraic characterization of superstring dualities ]exp[/SolvHG

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In D < 10 Superstring In D < 10 the structure of Superstring Theory is governed... The geometry of the scalar manifold M M = G/H is mostly a non compact coset manifold Non compact cosets admit an algebraic description in terms of solvable Lie algebras

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For instance, the Bose Lagrangian of any SUGRA theory in D=4 is of the form:

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By compactification from higher dimensions. In this case the scalar manifold is identified as the moduli space of the internal compact manifold By compactification from higher dimensions. In this case the scalar manifold is identified as the moduli space of the internal compact manifold By direct construction of each supergravity in the chosen dimension. In this case one uses the a priori constraints provided by supersymmetry. In particular holonomy and the need to reconcile p+1 forms with scalars By direct construction of each supergravity in the chosen dimension. In this case one uses the a priori constraints provided by supersymmetry. In particular holonomy and the need to reconcile p+1 forms with scalars Two ways to determine G/H or anyhow the scalar manifold DUALITIES Special Geometries The second method is more general, the first knows more about superstrings, but the two must be consistent

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The scalar manifold of supergravities is necessarily a non compact G/H, except: In the exceptional cases the scalar coset is not necessarily but can be chosen to be a non compact coset. Namely Special Geometries include classes of non compact coset manifolds

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Scalar cosets in d=4

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In D=10 there are 5 consistent Superstring Theories. They are perturbative limits of just one theory M Theory D=11 Supergravity Type IIA superstring in D=10 Type II B superstring in D=10 Type I Superstring in D=10 Heterotic Superstring SO(32) in D=10 Heterotic Superstring E8 x E8 in D=10 This is the parameter space of the theory. In peninsulae it becomes similar to a string theory This is the parameter space of the theory. In peninsulae it becomes similar to a string theory

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The 5 string theories in D=10 and the M Theory in D=11 are different perturbative faces of the same non perturbative theory. Heterotic SO(32) Type I SO(32) Heterotic E 8 xE 8 Type II A Type II B M theory D=10 D=9 D=11

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TheoriesBOSE STATESFERMI STATES NS - NSR - RLeft handed Right handed Type II A Type II B Heterotic SO(32) Heterotic E8 x E8 Type I SO(32) Table of Supergravities in D=10

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The Type II Lagrangians in D=10

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Scalar manifolds by dimensions in maximal supergravities Rather then by number of supersymmetries we can go by dimensions at fixed number of supercharges. This is what we have done above for the maximal number of susy charges, i.e. 32. These scalar geometries can be derived by sequential toroidal compactifications.

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How to determine the scalar cosets G/H from supersymmetry

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.....and symplectic or pseudorthogonal representations

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How to retrieve the D=4 table

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Essentials of Duality Rotations The scalar potential V( is introduced by the gauging. Prior to that we have invariance under duality rotations of electric and magnetic field strengths

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Duality Rotation Groups

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The symplectic or pseudorthogonal embedding in D=2r

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continued This embedding is the key point in the construction of N-extended supergravity lagrangians in even dimensions. It determines the form of the kinetic matrix of the self-dual p+1 forms and later controls the gauging procedures. D=4,8 D=6,10

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The symplectic case D=4,8 This is the basic object entering susy rules and later fermion shifts and the scalar potential

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The Gaillard and Zumino master formula We have: A general expression for the vector kinetic matrix in terms of the symplectically embedded coset representatives. This matrix is also named the period matrix because when we have Calabi Yau compactifications the scalar manifold is no longer a coset manifold and the kinetic matrix of vectors can instead be determined form algebraic geometry as the period matrix of the Calabi Yau 3-fold

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Summarizing: The scalar sector of supergravities is “mostly” a non compact coset U/H The scalar sector of supergravities is “mostly” a non compact coset U/H The isometry group U acts as a duality group on vector fields or p-forms The isometry group U acts as a duality group on vector fields or p-forms U includes target space T-duality and strong/weak coupling S-duality. U includes target space T-duality and strong/weak coupling S-duality. For non compact U/H we have a general mathematical theory that describes them in terms of solvable Lie algebras..... For non compact U/H we have a general mathematical theory that describes them in terms of solvable Lie algebras.....

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Solvable Lie algebra description...

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Differential Geometry = Algebra

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

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The relevant Theorem

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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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Definition of the cocycle N

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String interpretation of scalar fields

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

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Sequential Embeddings of Subalgebras and Superstrings

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

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

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

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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

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

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A second painting possibility - Type IIA painting

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

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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

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

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

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An application: searching for cosmological solutions in D=10 via D=3 E8E8 E8E8 D=10 SUGRA (superstring theory) D=3 sigma model dimensional reduction dimensional oxidation Since all fields are chosen to depend only on one coordinate, t = time, then we can just reduce everything to D=3 E 8 maps D=10 backgrounds into D=10 backgrounds

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What follows next is a report on work to be next published Based on the a collaboration: Based on the a collaboration: P. F., F. Gargiulo, K. Rulik (Torino, Italy) M. Trigiante (Utrecht, The Nederlands) V. Gili (Pavia, Italy) A. Sorin (Dubna, Russian Federation)

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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!!!

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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:

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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 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 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.

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The paradigma of the A2 Lie Algebra

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The A2 differential system

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Searching the normal form for the J=2 representation

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The normal form is a diagonal traceless matrix, obviously!!!

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Fixing the normal tangent vector

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NORMAL FORM of the 5-vector

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Explicit solution for the tangent vectors

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Which are solved by:

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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 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.

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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 Simons term

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The type IIB field equations

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Inequivalent embeddings PROBLEM: There are several inequivalent ways, due to the following graded structure of the Solvable Lie algebra of E 8 where

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5 physically inequivalent embeddings

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

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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:

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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

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Explicit Oxidation: The Metric and the Ricci tensor Non vanishing components

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

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

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

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

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