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

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

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

Two ways to determine G/H or anyhow the scalar 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 DUALITIES Special Geometries The second method is more general, the first knows more about superstrings, but the two must be consistent

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

Scalar cosets in d=4

In D=10 there are 5 consistent Superstring Theories In D=10 there are 5 consistent Superstring Theories. They are perturbative limits of just one theory Heterotic Superstring E8 x E8 in D=10 Heterotic Superstring SO(32) in D=10 M Theory D=11 Supergravity Ora che abbiamo vagamente intuito che cosa è una stringa, possiamo dare uno sguardo alla seguente figura che è divenuta celeberrima dopo l’introduzione della M teoria nel 1995. Non è facile da spiegare, ma era un risultato noto da circa 15 anni che di teoriequantisticamente consistenti delle superstringhe ne esistessero cinque e solo cinque. Esse hanno buffi nomi come la tipo IIA, la tipo IIB, la tipo I, l’eterotica SO(32) ovvero l’eterotica E8 x E8. Nonostante i buffi nomi queste teorie si distinguono per proprietà piuttosto semplici quale l’essere fatte di lacci chiusi ovvero sia di lacci che di corde aperte. Altro elemento di distinzione è il permettere che certe vibrazioni circolino sul foglio di mondo sia in senso destrorso che sinistrorso ovvero solo in uno dei due sensi. A prima vista le proprietà di queste cinque teorie sono radicalmente differenti. La grande scoperta del 1995 fu che questa diversità dipende solo dal fatto che esse sono studiate in un regime di deboli interazioni mutue tra stringhe. Quando le cinque teorie vengono analizzate in un regime di accoppiamento forte le differenze scompaiono ed anzi si scopre che ciascuna di esse descrive ad accoppiamento debole la fase di acoppiamento forte di un’altra teoria dello stesso insieme. Sorprendentement, il fatto stesso che sitratti di teorie di corde è un artefatto dell’acoppiamento debole. Man mano che ci inoltra nella fase di acoppiamento forte si scopre che vi sono altri oggetti estesi nella teoria, oltre quelli unidimensionali. Tali oggetti vengono chiamati brane e si parla di una p-brana per indicarne uno di dimensione p. Questo viaggio si fa muovendosi in uno spazio di parametri che controllano la teoria e che i fisici chiamano moduli. E’ la regione a forma di stella di questo disegno. Nella parte interna dello spazio dei moduli abbiamo la vera e propria teoria M e le eccitazioni sono p-brane per tutti i possibili valori di p da 0 a 9. Nelle regioni peninsulari la teoria M assomiglia sempre ad una teoria di stringa in dieci dimensioni, con una sola eccezione. Vi è una penisola in cui essa assomiglia ad una teoria di membrane o 2-brane in undici dimensioni. E’ questa penisola una delle possibili interpretazioni della M di teoria M. Type I Superstring in D=10 This is the parameter space of the theory. In peninsulae it becomes similar to a string theory Type II B superstring in D=10 Type IIA superstring in D=10

The 5 string theories in D=10 and the M Theory in D=11 are different perturbative faces of the same non perturbative theory. M theory D=11 Heterotic E8xE8 Type I SO(32) Heterotic SO(32) Type II B Type II A D=10 D=9

Table of Supergravities in D=10 Theories BOSE STATES FERMI STATES   NS - NS R - R Left 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

The Type II Lagrangians in D=10

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.

How to determine the scalar cosets G/H from supersymmetry

.....and symplectic or pseudorthogonal representations

How to retrieve the D=4 table

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

Duality Rotation Groups

The symplectic or pseudorthogonal embedding in D=2r

.......continued D=4,8 D=6,10 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.

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

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

Summarizing: 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 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.....

Solvable Lie algebra description...

Differential Geometry = Algebra

Maximal Susy implies Er+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

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 ST algebra W is a nilpotent algebra including no Cartan

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

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

If we compactify down to D=3 we have E8(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 Type II B painting + Spinor weight

A second painting possibility - Type IIA painting

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

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

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

An application: searching for cosmological solutions in D=10 via D=3 Since all fields are chosen to depend only on one coordinate, t = time, then we can just reduce everything to D=3 E8 D=10 SUGRA (superstring theory) D=10 SUGRA (superstring theory) E8 maps D=10 backgrounds into D=10 backgrounds dimensional reduction dimensional oxidation E8 D=3 sigma model D=3 sigma model

What follows next is a report on work to be next published 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)

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!!!

Orthogonal decomposition The orbits of geodesics are parametrized by as many parameters as the rank of U Indeed we have the following identification of the representation K to which the tangent vectors belong: Orthogonal decomposition Non orthogonal decomposition

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 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. 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 E8. 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

Searching the normal form for the J=2 representation

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

Fixing the normal tangent vector

NORMAL FORM of the 5-vector

Explicit solution for the tangent vectors

Which are solved by:

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

The type IIB field equations

Inequivalent embeddings PROBLEM: There are several inequivalent ways, due to the following graded structure of the Solvable Lie algebra of E8 where

5 physically inequivalent embeddings

Choosing an example of type 4 embedding Physically this example corresponds to a superposition of three extended objects: An euclidean NS 1-brane in directions 34 or NS5 in directions 1256789 An euclidean D1-brane in directions 89 or D5 in directions 1234567 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 a1,2 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 5 brane contribution to the stress energy tensor D3 brane contribution to the stress energy tensor and one can easily check that they are explicitly satisfied by use of the A2 model solution with the chosen identifications

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

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 567 P in 89 P in 12 P in 34

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

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