Cosmic Billiards are fully integrable: Tits Satake projections and Kac Moody extensionsSeminar by Pietro Fr atMilano I April 19th 2007

Introduction to cosmic billiardsI begin by introducing, somewhat heuristically, the idea of cosmic billiardsThen I will illustrate the profound relation between this pictorial description of cosmic evolution and the fundamental duality symmetries of string theory

The Universe is expanding, in the presence of matter the Universe cannot be static,

all directions of a FRW universe expand in the same way, (We introduce only one scale factor)

going back in time we turn to the moment when the Universe was very small and matter was concentrated in infinetely small region of space, matter density was infinite: Big Bang,

the character of the expansion depends on the equation of state: P = w

FRW : The Observed universe is homogeneous and isotropicTmn $ homogeneous isotropic medium with pressure P and density r

Standard FRW cosmology is concerned with studying the evolution of specific general relativity solutions, but we want to ask what more general type of evolution is conceivable just under GR rules. What if we abandon isotropy?Some of the scale factors expand, but some other have to contract: an anisotropic universe is not static even in the absence of matter!

The Kasner universe: an empty, homogeneous, but non-isotropic universeThese equations are the Einstein equations

Introducing Billiard WallsIf Fij = const this term adds a potential to the balls hamiltonianFree motionInaccessible region Wall position or bounce conditionAsymptoticaly

Billiard: a paradigm for multidimensional cosmology

The Rigid billiardball trajectoryWhen the ball reaches the wall it bounces against it: geometric reflectionIt means that the space directions transverse to the wall change their behaviour: they begin to expand if they were contracting and vice versaBilliard table: the configuration of the walls-- the full evolution of such a universe is a sequence of Kasner epochs with bounces between them-- the number of large (visible) dimensions can vary in time dynamically-- the number of bounces and the positions of the walls depend on the field content of the theory: microscopical input

Smooth Billiards and dualitiesh-spaceCSA of the U algebrawallshyperplanes orthogonalto positive roots (hi)bouncesWeyl reflectionsbilliard regionWeyl chamberThe Supergravity billiard is completely determined by U-duality groupSmooth billiards:Asymptotically any timedependent solution defines a zigzag in ln ai spaceExact cosmological solutions can be constructed using U-dualitybouncesSmooth Weyl reflectionswallsDynamical hyperplanes

To understand the previous statements we needa step backA bird flight overview ofSuperstring dualities

M-theoryD=11 SUPERGRAVITYType ISO(32) HeteroticE8 E8 HeteroticType II AType II BSTTSLow-energy effective descriptionBosonic U-dualityM-theory should admit all dualities as exact symmetries

- global duality symmetries UDN(Q)=32 D=10Type II A:

g , B ,

A , A

Type II B:

g , B ,

C , C , C0 O(1,1)SL(2,R) - local supersymmetry N(Q) 32Supergravities are supersymmetric field theories describing gravity + matter- general coordinate reparametrization invariance Diff(D) - local SO(D-1,1) invariance

Dualities: their action on the bosonic fieldsThe scalars typically parametrizea coset manifoldIn D=2n duality is a symmetry of the equations of motionfor (n-1)-forms UD

Supergravities in different dimensions are connected by dimensional reductiona , gMN , M = 0, .., D=1a ( g , g i , gi,j )More scalars and more global symmetries ! = 0,.. , di = i,..,N

Smooth supergravity and superstring billiards....THE MAIN IDEAfrom a D=3 viewpointP. F. ,Trigiante, Rulik, Gargiulo, SorinVan Proeyen and Roossel2003,2004, 2005various papers

Starting from D=3 (D=2 and D=1, also) all the (bosonic) degrees of freedom are scalarsThe bosonic Lagrangian of any Supergravity, can be reduced in D=3, to a gravity coupled sigma model

NOMIZU OPERATORSOLVABLE ALGEBRASince 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 scalarTime dep. backgroundsNomizu connection = LAX PAIRRepresentation. INTEGRATION!

With this machinery.....We can obtain exact solutions for time dependent backgroundsWe can see the bouncing phenomena (=billiard) We have to extend the idea to lower supersymmetry # QSUSY < 32 and...We do not have to stop at D=3. For time dependent backgrounds we can start from D=2 or D=1In D=2 and D=1 we have affine and hyperbolic Kac Moody algebras, respectively.....!

The cosmic ballString Theory implies D=10 space-time dimensions. In general in dimension DA generalization of the standard cosmological metric is of the type:In the absence of matter the conditions for this metric to be Einstein are:Now comes an idea at first sight extravagant.... Let us imagine that are the coordinates of a ball moving linearly with constant velocity What is the space where this fictitious ball moves

ANSWER:The Cartan subalgebra of a rank D-1 Lie algebra.What is this rank D-1 Lie algebra?It is UD=2 the Duality algebra in D=2 dimensions. This latter is the affine extension of UD=3

Now let us introduce also the roots......There are infinitely many, but the time-like ones are in finite number. There are as many of them as in UD=3. All the others are light-likeTime like roots, correspond to the light fields of Superstring Theory different from the diagonal metric: off-diagonal components of the metric and p-form fieldsWhen we switch on the roots, the fictitious cosmic ball no longer goes on straight lines. It bounces!!

The cosmic BilliardThe Lie algebra roots correspond to off-diagonal elements of the metric, or to matter fields (the p+1 forms which couple to p-branes) Switching a root we raise a wall on which the cosmic ball bounces

Or, in frontal view

Differential Geometry = Algebra

How to build the solvable algebraGiven 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

Maximal Susy implies Er+1 seriesScalar fields are associated with positive roots or Cartan generators

From the algebraic view point.....Maximal SUSY corresponds to...MAXIMALLY non-compact real forms:i.e. SPLIT ALGEBRAS.This means:All Cartan generators are non compactStep operators E 2 Solv , 8 2 + The representation is completely realThe billiard table is the Cartan subalgebra of the isometry group!

Explicit Form of the Nomizu connection for the maximally split caseThe components of the connection

Let us briefly surveyThe use of the solvable parametrization as a machinary to obtain solutions, in the split case

The general integration formulaInitial data at t=0 areA) , namely an element of the Cartan subalgebra determining the eigenvalues of the LAX operatorB) , namely an element of the maximal compact subgroupThen the solution algorithm generates a uniquely defined time dependent LAX operator

Properties of the solutionFor each element of the Weyl group

The limits of the LAX operator at t=1 are diagonal

At any instant of time the eigenvalues of the LAX operator are constant 1, ...,n

where wi are the weights of the representation to which the Lax operator is assigned.

Disconnected classes of solutionsProperty (2) and property (3) combined together imply that the two asymptotic values L1 of the Lax operator are necessarily related to each other by some element of the Weyl groupwhich represents a sort of topological charge of the solution:

The solution algorithm induces a map:

A plotted example with SL(4,R)/O(4)The U Lie algebra is A3The rank is r = 3.The Weyl group is S4 with 4! elementsThe compact subgroup H = SO(4)

The integration formula can be easily encoded into a computer programme and for any choice of the eigenvalues and for any choice of the group element 2 O(4)The programme CONSTRUCTS the solution

Example (1=1, 2=2 , 3=3)Indeed we have:

Plots of the (integrated) Cartan Fields alongthe simple roots This solution has four bounces

Explicit analytic form of the toy A2-model: M5 = SL(3,R)/SO(3)

The generating solution = Kasner epoch solutionCan be transformed by A2-duality to the general solutionwhich is equivalent to the general integral obtained byToda integration

Explicitly the full A2 model integral has the formThe walls are given byThe billiard is defined by a trajectory in the (h1,h2) space

Let us now introduce more structure of SUGRA/STRING THEORYThe first point:Less SUSY (NQ < 32) and non split algebras

Scalar Manifolds in Non Maximal SUGRAS and Tits Satake submanifoldsWHAT are these new manifolds (split!) associated with the known non split ones....???

The Billiard Relies on Tits Satake TheoryTo each non maximally non-compact real form U (non split) of a Lie algebra of rank r1 is associated a unique subalgebra UTS U which is maximally split.UTS has rank r2 < r1The Cartan subalgebra CTS UTS is the true billiard tableWalls in CTS now appear painted as a memory of the parent algebra U

root systemof rank r1ProjectionSeveral roots of the higher system have the same projection.These are painted copies of the same wall.The Billiard dynamics occurs in the rank r2 system

Two type of roots123

To say it in a more detailed way:Non split algebras arise as duality algebras in non maximal supergravities N< 8 Under the involutive automorphism that defines the non split real sectionNon split real algebras are represented by Satake diagramsFor example, for N=6 SUGRAwe have E7(-5)

The Paint Group1234

Why is it exciting?Since the Nomizu connection depends only on the structure constants of the Solvable Lie algebra

Paint group in diverse dimensionsThe paint group survives under dimensional reduction, that adds only non-compact directions to the scalar manifoldIt means that the Tits Satake projection commutes with the dimensional reduction

And now let us go the next main point..Kac Moody Extensions

Affine and Hyperbolic algebrasand the cosmic billiardWe do not have to stop to D=3 if we are just interested in time dependent backgroundsWe can step down to D=2 and also D=1In D=2 the duality algebra becomes an affine Kac-Moody algebraIn D=1 the duality algebra becomes an hyperbolic Kac Moody algebra Affine and hyperbolic symmetries are intrinsic to Einstein gravity

(Julia, Henneaux, Nicolai, Damour)

Structure of the Duality Algebra in D=3 (P.F. Trigiante, Rulik and Gargiulo 2005)

Let me remind youabout the relation betweenthe Cartan presentationand the Chevalley Serre presentation of a Lie algebra

Chevalley-Serre presentationCartan presentation

TA(L0 L+ L-) (W+ W-)The new Chevalley-Serre triplet: we take the highest weight of thesymplectic representation hnew symmetries

The Kac Moody extension of the D=3 Duality algebraIn D=2 the duality algebra becomes the Kac Moody extension of the algebra in D=3.Why is that so?

The reason is...That there are two ways of stepping down from D=4 to D=2The Ehlers reductionThe Matzner&Misner reductionThe two routes give two different lagrangians with two different finite algebra of symmetriesThere are non local relations between the fields of the two lagrangiansThe symmetries of one Lagrangian have a non local realization on the other and vice versaTogether the two finite symmetry algebras provide a set of Chevalley generators for the Kac Moody algebra

Let us seeHow the two type of dimensional reductionsEhlersMaztner MissnerYield the necessary set of Chevalley Serre triplets to generate the Kac Moody extension of the U algebra

Ehlers reduction of pure gravity+

Matzner&Misner reduction of pure gravityNO DUALIZATION OF VECTORS !!DIFFERENT SL(2,R) fields non locally related

General Matzner&Misner reduction (P.F. Trigiante, Rulik e Gargiulo 2005)

The reduction is governed by the embedding

Symmetries of MM LagrangianGD=4 through pseudorthogonal embeddingSL(2,R)MM through gravity reductionLocal O(2) symmetry acting on the indices A,B etc [ O(2) 2 SL(2,R)MM ]Combined with GD=3 of the Ehlers reduction these symmetries generate the affine extension of GD=3 ! GD=3

Let us reviewThe algebraic mechanism of this extension

N=8E8(8)N=6E7(-5)N=5E6(-14)N=4SO(8,n+2) N=3SU(4,n+1)Duality algebras for diverse N(Q) from D=4 to D=3E7(7)SO*(12)SU(1,5)SL(2,R)SO(6,n) SU(3,n) U(1)Z

What happens for D

This extensions is affine! The new triplet is connected to the vector root with a single line,since the SL(2)MM commutes with UD=42 exceptions: pure D=4 gravity and N=3 SUGRA 01The new affine triplet: (LMM0, LMM+, LMM-)

N=8E8(8)N=6E7(-5)N=5E6(-14)N=4SO(8,n+2) N=3SU(4,n+1)D=4E7(7)SO*(12)SU(1,5)SL(2,R)SO(6,n) SU(3,n) U(1)ZE9(9)E7E6SO(8,n+2)D=3D=2

ConclusionsThe algebraic structure of U duality algebras governs many aspects of String TheoryIn particular it is responsible for the cosmic billiard paradigma of multidimensional cosmologiesThe integrability of the maximal split case has to be extended to the non maximal split cases taking advantage of TS projection (work in progress)The exact integrability in D=3 has to be extended to the affine and hyperbolic casesThis is for ungauged supergravities

Gauged SupergravitiesGauging is equivalent to introducing fluxesIn gauged supergravities we have -models with potentialsThe integrability of these cases is an entirely new chapter.May be for next year seminar.!