The arrow of time and the Weyl group: all supergravity billiards are integrable

DFTT07/23 JINR-E2-2007- 145 The arro w of time and the W eyl group: all sup ergra vit y billiards are in tegrable † Pietr o F r ´ e a and Alexander S. Sorin b a Dip artimento di Fisic a T e oric a, Universit´ a di T orino, & INFN - Sezione di T orino via P. Giuria 1, I-10125 T orino, Italy fre@to.infn .it b Bo goliub ov L ab or atory of The or etic al Physics, Joint Institute for Nucle ar R ese ar ch, 141980 Dubna, Mosc ow R e gi o n, R ussia sorin@theor .jinr.ru Abstract In this pap er w e sho w that all su p ergra vit y billiards corresp onding to σ -mo dels on an y U / H non compact-symmetric spac e and obtained by compactifying sup ergravit y to D = 3 are fully in tegrable. The k ey p oin t in establishing th e integ ration algorithm is provided b y an up p er triangular emb edding of the s olv able Lie algebra asso ciated with U / H into sl (N , R ) w h ic h alw ays exists. In th is conte xt w e establish a remark- able relation b et w een the arro w of time and the prop erties of the W eyl group. The asymptotic states of the dev eloping Universe are in one-to-one corresp ondence with the elemen ts of the W eyl group w h ic h is a prop ert y of the Tits S atak e un iv ersalit y classes and not of their single representat iv es. F urthermore the W eyl group adm its a natural ordering in terms of ℓ T , the num b er of reflections with resp ect to the simp le ro ots and the direction of time flo ws is alw ays to wards increasing ℓ T , w h ic h pla ys the u nexp ected role of an en tropy . † This work is s upp or ted in par t by the Eur op e an Union R TN contract MR TN-CT-2004 -00510 4 and by the Italian Ministry of Universit y (MIUR) under contracts PRIN 20 05-024 045 and PRIN 20 05-02 3102. F urther- more the work of A.S. was partially supp orted by the RFBR Grant No. 06-01-006 27-a, RFBR- DF G Grant No. 06-02 -04012 -a, DF G Grant 436 RUS 113/6 69-3, the Prog ram for Supp orting Lea ding Scientific Scho ols (Grant No. NSh-5332 .2006.2 ), and the Heisenberg-La ndau Progr am. 1 F ore w ord Not withstanding its length a nd its somewhat p edagogical o rganization, the presen t one is a researc h a r ticle and not a review. All the presen ted material is, up to our kno wledge, new. Due to the com bination of sev eral differen t mathematical results a nd techn iques necessary to mak e our p oin t, whic h is instead ph ysical in spirit and relev ant to ba sic questions in sup ergra vit y and sup erstring cosmology , w e considered it appro pr ia te to c ho ose the presen t somewhat uncon v en tional format for our pap er. After the theoretical statemen t of our result, w e hav e illustrated it with the detailed study of a few examples. Thes e case-studies w ere essen tial for us in order to understand the main p oint which w e ha ve formalized in mathematical terms in part I and we think that they will be similarly esse n tia l for the ph ysicist reader. The table o f con t en ts helps the reader to get a comprehensiv e view of the article and of its structure. Con t en ts 1 F orew ord 1 I Theory: Stati ng the principles 2 2 Supergravit y billiards: a paradigm for cosmology 2 3 The pain t group and the T it s Satake pro jection 4 3.1 The solv able algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 The pain t group and its Lie algebra . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 The subpain t group and the Tits Satake subalgebra . . . . . . . . . . . . . . 6 4 T riangular em b edding in SL(N , R ) / SO(N) and in t egrabilit y 6 4.1 The in tegratio n algorithm for the La x Equation . . . . . . . . . . . . . . . . 8 5 Prop erties of the general in tegral and the parameter space 9 5.1 Discussion of the generalized W eyl group . . . . . . . . . . . . . . . . . . . . 10 5.2 The arro w of time, trapp ed and critical surfaces . . . . . . . . . . . . . . . . 12 I I Examples illustrating the principles 16 6 Choice of t he examples 17 7 The simplest maximally split case: SL(3 , R ) / SO(3) 17 7.1 Discussion of the generalized W eyl group . . . . . . . . . . . . . . . . . . . . 22 7.2 The flo w diagram and the critical surfaces fo r SL(3 , R ) . . . . . . . . . . . . 24 1 8 The maximally split case Sp(4 , R ) / U(2) 31 8.1 The W eyl group and the generalized W eyl group of sp (4 , R ) . . . . . . . . . 32 8.2 Construction of the sp (4 , R ) Lie algebra . . . . . . . . . . . . . . . . . . . . . 36 8.3 P ara meterization of the compact group U(2) a nd critical submanifolds . . . . 37 8.4 Examples for sp (4 , R ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 8.4.1 An example of flo w in the bulk of para meter space: Ω 5 ⇒ Ω 8 . . . . 44 8.4.2 An example of flo w on the sup er-critical surface Σ 9 : Ω 6 ⇒ Ω 8 . . . . 45 8.4.3 An example of flo w on the sup er-critical surface Σ 2 : Ω 1 ⇒ Ω 8 . . . . 47 9 The case of t he so ( r, r + 2 s ) algebra 49 9.1 The correspo nding complex Lie alg ebra and ro ot system . . . . . . . . . . . 49 9.2 The real form so ( r , r + 2 s ) of the D r + s Lie algebra . . . . . . . . . . . . . . . 51 10 A case study for the Tits Satak e pro jection: SO(2 , 4) 56 10.1 The generalized W eyl group for SO(2 , 4) . . . . . . . . . . . . . . . . . . . . 59 10.2 V ertices, edges and trapp ed surfaces . . . . . . . . . . . . . . . . . . . . . . . 6 1 10.3 Examples of flows for SO(2 , 4) . . . . . . . . . . . . . . . . . . . . . . . . . . 62 I I I Pe rsp ectiv es 68 11 Summary of results 68 12 Op en problems and directions t o b e pursued 69 P art I Theory: Stating the pri nciples 2 Sup ergra vit y b i lliards: a p aradigm for co smology Cosmological implications o f sup erstring theory hav e b een under atten tiv e consideration in the last few years from v arious viewpoints [1]. This inv olv es the classification and the study of p ossible time-ev o lving string bac kgro unds whic h amounts to the construction, classification and analysis of sup ergrav it y solutions dep ending only on time o r, more generally , on a low n umber of co ordinates including time. In this con text a quite challe nging and p otentially highly relev a n t phenomenon for the o v erall in terpretation of extra–dimensions and string dynamics is prov ided b y the so named c osmic bil liar d phenomenon [2], [3], [4], [5]. This is based on the relat io n b etw een the cosmo- logical scale factors and the duality gro ups U of string theory . The g r o up U app ears a s isom- etry group of the scalar manifold M scalar emerging in compactifications of 10 –dimensional sup ergra vit y to low er dimensions D < 10 and dep ends b oth on the g eometry o f the compact dimensions a nd on the num b er of preserv ed supersymmetries N Q ≤ 32. F or N Q > 8 the scalar manifold is alwa ys a homogeneous space U / H. The cosmological scale factor s a i ( t ) 2 asso ciated with the v ario us dimens ions of sup ergra vit y are in terpreted as exp o nen tials of those scalar fields h i ( t ) whic h lie in the Cartan subalgebra of U , while the other scalar fields in U / H correspo nd to p ositiv e ro ots α > 0 of the Lie algebra U . The cosmological ev olution is describ ed b y a fictitious b al l that mov es in the C SA of U and o ccasionally b ounces on the hy p erplanes orthogona l to the v arious ro ots: the billiard walls . Such b ounces represen t in vers ions in the time evolution of scale factors. Suc h a scenario w as introduced b y D amour, Henneaux, Julia and Nicolai in [2 ], [3], [4], [5], generalizing classical results obtained in the con text of pure General Relativity [6]. In these pap ers the billia r d phenomenon w as mainly considered as an asymptotic regime near singularieties. In a series of pap ers [7], [8, 9, 10] inv olving b oth the presen t authors and other collab- orators it w as started and dev elop ed what can b e describ ed as the smo oth c osm i c bil liar d pr o gr amme . This amoun ts to the study of the bil liar d fe atur es within the framew ork of ex- act a na lytic solutio ns o f sup ergravit y rat her than in asymptotic regimes. Crucial star t ing p oin t in this prog ramme was the observ at ion [7] that the fundamental mathematical setup underlying the app earance of t he billiard phenomenon is the so named So l v a ble Lie algebr a p ar ametrization of sup ergra vit y scalar manifolds, pioneered in [11] and la ter applied to the solution of a large v ariety of sup erstring/sup ergrav it y problems [12], [13], [16], [14], [15] (for a comprehensiv e review see [17]). Thanks to the solv able pa r ametrization, one can establish a precise alg o rithm to imple- men t the follow ing programme: a R educe the original sup ergrav it y in higher dimensions D ≥ 4 (for instance D = 10 , 11) to a g r a vity-coupled σ –mo del in D ≤ 3 where gravit y is non–dynamical and can b e eliminated. The tar get manifold is the non compact coset U / H ∼ = exp [ S ol v (U / H)] metrically equiv a lent to a solv able group manifold. b Ut ilize v arious group theoretical tech niques in order to in tegrate analytically the σ –mo del equations. c Dimensionally oxide the solutions obtained in this w a y to extract time dep enden t solutions of D ≥ 4 sup ergravit y . In view of the ab ov e observ ation w e will use the fo llowing definition of sup ergra vit y billiards: Definition 2.1 << A sup er gr avity bil liar d is a one-dim ensional σ -mo del whose tar get sp ac e is a n o n -c omp act c oset manifold U / H , metric al ly e quivalent, in for c e of a gener al the or em, to a solvable gr oup ma nifold exp [ S olv (U / H)] . >> There exists a complete classification [17, 18, 10] of all non-compact coset ma nif o lds U / H relev ant to the v arious instances of sup ergra vities in all space-time dimensions D and for all n umbers N Q of sup erc harges. A general imp orta n t feature is that maximal sup ersymmetry N Q = 32 corresp onds t o maximally split symmetric cosets. Definition 2.2 << A s ymm etric c oset manifold U / H is maximal ly s plit w hen the Lie alge- br a U of U is the maximal ly no n c om p act r e al se ction of its o wn c o mplexific ation and H ⊂ U is the unique maximal c omp act s ub a l g ebr a. In this c ase the Cartan sub algebr a C is c ompletely 3 non-c omp act, namely the non-c omp act r a n k r n.c. = r e quals the r ank and the solvab l e Lie al- gebr a S ol v (U / H) is mad e by a l l the Cartan gener ators H i plus the step op er ators E α for al l the p ositive r o ots α > 0 . >> In [19] the presen t authors shew that for maximally split cosets the o ne-dimensional σ - mo del is fully in tegrable and the general in tegral can b e constructed using a well established algorithm endow ed with a series of distinctiv e and quite inspiring features. In the presen t pap er w e demonstrate that the algo r ithm of [19] can b e actually extended to all the other cases, also t ho se not maximally split, so that a ll sup ergravit y billiards are in fact completely in tegrable as claimed in the title. Besides demonstrating the in tegra bility w e will illustrate the main features of the g eneral in tegra l whic h reve al a v ery ric h and highly in t eresting geometrical structure of the parameter space. In t his con text it will emerge a c hallenging new concept. The time flows app earing as exact analytical solutions of sup ergr avit y billiards hav e a preferred orientation whic h is in trinsically determined in group theoretical terms. There emerges a similarit y b et w een the second law of thermo dynamics and t he prop erties of cosmological ev olutions just as there is suc h a similarit y in the case of black -hole dynamics. W e establish the fo llo wing principle Principle 2.1 << The asymptotic states of the c osmic bil liar d at p a s t and futur e infinity t = ±∞ ar e in one-to-one c orr esp ondenc e w ith the ele m ents w i of the duality algebr a Weyl gr oup W eyl( U ) . The Weyl gr oup, which for suitable cho ic e of N is a sub gr oup of the symm e tric gr oup S N admits a natur al or dering in terms of the minimal numb er ℓ T of r efle ctions with r esp e ct to simple r o ots α i ne c essary to r epr o duc e any c onsid er e d element w . The numb er ℓ T ( w ) , na m e d the he i g h t of w ∈ W eyl(U) , is the s a me as the numb er of tr ansp ositions of the c orr esp onding p ermutation when W eyl(U) is emb e dde d in the symmetric gr oup. Time flows go es a l w ays in the dir e ction of in c r e a sing ℓ T which, ther efor e, pl a ys the r ole of e n tr op y. >> 3 The p aint group and the Tit s Satak e pro je ction In [9 ] fir st and t hen more systematically in [18] it w as observ ed that the Tits-Satak e theory of non-compact cosets, whic h is a classical chapter of mo dern differential geometry , provides a natural frame to discuss the structure of the U / H cosets a pp earing in sup ergravit y with particular reference to their ro le in billiard dynamics. In [9] a new concept w as introduced, that of pain t group , whic h pla ys a fundamen tal role in classifying the relev ant U / H mani- folds and grouping them in to univers alit y classes with resp ect to the Tits Satake pro jection. The systematics of these univ ersalit y classes was dev elop ed in [18]. In the presen t pap er w e will clarify and illustrate by means of explicit examples the meaning of these unive rsalit y classes sho wing that the essen tial features of billiard dynamics are just a pro p ert y of the class, indep enden tly from t he c hoice of the represen ta tiv e, namely indep enden tly f rom the c hoice of t he paint gro up. In particular the W eyl gr o up and the asymptotic states are common to the whole class. On the other hand the notion of the pain t group enters in the precise definition of the parameter space for the general in tegral. Let us therefore recall the essen tial notions relev an t to our subsequen t discussion. 4 3.1 The solv able algebra F ollo wing the discussion of [9] let us recall that in the case the scalar manifo ld o f sup ergravit y is a non maximal ly non-c omp act manifold U / H the Lie alg ebra U of the n umerato r group is some appropriate real form U = U R (3.1) of a complex L ie algebra U ( C ) of rank r = rank( U ). The Lie algebra H o f the denominator H is the ma ximal compact subalgebra H ⊂ U R , whic h has t ypically ra nk r c < r . Denoting , as usual, b y K the orthogo nal complemen t of H in U R U R = H ⊕ K (3.2) and defining as non-compact rank , or rank of the c oset U / H, the dimension of the noncompact Cartan subalgebra r nc = rank (U / H) ≡ dim H nc ; H nc ≡ CSA U ( C ) \ K , (3.3) w e obtain that r nc < r . The manifold U R / H is alwa ys metrically equiv alent to a solv a ble group manifo ld M S ol v ≡ exp[ S ol v (U R / H)] although the form of t he solv able Lie algebra S ol v (U R / H), whose structure constan ts define the Nomizu conne ction, is more complicated when r nc < r than in the maximal ly split c ase r nc = r . F or the details on the construction of the solv able Lie algebra w e refer to the literature [11, 17]. The important thing in our presen t con text is that it exists. F urthermore, using a general theorem pro v en in suc h textb o oks like [20] w e kno w that eve ry linear represen tation of a solv able Lie algebra can be written in a basis where all of its elemen t s are given b y upp er triangular matrices. Hence f or an y of the U / H cosets of sup ergravit y we can c ho ose a coset represen tativ e L ( φ ) giv en b y the matrix exp onen tial of an upp er triangular matrix. This is the so named solv able parametrization o f the coset manifold whic h plays a fundamen ta l role in our subsequen t discussion of the general in tegral. 3.2 The pain t group and its Lie algebra Naming M = U / H the cons idered coset manifold and S o lv M ⊂ U the corres p onding solv able algebra, there exists a c omp ac t algebr a G pain t whic h acts as an algebra of outer automor- phisms ( i.e. outer deriv ativ es) of the solv able algebra S ol v M Aut [ S ol v M ] = { X ∈ U | ∀ Ψ ∈ S o lv M : [ X , Ψ ] ∈ S ol v M } . (3.4) By its o wn definition the alg ebra Aut [ S olv M ] contains S ol v M as an ideal. Hence w e can define the algebra of external automo r phisms as the quotien t Aut Ext [ S ol v M ] ≡ Aut [ S ol v M ] S ol v M , (3.5) and w e iden tify G pain t as the maximal compact subalgebra of Aut Ext [ S ol v M ]. Actually we immediately see that G pain t = Aut Ext [ S ol v M ] . (3.6) 5 Indeed, as a consequence of its o wn definition the algebra Aut Ext [ S ol v M ] is comp osed of isometries whic h b elong to the stabilizer subalgebra H of an y p oint of the manifold, s ince S ol v M acts transitiv ely . In v irtue of the Riemannian structure o f M we ha v e H ⊂ so ( n ) where n = dim ( S ol v M ) and hence also Aut Ext [ S ol v M ] ⊂ so ( n ) is a compact Lie algebra. The pain t group is no w defined b y exp onen tiatio n of the paint alg ebra G pain t ≡ exp [ G pain t ] . (3.7) The notion of maximally split a lg ebras can b e formulated in terms of the pain t algebra by stating that U = maximally split ⇔ Aut Ext  S ol v U / H  = ∅ . (3.8) Namely U is maximally split if and only if the paint group is just the trivial iden tity group. 3.3 The subp ain t group and the Tits Satak e subalgebra Making a long story short, once the pain t algebra has b een defined, the solv able Lie algebra falls in to a linear represen tatio n of G pain t and one can define its little gr oup, generated by the stability subalgebra of a generic eleme n t X ∈ S ol v M . In other words, view ed as a represen tation of G pain t , under the subalgebra G subpain t ⊂ G pain t (3.9) the solv able Lie alg ebra decomp oses into a singlet subalgebra S ol v T S plus a bunch of non trivial ir r educible represen tatio ns of G subpain t . W e name suc h a Lie subalgebra the subpain t algebra . Then the Tits Sata k e subalgebra of the orig inal algebra U is defined a s t he set of all elemen ts whic h are in v ar ia n t with resp ect to G subpain t : X ∈ U T S ⊂ U ⇔ ∀ Ψ ∈ G subpain t : [ X , Ψ] = 0 . (3.10) By construction t he Tits Satak e subalgebra U TS is maximally split and the Tits Satak e pro jection is defined as the following mapping of coset manifolds: Π TS : U H → U TS H TS . (3.11) In terms of ro o t systems the Tits Satak e pro jection has a na tural and simple intepretation. The ro ot system ∆ U of the origina l a lgebra is comp osed by a set o f v ectors in r -dimension where r is the rank of U . This system of ve ctors can b e pro jected on to the r nc -dimensional subspace dual to the non-compact Cartan subalgebra. Somewhat surprisingly , with just one exception, the pro jected set of v ectors is a new ro o t system in rank r nc , which we name ∆ TS . Indeed the corresp onding Lie a lg ebra is precisely the Tits Satak e subalgebra U TS ⊂ U of the original algebra. 4 T riang ular em b e dding in SL(N , R ) / SO(N) and in tegra- bilit y As a consequence of all the algebraic structures we hav e describ ed w e can conclude with the follo wing statemen t. 6 Statemen t 4.1 << L et N b e the r e al di m ension of the fundamental r epr esentation of U . Then ther e is a c an o nic al em b e dding U ֒ → sl ( N , R ) , U ⊃ H ֒ → so ( N ) ⊂ sl ( N , R ) . (4.1) This emb e dding is determine d by the choic e of the b asis wh er e S ol v (U / H) is made by upp er triangular matric es. I n the sa me b asis the elements o f K ar e symmetric matric es while those of H ar e antisymmetric ones. >> The em b edding (4.1) defines also a canonical embedding of the relev an t W eyl group W ey l ( U ) of U in to that of sl ( N , R ) namely in to the symmetric group S N . The existence of (4.1 ) is the key -p oint in order to extend the in tegration algo r ithm of sup ergra vit y billiards presen ted in [19 ] from the case of maximally-split cosets to the generic case. Indeed tha t algorithm is defined fo r SL(N , R ) / SO(N) and it has the prop ert y that if initial data are defined in a submanifold U / H where U ⊂ SL(N , R ) and H ⊂ SO( N), then the en tire time flo w o ccurs in the same submanifold. Hence the embedding (4.1) suffices to define explicit in tegration for mulae f or all sup ergra vit y billiards. Let us review the steps of the pro cedure. 1. First one defines a coset represen tativ e fo r U / H in the solv able pa r ametrization as follo ws: L ( φ ) = I =1 Y I = m exp [ ϕ I E α I ] exp  h i H i  (4.2) where the ro ots p erta ining to the solv a ble Lie algebra ar e or dered in ascending order of heigh t ( α I ≤ α J if I < J ), H i denote t he non compact Cartan generators and the product of matr ix exp onentials app earing in (4.2) go es from the highest on the left, to lo wes t r o ot on the righ t. In this w a y the para meters { φ } ≡ { ϕ I , h i } hav e a precise a nd uniquely defined corresp ondence with the fields of sup ergrav it y b y means of dimensional ox idation [7, 8]. 2. Restricting all the fields φ of sup ergr avit y to pure t ime dep endence φ = φ ( t ), the coset represen tativ e b ecomes also a function of time L ( φ ( t )) = L ( t ) and w e define the Lax op erator L ( t ) and the connection W ( t ) as follow s: L ( t ) = X i T r  L − 1 d dt L K i  K i , W ( t ) = X ℓ T r  L − 1 d dt L H ℓ  H ℓ (4.3) where K i and H ℓ denote an orthonormal basis of generators for K and H , resp ectiv ely . 3. With these definitions the field equations of sup ergra vit y , whic h are just the geo desic equations for the manifold U / H in the solv able parametrization, reduce to the single matrix v alued Lax equation [19] d dt L = [ W , L ] . (4.4) 7 4. If we are able to w rite the g eneral in tegr a l of the Lax equation, dep ending on p = dim(U / H) in tegration constan ts, then comparison of the definition of t he Lax op erator (4.3,4.2) with its explicit form in the integration reduces the differen tia l equations of sup ergra vit y to quadratures d dt φ ( t ) = F ( t ) = kno wn function of time. (4.5) 4.1 The in tegration algorithm for the Lax Equation Let us assume that w e ha v e explicitly constructed the em b edding (4.1). In this case, in the decomp osition U = K ⊕ H (4.6) of the relev ant Lie a lgebra U , the matrices represen ting the elemen ts of K are a ll symmetric while those represen ting the elemen ts of H are all an tisymmetric as w e ha ve already p o in ted out. F urthermore the matrices represen ting the solv able Lie a lgebra S ol v (U / H) are all upp er triangular. These are the necessary a nd sufficien t c onditions to a pply to t he relev an t Lax equation (4.4) the in tegra tion algorithm originally described in [21] and review ed in [19]. Th e k ey p oin t is that the connection W ( t ) app earing in eq.(4.4) is related to the Lax op erator b y means of an algebraic pro jection op era t o r as follo ws: W = Π( L ) := L > 0 − L < 0 , (4.7) L > 0 ( < 0) denoting the strictly upp er (low er) triangular part o f the N × N matr ix L . The relation (4.7) is nothing else but the statement that the coset represen tat iv e L ( φ ) from whic h the Lax op erator is extracted is tak en in the solv able parametrization. This es tablished, we can pro ceed to a pply the in tegration algorithm. Actually this is nothing else but an instance of the in vers e scattering metho d. Indee d equation (4.4 ) repre- sen ts the compatibility condition f o r the following linear system exhibiting the iso-sp ectral prop ert y of L : L Ψ = ΨΛ , d dt Ψ = P Ψ (4.8) where Ψ( t ) is the eigenmatrix, namely the matrix whose i -th row is the eigen v ector ϕ ( t, λ i ) corresp onding to the eigenv alue λ i of the Lax op erator L ( t ) at time t and Λ is the diagonal matrix of eigen v alues, whic h are constan t throughout the whole time flo w Ψ = [ ϕ ( λ 1 ) , . . . , ϕ ( λ n )] ≡ [ ϕ i ( λ j )] 1 ≤ i,j ≤ n , Ψ − 1 =  ψ ( λ 1 ) , . . . , ψ ( λ n )] T ≡ [ ψ j ( λ i )  1 ≤ i,j ≤ n , Λ = diag ( λ 1 , . . . , λ n ) . (4.9) The solution of (4.8) for the Lax op erator is giv en b y the following explicit fo rm o f the mat r ix elemen ts: [ L ( t )] ij = n X k =1 λ k ϕ i ( λ k , t ) ψ j ( λ k , t ) . (4.10) 8 The eigen ve ctors o f the Lax op erat o r at eac h instan t of time, whic h define the eigenmatrix Ψ( t ), and the columns of its in vers e Ψ − 1 ( t ), are expressed in closed form in terms of the initial data at some conv en tional instan t of time, say at t = 0. Explicitly w e ha v e ϕ i ( λ j , t ) = e − λ j t p D i ( t ) D i − 1 ( t ) Det     c 11 . . . c 1 ,i − 1 ϕ 0 1 ( λ j ) . . . . . . . . . . . . c i 1 . . . c i,i − 1 ϕ 0 i ( λ j )     , ψ j ( λ i , t ) = e − λ i t p D j ( t ) D j − 1 ( t ) Det        c 11 . . . c 1 ,j . . . . . . . . . c j − 1 , 1 . . . c j − 1 ,j ψ 0 1 ( λ i ) . . . ψ 0 j ( λ i )        (4.11) where the time dep enden t matrix c ij ( t ) is defined b elow c ij ( t ) = N X k =1 e − 2 λ k t ϕ 0 i ( λ k ) ψ 0 j ( λ k ) (4.12) and ϕ 0 i ( λ k ) := ϕ i ( λ k , 0 ) , ψ 0 i ( λ k ) := ψ i ( λ k , 0 ) (4.13) are the eigenv ectors and t heir adjo ints calculated at t = 0. These constan t v ectors as we ll as eigen v alues λ k constitute the initial data of the problem and provide the in tegration constan ts. Finally D k ( t ) denotes the determinan t of the k × k matrix with en tr ies c ij ( t ) D k ( t ) = D et   c ij ( t )  1 ≤ i,j ≤ k  . (4.14) Note that c ij (0) = δ ij and D k (0) = 1. 5 Prop erties of the gener al in tegral and the paramete r space The algorithm w e hav e describ ed in the previous section realizes a map I K : L 0 7→ L ( t, L 0 ) (5.1) whic h, starting from the initial data, i.e. the Lax op erator L (0) = L 0 ∈ K at some con ven tional time t = 0, pro duces a flow, namely a map of the infinite time line into the subspace K ⊂ U L ( t, L 0 ) : R |{z} −∞ ≤ t ≤ + ∞ 7→ K . (5.2) It is o f the outmost in terest to en umerate the prop erties of the maps (5.1,5.2). A first set o f four fundamen tal prop erties are listed b elo w: 9 1. The flow L ( t, L 0 ) is iso-sp ectral. This means the follo wing. The Lax op erator is a symmetric matr ix and therefore can b e diagonalized at ev ery instan t of time. Calling λ 1 . . . λ N the set of its N eigen v alues, w e ha v e that this set is time–indep enden t, namely the nume rical v alues of the eigenv alues remain the same throughout the en t ir e mot ion. 2. If t he Lax op erator L ( t ) is diagona l at an y finite time t 6 = ±∞ , then it is actually constan t L ( t ) = L 0 3. The asymptotic limits of the Lax op erat o r for t 7→ ±∞ are diagonal ma t rices L ± ∞ . 4. If L 0 ∈ K U b elongs to the symmetric part of a prop er Lie subalgebra U ⊂ sl ( N , R ), then the en tire motion remains in that subalgebra, namely ∀ t , L ( t ) ∈ K U . Relying on this first set of prop erties w e can refine o ur form ulation of the initial conditions and of the asymptotic limits in terms of the generalized W eyl group and of its Tits Satake pro jection. This leads to state further prop erties of the map (5.1) whic h are ev en more striking. 5.1 Discussion of the generalized W eyl group Diagonal matrices are just elemen ts of the non-compact Cartan subalgebra C ⊂ K ⊂ U . The Lax op erator at t = 0 can b e diagonalized by means of an o r t hogonal matrix O ∈ SO(N) whic h actually lies in the subgroup H ⊂ SO(N). Hence b y writing L 0 = O T C 0 O (5.3) initial data can b e given as a pair C 0 ∈ CSA \ K ; O ∈ H . (5.4) Let us now in tro duce the notion of generalized W eyl gr oup W ( U ). T o understand its def- inition let us review the definition of t he standard W eyl group. This latter is an intrinsic attribute of a complex Lie algebra. F or a complex Lie algebra U C , the W eyl group W eyl( U C ) is the finite group generated b y the reflections σ α with resp ect to all the ro ot s α . Actually as generators of W eyl( U ) it suffices to consider the reflections with resp ect to the simple ro ots σ α i . It turns out tha t if we consider the maximally split real section U split of the complex Lie algebra U C then the W eyl group W eyl( U C ) is realized as a subgroup of the maximal compact s ubgroup H split ⊂ U split . This isomorphism is realized as fo llows. Consider the in teger v alued elemen ts of H defined b elo w H ∋ γ α ≡ exp h π 2  E α − E − α  i , α > 0 (5.5) and tak e them as generators. These generators pro duce a finite subgroup W ( U ) whic h w e name gen e r al i z e d Weyl gr oup . It con tains a normal subgroup N( U ) ⊂ W ( U ) whose adjoin t action on an y Cartan Lie algebra elemen t is just the iden tit y . The factor group W ( U ) / N( U ) ∼ W eyl( U C ) is isomorphic to the abstract W eyl gr o up of the complex Lie algebra. Imitating suc h a construction also in the no n maximally split cases w e can in tro duce the follo wing 10 Definition 5.1 << L et U b e a not ne c essarily maximal ly split r e al se ction of the c o m plex Lie algebr a U C and H ⊂ U its maximal c omp ac t sub a l g ebr a. L et  α [ K ]  b e the set o f p ositive r o ots which ar e not in the kernel of the Tits Satake p r oj e c tion and whic h ther efor e p articip ate in the c onstruction of the s o lvable Lie algebr a of U / R . The gener ali z e d Weyl gr oup W ( U ) is the finite sub gr oup of H ge ner ate d by the fol lowing gener ators: H ∋ γ α [ K ] ≡ exp h π 2  E α [ K ] − E − α [ K ]  i , α [ K ] > 0 (5.6) whose numb er is d im (U / H) − r ank (U / H) . >> As w e already noted, the generalized W eyl gro up is ty pically big g er and has more elemen ts than the ordinary W eyl group. By construction the adjo in t a ction of the generalized W eyl group maps the non- compact Cartan subalgebra into itself ∀ O w ∈ W( U ) and ∀ C ∈ CSA \ K : O T w C O w ∈ CSA \ K . (5.7) This can b e ve rified b y means of the same calculation whic h shows that the o r dinary W eyl group, as defined in eq.(5.5), maps the Cartan subalgebra into itself for t he maximally split case. This observ atio n sho ws that giving the initia l data as w e did in eq.(5.4) actually corre- sp onds to an ov er-coun ting. Indeed the g eneralized W eyl group should b e mo dded out since it amounts to a redefinition of t he Cartan subalgebra data C 0 . So w e are led to guess that for eac h choice of the eigen v alues of the Lax op erat o r, namely at fix ed C 0 , the parameter space of the Lax equation is P = H / W ( U ). This how ev er is not ye t the complete t r ut h. Indeed there is also a con tin uous group, whose adjoin t action o n the non-compact Cartan subalgebra is the identit y map. This is the paint group G pain t . Hence the true parameter space of the Lax equation is the orbifold with respect to the generalized W eyl group, no t of a group, rather of a compact coset manifold. Indeed w e can write P = H G pain t / W ( U ) . (5.8) F urthermore w e can consider a normal subgroup N W ( U ) ⊂ W ( U ) of the generalized W eyl group defined by the following condition: γ ∈ N W ( U ) ⊂ W ( U ) iff ∀ C 0 ∈ CSA \ K γ T C 0 γ = C 0 (5.9) and w e can state the prop osition whic h is true f o r all non-compact cosets U / H: Statemen t 5.1 << The factor gr oup of the gener alize d Weyl gr oup with r esp e ct to its nor- mal sub gr oup stabilizing al l elements of the non-c omp act Ca rtan sub alge b r a i s j ust isomorphic to the or dinary Weyl gr oup of the Tits Sa take sub algebr a: W ( U ) N W ( U ) ≃ W eyl ( U T S ) . (5.10) >> 11 This sho ws that the only relev an t W eyl group is just the W eyl group of the Tits-Satak e subalgebra U TS . In view of the iso-sp ectral prop ert y a nd of the asymptotic prop ert y o f the Lax opera- tor whic h b ecomes diago na l at t = ±∞ w e conclude that, once C 0 is chose n, the av ailable end-p oin ts of the flow s at t he remote past and a t the remote future are in one-to- one corre- sp ondence with the elemen ts of the W eyl group W eyl ( U T S ). Indeed diagonal matrix means an elemen t of the non- compact Cartan subalgebra and, since the eigenv alues are n umerically fixed by the original choice of C 0 , the only thing whic h can happ en is a p erm uta tion. The a v aila ble p ermutations are on the other hand dictated b y the em b edding of the W eyl group in to the symmetric group: W eyl ( U TS ) ֒ → S N ≃ W eyl ( A N − 1 ) (5.11) whic h is induced b y the embedding (4.1) of the Lie algebra U into sl (N , R ) . The latter, as w e already stressed, fo llo ws by the choice of the upp er tria ngular basis for the solv able Lie algebra in the fundamen t a l represen ta t ion of U . 5.2 The arro w of time, trapp ed and critical surfaces In view of the ab ov e discussion w e conclude that the integration algo rithm (5.1) realizes a map of the follow ing ty p e: T K : H / G pain t W ( U ) = ⇒ W eyl ( U TS ) − ⊗ W eyl ( U TS ) + (5.12) where ∓ refer to the choice of a W eyl group elemen t at ∓∞ realized b y the asymptotic limits of the Lax op erator. It is of the outmost in terest to explore the general prop erties of the map T K . Let − → w I , ( I = 1 , . . . , N) b e the w eights of U in its fundamen tal N − dimensional repre- sen tation R N and let − → h =        h 1 , .., h r nc | {z } − → h T S , 0 , 0 , . . . , 0 | {z } r − r nc        (5.13) b e the r - v ector of parameters iden tifying the C 0 elemen t in the non compact Cartan subal- gebra CSA \ K ∋ C 0 = r nc X i =1 h i H i . (5.14) The r − r nc zeros in eq.(5.13) cor r esp ond to the statemen t that all comp onen ts of C 0 in the compact directions of the Cartan subalgebra v anish. The sub-ve ctor − → h T S is the only non v anishing one and it is the same as we would ha v e in the Tits Satake pro jected case. With these not a tions the N eigen v alues of the Lax op erator λ 1 , λ 2 , . . . , λ N are represen ted a s follo ws: λ I = − → w I ( C 0 ) = − → w I · − → h . (5.15) 12 Consider no w the branc hing o f the fundamen ta l represen tat io n of U with resp ect t o the Tits Satak e subalgebra times the G subpain t algebra: R N U TS × G subpaint = ⇒ ( R N TS , 1 ) ⊕ ( p , q ) . (5.16) By definition R N TS is the fundamental represen tation of the Tits Satak e subalgebra of dimen- sion N TS < N whic h is a singlet under the subpain t alg ebra G subpain t , while the remaining represen tation ( p , q ) is non trivial both with respect to the Tits-Satak e a nd with resp ect to the subpaint Lie algebra. Obviously w e ha v e p × q = N − N TS . Corresp ondingly the eigen v alues of the Lax op erator organize in the w a y w e are go ing to describ e. Let w i −| T S < 0 ( i = 1 , . . . , m ) b e the negativ e w eights of the represen ta tion R N TS , let w 0 T S = 0 b e the n ull- weigh t of the same represen tation (if it exists) and let w i + | T S > 0 ( i = 1 , . . . , m ) b e the p ositiv e w eights . If there is a n ull- weigh t w e hav e N TS = 2 m + 1 , o t herwise N TS = 2 m . A con ven tional order for the N eigen v alues is giv en by the following vec tor : − → λ [1] =                            λ 1 = − → w 1 −| T S · − → h T S λ 2 = − → w 2 −| T S · − → h T S . . . . . . . . . λ m = − → w m −| T S · − → h T S λ m +1 = 0 λ m +2 = 0 . . . . . . . . . λ m + pq +1 = 0 λ m + pq +2 = − → w 1 + | T S · − → h T S λ m + pq +3 = − → w 2 + | T S · − → h T S . . . . . . . . . λ N = − → w m + | T S · − → h T S                            (5.17) where the w eights are org a nized fr om the low est t o the highest. The v ector − → λ [1] corresp onds to the dia gonal en tries o f the matrix C 0 defined in eq.(5.4 ). All the other p ossible orders of the same eigen v alues are obtained fro m the actio n o f the W eyl group W eyl ( U TS ) on the w eights of R N TS . By construction, suc h an action p erm utes the p ositions of the non-v anishing eigen v alues while a ll the zeros sta y at their place. In this wa y starting f rom − → λ [1] w e obtain n = | W eyl ( U TS ) | suc h vec tors − → λ [ x ] in o ne-to-one corresp ondence with the elemen ts o f the Tits Satak e W eyl group. Sc hematically , naming Ω x the elemen ts of W eyl ( U TS ), w e obtain − → λ [ x ] = Ω x − → λ [1] (5.18) with the understanding that Ω 1 is the iden tity elemen t of the W eyl group. Eac h elemen t Ω x is represen t ed by a p erm utation o f the non v anishing eigenv alues, hence by an elemen t P (Ω x ) ∈ S N TS ⊂ S N . Among the v ectors − → λ [ x ] there will b e one − → λ [ min ] where the eigen v alues are organized in decreasing o rder λ 1 [ min ] ≥ λ 2 [ min ] ≥ . . . ≥ λ N − 1 [ min ] ≥ λ N [ min ] (5.19) 13 and there will b e another one − → λ [ max ] where the eigen v alues are instead organized in increasing order λ 1 [ max ] ≤ λ 2 [ max ] ≤ . . . ≤ λ N − 1 [ max ] ≤ λ N [ max ] . (5.20) Name Ω min/max the c orresp onding W eyl e lemen ts. I t follo ws that equation (5.18) can b e rewritten as − → λ [ x ] = Ω x Ω − 1 min − → λ [ min ] . (5.21) The symmetric gro up a dmits a partial ordering of its elemen ts given by the num b er ℓ T ( P ) of elemen ta ry transp ositions necessary to obtain a giv en p erm uta tion P starting from the fundamen tal one P 0 . The e m b edding of the W eyl group in to the s ymmetric gro up allo ws to transfer this partial ordering to the W eyl group as we ll. W e define the length of a W eyl elemen t as follo ws. T aking the p erm utation of λ [ min ] as fundamen ta l w e set ∀ Ω x ∈ W eyl ( U T S ) : ℓ T (Ω x ) ≡ ℓ T  P  Ω x Ω − 1 min  . (5.22) With this definition the W eyl elemen t Ω min has length ℓ T = 0 while the W eyl eleme n t Ω max has the maxim al length ℓ T = 1 2 (N TS − 1) N TS and an elemen t Ω x is higher than an elemen t Ω y if ℓ T (Ω x ) > ℓ T (Ω y ). W e can observ e that the partial ordering induced b y the immersion in the symmetric gro up is, up to some r ear r angemen t, t he intrinsic o rdering of the W eyl group pro vided by coun ting the minimal num b er of reflections with resp ect to simple ro ots necess ary to construct the considered elemen t. In our conte xt formalising the precise corresp ondence b et w een the t w o ordering pro cedures is not necessary since the relev an t one is that with resp ect to p ermutations and this is w ell and uniquely defined. Ha ving in tro duced the ab ov e ordering of W eyl elemen ts we can no w state the main and most significant prop ert y of the map (5.12) and of the T o da flo ws realized b y the integration algorithm (5.1). Principle 5.1 << In an y fl ow the arr ow of time is so dir e cte d that the state at t = −∞ c orr esp onds to the lowest ac c essible Weyl elem ent and the state at t = + ∞ c orr esp onds to the highest a c c essible one. >> T o make the principle 5.1 precise w e need to define the notion o f accessible W eyl elemen ts. This latter relies on a no ther remark a ble and striking prop erty of the T o da flows (5 .1) whic h w e ha v e n umerically v erified in a large v ariet y of cases nev er finding any coun terexample. Prop erty 5.1 << At any ins tant of time the L ax op er ator L ( t ) c an b e diagonalize d by a time dep enden t ortho gonal matrix O ( t ) ∈ H , writing L ( t ) = O T ( t ) C 0 O ( t ) . Consider now the N 2 − 1 minors of O ( t ) ob taine d by interse cting the first k c olumns with any set of k - r ow s, for k = 1 , . . . , N − 1 . If any of these minors vanishes a t any finite time t 6 = ±∞ then it i s c onstant and vanishes at al l times. >> The remark able conserv ation law stated in pro p ert y 5.1 whic h has the mathematical status of a conjecture implies that there are generic initial dat a , namely p oints of the parameter space P defined in eq.(5.8) and N 2 − 1 trapp ed hypersurfaces Σ i ⊂ P defined b y the v a nishing of one of the minors. The se trapp ed surfaces can also b e inters ected creating trapp ed sub-v arieties of equal or lo w er dimensions. If the initial data are generic, then principle 5.1 implies that the flow will necessarily b e fro m Ω min to Ω max . On the other hand 14 if w e are on a trapp ed surface w e hav e to see which elemen ts o f the generalized W eyl gro up W ( U ) b elong to that surface. As w e know fr om (5.10) eac h elemen t of W ( U ) is equiv alent to an elemen t of W eyl( U TS ) mo dulo an elemen t in the normal subgroup. Hence w e can in tro duce the follow ing definition: Definition 5.2 << A Weyl gr oup elem e nt γ ∈ W eyl ( U T S ) i s ac c essibl e to a tr app e d surfac e Σ i f ther e exists a r epr es e ntative µ ∈ W ( U ) of its e quiva l e n c e class in the gener alize d Weyl gr oup which b elongs to Σ . >> The set of W eyl elemen ts A Σ accessible to a trapp ed surface Σ inherits an o r dering from the general ordering of W eyl ( U T S ), namely w e can write: A Σ = { Ω x 1 , Ω x 2 , . . . , Ω x σ } (5.23) where σ is the cardinality of the set σ = card A Σ and Ω x i ≤ Ω x j if i < j . Then the flow is alw ays from the low est W eyl elemen t of A Σ at t = −∞ (i.e., from Ω x 1 ) to the highest one at t = ∞ (i.e., to Ω x σ ) as stated in principle 5 .1. If w e consider lo wer dimensional trapp ed surfa ces obtained b y in tersection, then the set of W eyl elemen ts accessible t o the intersec tion is simply giv en by the in tersection of the accessible sets: A Σ T Π = A Σ \ A Π (5.24) and the flo w is from the low est elemen t of A Σ T Π to its highest one. W e can now in tro duce a further Definition 5.3 << A tr app e d surfac e Σ is name d critic al if the s et o f Weyl elements A Σ ac c essible to the surfac e is a pr op er subse t of the Weyl gr oup, in other wor ds if card A Σ < | W eyl ( U T S ) | . (5.25) >> Note that the property o f criticalit y do es not necessarily imply a v ariance of a symptotics from the generic case Ω min → Ω max . Indeed, altho ugh the cardinalit y of A Σ is lo w er than the order of the W eyl group so that some eleme n ts are mis sing, y et it suffices that both Ω min ∈ A Σ and Ω max ∈ A Σ to guaran tee tha t the infinite past and infinite future states of the Univers e will b e the same as in the generic case. This observ ation motiv ates the further definition: Definition 5.4 << A tr a p p e d surfac e Σ is name d sup er-critic al if it is critic al and mor e- over either the maximal or the min imal Weyl ele m ents ar e mis s ing fr om A Σ : Ω min / ∈ A Σ and/or Ω max / ∈ A Σ . (5.26) >> This discussion shows that the truly relev a nt concept is that of tra pp ed surface whic h streams from the remark able conserv atio n la w given by prop ert y 5.1, criticalit y or sup er-criticalit y b eing, from the mathematical p oint of view, just accessory features alt ho ugh of the highest 15 ph ysical relev ance. If we just fo cus our attention o n the initial and final states the interme - diate concept of critical surface seems to b e unmotiv ated. The reason why it is useful is that critical surfaces as defined in 5.3 can b e computed in an in t r insic w ay ta king a dual p oin t of view. Rather than computing accessible W eyl elemen ts one can define forbidden ones b y using the em b edding of the W eyl g roup into the symmetric group men tioned in eq.(5.11). It follows from this that to eac h elemen t Ω x ∈ W eyl ( U T S ) w e can asso ciate a p ermutation P x ∈ S N , where N is the dimension of the fundamen tal represen tation of U and hence of the orthogonal matrix O we are discussin g. This fa ct allows to asso ciate to Ω x a set of N − 1 minors defined as follows : W eyl ( U T S ) ∋ Ω x →  min (1) x [ O ] , min (2) x [ O ] , . . . , min ( N − 1) x [ O ]  (5.27) min ( k ) x [ O ] = D et ( O [( P x (1) , . . . , P x ( k )) , (1 , . . . , k )]) (5.28) where M [( a 1 , . . . , a k ) , (1 , . . . , k )] denotes the minor of the matrix M obtained b y inte rsecting the k -ro ws a 1 , . . . , a k with the fir st k - columns. Using a dual view-p oin t it w as sho wn in [22] that in an y flow, in order for a p erm utation P of the eigen v alues to be a candidate for asymptotics (i.e. to b e a v ailable), its asso ciated minor s should all b e non zero. Hence relying on the em b edding of the W eyl group in to the s ymmetric group w e conclude that if a n y minor min ( k ) x [ O ] v anishes then Ω x is excluded from the set A Σ x | k of W eyl elemen ts accessible to t he surface Σ x | k defined by the v anishing o f the minor min ( k ) x [ O ]. W e can write: Σ x | k ≡  O ∈ H \ min ( k ) x [ O ] = 0  ⇒ Ω x / ∈ A Σ x | k . (5.29) The same minor is pro duced by more than one elemen t and hence iden tif ying all the Ω x for whic h min ( k ) x [ O ] = min ( k ) [0] w e immediately calculate the set of W eyl elemen ts excluded from A Σ 0 and b y complemen t w e also kno w the set A Σ 0 . If all the p ossible minors considered in prop erty (5.1) could b e pro duced b y W eyl ele- men ts, then what w e ha v e just describ ed would b e a quic k and efficien t w ay to obtain all trapp ed surfaces. In that case all t r app ed surfaces w o uld also b e critical. The fact is that not all minors can b e obtained from W eyl elemen ts and this implies that there are tr app e d surfac es which ar e n o t critic al . The reason of this difference is eviden t from our discussion. It is due to the fact tha t the W eyl g roup is in general only a prop er subgroup of the symmetric group S N . Therefore there a re p erm uta tions and therefore minors whic h do not corresp ond to any W eyl elemen t and fo r that reason they define non-critical tr a pp ed surfaces. In the case o f SL(N , R ) / SO(N) flo ws the W eyl gro up is just the full S N and a ll tra pp ed surfaces are critical. In conclusion t r a pp ed but not critical surfaces are critical surfaces of the em b edding SL(N) where the missing W eyl elemen ts are in the kernel of the pro jection S N 7→ W eyl( U ). This concludes the general presen tation of our results. By means of some case studies the next part illustrates the principles f orm ulat ed in this part. 16 P art I I Exampl es i l lustrating the princi pl es 6 Choice of the example s In this part w e mak e three case studies: 1 W e surv ey the flows on the simplest example SL( 3 , R ) / SO(3) of maximally split coset manifolds in order to demonstrate the relation b et we en t he W eyl group and the arrow of time by calculating explicitly all the critical surfaces whic h are t w o- dimensional and can b e visualized. The pa rameter space is a three dimensional cub e with some vertice s iden tified and can also b e visualized. 2 Next we mak e a detailed study of the flow s on Sp(4 , R ) / U(2). This manifold is the Tits Satak e pro jection of an en tire unive rsalit y class of manifolds, SO(2 , 2 + 2s) / SO(2) × SO(2 + 2s) whic h, on the other hand, is for r = 2 of the t yp e SO(r , r + 2s) / SO(r ) × SO(r + 2s). F or the lat t er w e mak e the general construction of the triangular basis of the solv a ble algebra illustrating the embedding: so ( r , r + 2 s ) ֒ → sl (2 r + 2 s ) (6.1) whic h is crucial in order to establish the in tegration algo rithm. In the case of sp (4) ∼ so (2 , 3), whic h is maximally split of rank tw o, the para meter space is a 4- dimensional h yp ercub e also with v ertices iden tified. 3 Finally w e study the case of SO(2 , 4) in comparison with that of SO(2 , 3) in o rder to illustrate the prop erties of the Tits Sata ke pro jection and the meaning of Tits Satak e univ ersality classes. 7 The s implest maximally split case: SL (3 , R ) / SO(3) In or der to illustrate the general ideas discusse d in the previous part and as a preparation to the study of more general cases, we b egin with a detailed analysis of the time flow s in the simplest instance of maximally split coset manifolds, namely for M 5 = SL(3 , R ) SO(3) . (7.1 ) The sl (3 , R ) Lie a lg ebra is the maximally split real section of the A 2 Lie algebra, enco ded in the D ynkin diag ram of fig.1. The ro ot system has rank t wo and it is comp o sed b y the six 17 Figure 1: The Dynkin diagr am of the A 2 Lie algebr a. A 2 ✐ α 1 ✐ α 2 v ectors displa ye d b elo w and pictured in fig.2: ∆ A 2 =                            α 1 =  √ 2 , 0  , α 2 =  − 1 √ 2 , q 3 2  , α 1 + α 2 =  1 √ 2 , q 3 2  , − α 1 =  − √ 2 , 0  , − α 2 =  1 √ 2 , − q 3 2  , − α 1 − α 2 =  − 1 √ 2 , − q 3 2  . (7.2) The simple ro ots are α 1 and α 2 . Α 1 Α 2 Α 1 + Α 2 Figure 2: The A 2 r o ot system. 18 A complete set of generators for the Lie algebra is prov ided by the followin g 3 × 3 mat r ices: H 1 =     1 √ 2 0 0 0 − 1 √ 2 0 0 0 0     ; H 2 =     − 1 √ 6 0 0 0 − 1 √ 6 0 0 0 q 2 3     , E α 1 =     0 1 0 0 0 0 0 0 0     ; E α 2 =     0 0 0 0 0 1 0 0 0     , E α 1 + α 2 =     0 0 1 0 0 0 0 0 0     , E − α 1 = ( E α 1 ) T ; E − α 2 = ( E α 2 ) T ; E − α 1 − α 2 =  E α 1 + α 2  T (7.3) where H 1 , 2 are the tw o Cartan generator s and E α are the step op erators asso ciated to the corresp onding ro ots. The solv able Lie alg ebra generating the coset (7.1) is comp o sed b y the follo wing fiv e op erators: S ol v  SL(3 , R ) SO(3)  = span  H 1 , H 2 , E α 1 , E α 2 , E α 1 + α 2  (7.4) and it is clearly represen ted by upp er t riangular matrices. The orthogonal decomp osition G = H ⊕ K (7.5) of the Lie algebra with resp ect to its maximal compact subalgebra: so (3) ≡ H ⊂ G ≡ sl (3 , R ) (7.6) is p erformed b y defining the following g enerators: K = span { K 1 , . . . , K 5 } ≡ n H 1 , H 2 , 1 √ 2  E α 1 + E − α 1  , 1 √ 2  E α 2 + E − α 2  , 1 √ 2  E α 1 + α 2 + E − α 1 − α 2  o , (7.7) H = span { J 1 , . . . , J 3 } ≡ n 1 √ 2  E α 1 − E − α 1  , 1 √ 2  E α 2 − E − α 2  , 1 √ 2  E α 1 + α 2 − E − α 1 − α 2  o . (7.8) By definition the Lax o p erator L ( t ) is a symmetric 3 × 3 matr ix whic h can b e decomp osed along the generators of the subspace K : L ( t ) = 5 X i =1 k i ( t ) K i (7.9) 19 and once the functions k i ( t ) ha v e b een determined, b y means of a n o xidatio n pro cedure which w as fully describ ed in [7], the fields of sup ergravit y can b e extracted b y simple quadratures. As w e explained in [19] and w e recalled in t he in tro duction, the initial data for the in tegra tion of the Lax equation are pro vided b y the c hoice of an elemen t of the Cartan subalgebra, namely by a diago nal matrix of the form: CSA ∋ C ( { λ 1 , λ 2 } ) =     λ 1 0 0 0 λ 2 0 0 0 − λ 1 − λ 2     (7.10) and b y a finite elemen t O ∈ SO( 3 ) of t he compact subgroup whic h to gether with C deter- mines the v alue of the Lax op erator at time t = 0: L 0 = O T C ( { λ 1 , λ 2 } ) O . (7.11) W e stressed that t he c hoice of the group elemen t O is actually defined mo dulo m ultiplicatio n on the left by an y elemen t w ∈ W ⊂ H ≃ exp H of the discrete W eyl subgroup. By definition the W eyl group maps the Cartan subalgebra in to itself, so that w e ha v e: ∀ w ∈ W ⊂ SO(3) : w T C ( { λ 1 , λ 2 } ) w = C ( w { λ 1 , λ 2 } ) ∈ CSA (7.12) where C ( w { λ 1 , λ 2 } ) denotes the diago nal matrix of type (7.10) with eigen v a lues λ ′ 1 , λ ′ 2 , − λ ′ 1 − λ ′ 2 obtained from the action of the W eyl group on the orig inal ones. So the actual mo duli space of the Lax equation is not H but the quotien t H / W . In the case of the Lie algebras A n the W eyl group is the symmetric group S n +1 and its action on the eigenv alues λ 1 , λ 2 , . . . , λ n , λ n +1 = − P n i =1 λ i is just t hat of p erm utations on these n + 1 -eigen v alues. F or A 2 w e hav e S 3 whose order is six. The six group elemen ts can b e en umerated in the following wa y: w 1 =     1 0 0 0 1 0 0 0 1     ; ( λ 1 , λ 2 , λ 3 ) 7→ ( λ 1 , λ 2 , λ 3 ) , w 2 =     0 1 0 1 0 0 0 0 1     ; ( λ 1 , λ 2 , λ 3 ) 7→ ( λ 2 , λ 1 , λ 3 ) , w 3 =     0 0 1 0 1 0 1 0 0     ; ( λ 1 , λ 2 , λ 3 ) 7→ ( λ 3 , λ 2 , λ 1 ) , (7.13) 20 ℓ T W eyl group of SL(3 , R ) 0 w 2 1 w 1 1 w 5 2 w 3 2 w 4 3 w 6 T a ble 1: P a rtial ordering of the W eyl group of SL(3 , R ) . w 4 =     1 0 0 0 0 1 0 1 0     ; ( λ 1 , λ 2 , λ 3 ) 7→ ( λ 1 , λ 3 , λ 2 ) , w 5 =     0 0 1 1 0 0 0 1 0     ; ( λ 1 , λ 2 , λ 3 ) 7→ ( λ 2 , λ 3 , λ 1 ) , w 6 =     0 1 0 0 0 1 1 0 0     ; ( λ 1 , λ 2 , λ 3 ) 7→ ( λ 3 , λ 1 , λ 2 ) . (7.14) Let us now c ho ose as eigen v alues λ 1 , λ 2 , λ 3 defined at t he cen tral time t = 0 the con v en tional set λ 1 = 1 ; λ 2 = 2 ; λ 3 = − 3 . (7.15) In this case the decreasing sorting to be exp ected at past infinit y is g iv en b y: 2 , 1 , − 3 whic h, according to eq.(7.14), correspo nds to the W eyl elemen t w 2 . Hence we can use w 2 as the fundamental p erm utation a nd rate a ll the other W eyl group elemen ts according to the n umber of transp ositions ℓ T needed t o bring their corresp onding p ermutation to tha t of w 2 . In this wa y w e o bta in a partial ordering of the W eyl gro up where the highest elemen t is the unique w 6 corresp onding to the increasing sorting of eigenv alues − 3 , 1 , 2. Indeed we ha ve the result show n in table 1 and if all the W eyl elemen ts a re accessible there is a unique predetermined pro cess: the state of the univers e at past infinit y is the Ka sner era w 2 , while the state of the Univ erse at future infinity is t he Kasner era w 6 . If not all the W eyl elemen ts are a ccessible, the n w e can ha v e different situations. In order to discuss them we hav e to study the structure of the orbifo ld SO ( 3 ) / W . T o para metrize the SO(3) compact g r o up w e in t r o duce three Euler angles θ i ( i = 1 , 2 , 3) 21 and w e write O ( θ i ) ≡ exp [ θ 1 J 1 ] exp [ θ 2 J 2 ] exp [ θ 3 J 3 ] =     O 11 O 12 O 13 O 21 O 22 O 23 O 31 O 32 O 33     (7.16) where: O 11 = cos ( θ 1 ) cos ( θ 3 ) − sin ( θ 1 ) sin ( θ 2 ) sin ( θ 3 ) ; O 12 = cos ( θ 2 ) sin ( θ 1 ) ; O 13 = cos ( θ 3 ) sin ( θ 1 ) sin ( θ 2 ) + cos ( θ 1 ) sin ( θ 3 ) ; O 21 = − cos ( θ 3 ) sin ( θ 1 ) − cos ( θ 1 ) sin ( θ 2 ) sin ( θ 3 ) ; O 22 = cos ( θ 1 ) cos ( θ 2 ) ; O 23 = cos ( θ 1 ) cos ( θ 3 ) sin ( θ 2 ) − sin ( θ 1 ) sin ( θ 3 ) ; O 31 = − cos ( θ 2 ) sin ( θ 3 ) ; O 32 = − sin ( θ 2 ) ; O 33 = cos ( θ 2 ) cos ( θ 3 ) . (7.17) In this parametrization, if we introduce the notation O xy z = O  x π 2 , y π 2 , z π 2  (7.18) w e obtain: O 000 =     1 0 0 0 1 0 0 0 1     ; O 100 =     0 1 0 − 1 0 0 0 0 1     , O 010 =     1 0 0 0 0 1 0 − 1 0     ; O 001 =     0 0 1 0 1 0 − 1 0 0     , (7.19) O 110 =     0 0 1 − 1 0 0 0 − 1 0     ; O 101 =     0 1 0 0 0 − 1 − 1 0 0     , O 011 =     0 0 1 − 1 0 0 0 − 1 0     ; O 111 =     − 1 0 0 0 0 − 1 0 − 1 0     . (7.20) 7.1 Discussion of the generalized W eyl group Let us no w construct the generalized W eyl gr o up, a ccording to the definition 5 .1. This case is maximally split and all ro ots participate in the construction. Hence as generator s we take 22 the three matrices generators = {O 100 , O 010 , O 001 } (7.21) as defined ab o v e in eq.(7.19). Closing the shell of pro ducts w e find a gro up W ( sl (3)) con- taining 2 4 elemen ts organized in 6 equiv alence classe s w ith resp ect to a normal subgroup N ( sl (3 )) ∼ Z 2 × Z 2 . The four elemen ts of N ( sl (3)) are the follo wing four matrices: n 1 =     1 0 0 0 1 0 0 0 1     ; n 2 =     1 0 0 0 − 1 0 0 0 − 1     , n 3 =     − 1 0 0 0 1 0 0 0 − 1     ; n 4 =     − 1 0 0 0 − 1 0 0 0 1     . (7.22) The factor group is isomorphic to the W eyl gro up o f sl (3) W ( sl ( 3 )) N ( sl (3 )) ∼ W eyl ( sl (3) ) ≡ S 3 (7.23) and a represen tative of the six equiv a lence classes is listed b elow w 1 ∼     1 0 0 0 1 0 0 0 1     N ( sl (3)) ; w 2 ∼     0 1 0 1 0 0 0 0 − 1     N ( sl (3 )) , w 3 ∼     0 0 1 0 1 0 − 1 0 0     N ( sl (3 )) ; w 4 ∼     1 0 0 0 0 1 0 − 1 0     N ( sl (3 )) , w 5 ∼     0 0 1 1 0 0 0 1 0     N ( sl (3)) ; w 6 ∼     0 1 0 0 0 1 1 0 0     N ( sl (3 )) . (7.24) Hence mo dulo the normal subgroup the eigh t matrices listed in eq.s(7.19,7.20) can b e iden- tified with the six elemen ts of the W eyl group in the fo llowing w ay : O 000 ∼ w 1 ; O 100 ∼ w 2 ; O 010 ∼ w 4 O 001 ∼ w 3 ; O 110 ∼ w 5 ; O 101 ∼ w 5 O 011 ∼ w 6 ; O 111 ∼ w 4 . (7.25) Let us now consid er the general form of the SO(3) matrix as g iv en in eq.(7.16) and the mo dding b y the generalized W eyl group. Precis ely , with our con ve n t io ns this means the follo wing 1 : ∀ γ ∈ W ( sl (3)) a nd ∀ O ∈ SO(3) : γ O ∼ O . (7.26) 1 Mo dding is done by left multiplication b ecause, if sitting on the left of O the generalized W eyl group element γ will act on the Carta n element C by conjugation γ T C γ (see eq.(7.1 1)). 23 In terms of the matrix en tries O ij the op eration (7.26) is quite simple, it just implies that all orthogonal matrices whic h differ by an arbitra ry p erm ut a tion of the ro ws accompanied by o v erall changes of signs row s b y ro ws a r e to b e iden tified. On the other hand transferring the m ultiplication b y γ on the theta angles is a highly non trivial a nd complicated op eratio n. In other w o rds the map θ i → f i γ ( θ ) (7.27) defined b y: O  f i γ ( θ )  = γ O ( θ ) (7.28) is quite in volv ed and not handy . This implies that display ing a fundamen tal cell in θ -space is not an easy task a nd do es not lead to an y illuminating picture. This is no serious problem, since it is just a co ordinate art if act. F ur t hermore precisely since w e are finally in terested in equiv alence classes with respect to the algebraic W eyl group, i.e. in sets o f 4-matrices of the form N( sl (3)) O (7.29) then it just suffices to iden t if y a minimal neigh b orho o d of R 3 in the op en chart of the group manifold SO(3) defined b y the Euler angle para meterization (7.16) suc h that it con tains at least one cop y of eac h W eyl group eleme n t w i ∈ W eyl( sl (3)). An example of suc h a minimal submanifold is prov ided by the cub e 0 ≤ θ i ≤ π 2 sho wn in fig.3 whose v ertices ar e just the r equired represen tativ es o f the W eyl group elemen ts. In all cases w e can fo cus our atten t io n on t he h yp ercub e in Euler angle spaces defined b y the v ertices whic h corresp ond to the nearest copies of a ll the W eyl group elemen ts. W e stress that these h yp ercub es are not fundamental cells for the equiv a lence classes H / W ( U ) but are just sufficien t for our purp oses, in particular in order to study the flow diagram pro duced b y links, namely flow s on one dimensional tra pp ed surfaces. 7.2 The flo w diagram and the critical surfaces for SL(3 , R ) W e can no w explore the b ehaviour of Lax equation on the ve rtices, the edges and the interior of the parameter space w e hav e describ ed in the previous section. V ert ices As w e kno w from the general prop erties o f the in tegral discussed in section 5 if the Lax op erator lies in the Cartan subalgebra at the initial p oint t = 0 , namely it is diago na l it will remain constan t all the time from −∞ to + ∞ . Hence on eac h v ertex of the cub e, whic h corresponds to a W eyl group elemen t, w e hav e constan t Lax op erators, corresp onding to as man y Kasner ep o chs . 24 1 2 4 3 5 6 5 4 Figure 3: A thr e e-dimensio nal cub e 0 ≤ θ i ≤ π 2 ( i = 1 , 2 , 3 ) whos e eight vertic es ar e identifie d with the six Weyl gr oup elements a s shown in the pictur e. The 12 e dges of the cub e r epr es e n t one p a r ameter submanifolds of SO(3 ) wher e just o n e angle varies whil e the other two ar e at fixe d v alues, either 0 or π / 2 . Edges It is inte resting to see what happ ens on the tw elv e edges of the cub e. Let us display the form of the matrix O on each of these edges. 1) (000) ↔ (100) O =     cos ( θ 1 ) sin ( θ 1 ) 0 − sin ( θ 1 ) cos ( θ 1 ) 0 0 0 1     , 2) (000) ↔ (010) O =     1 0 0 0 cos ( θ 2 ) sin ( θ 2 ) 0 − sin ( θ 2 ) cos ( θ 2 )     , 3) (000) ↔ (001) O =     cos ( θ 3 ) 0 sin ( θ 3 ) 0 1 0 − sin ( θ 3 ) 0 cos ( θ 3 )     , (7.30) 25 4) (100) ↔ (110) O =     0 cos ( θ 2 ) sin ( θ 2 ) − 1 0 0 0 − sin ( θ 2 ) cos ( θ 2 )     , 5) (100) ↔ (101) O =     0 1 0 − cos ( θ 3 ) 0 − sin ( θ 3 ) − sin ( θ 3 ) 0 cos ( θ 3 )     , 6) (010) ↔ (011) O =     cos ( θ 3 ) 0 sin ( θ 3 ) − sin ( θ 3 ) 0 cos ( θ 3 ) 0 − 1 0     , (7.31) 7) (010) ↔ (110) O =     cos ( θ 1 ) 0 sin ( θ 1 ) − sin ( θ 1 ) 0 cos ( θ 1 ) 0 − 1 0     , 8) (001) ↔ (101) O =     0 sin ( θ 1 ) cos ( θ 1 ) 0 cos ( θ 1 ) − sin ( θ 1 ) − 1 0 0     , 9) (001) ↔ (011) O =     0 0 1 − sin ( θ 2 ) cos ( θ 2 ) 0 − cos ( θ 2 ) − sin ( θ 2 ) 0     , (7.32) 10) (110) ↔ (111) O =     − sin ( θ 3 ) 0 cos ( θ 3 ) − cos ( θ 3 ) 0 − sin ( θ 3 ) 0 − 1 0     , 11) (011) ↔ (111) O =     − sin ( θ 1 ) 0 cos ( θ 1 ) − cos ( θ 1 ) 0 − sin ( θ 1 ) 0 − 1 0     , 12) (101) ↔ (111) O =     − sin ( θ 2 ) cos ( θ 2 ) 0 0 0 − 1 − cos ( θ 2 ) − sin ( θ 2 ) 0     . (7.33) On eac h link w e ha v e one of the three one-parameter subgroups resp ectiv ely generated by J 1 , 2 , 3 m ultiplied o n t he left or on the righ t b y a W eyl group elemen t. By means of a computer programme w e can then easily ev aluate the general inte gral on each of these links. F or instance on the link num b er 1 w e obtain: L ( t ) =     L 11 ( t ) L 12 ( t ) 0 L 12 ( t ) L 22 ( t ) 0 0 0 − λ 1 − λ 2     , 26 L 11 ( t ) = e 2 tλ 2 λ 1 cos 2 ( θ 1 ) + e 2 tλ 1 sin 2 ( θ 1 ) λ 2 e 2 tλ 2 cos 2 ( θ 1 ) + e 2 tλ 1 sin 2 ( θ 1 ) , L 22 ( t ) = e 2 tλ 2 λ 2 cos 2 ( θ 1 ) + e 2 tλ 1 sin 2 ( θ 1 ) λ 1 e 2 tλ 2 cos 2 ( θ 1 ) + e 2 tλ 1 sin 2 ( θ 1 ) , L 12 ( t ) = e t ( λ 1 + λ 2 ) sin (2 θ 1 ) ( λ 1 − λ 2 ) ( − e 2 tλ 1 + e 2 tλ 2 ) cos (2 θ 1 ) + e 2 tλ 1 + e 2 tλ 2 . (7.34) W e can a lso calcu late the a symptotic limits of the Lax op erat o r at ±∞ f o r each o f these flo ws. As it fo llo ws from the prop erties of the general integral, at r emotely early or at remotely late times the Lax op erato r is alwa ys diag onal and it s eigen v a lues are organized in one of the p ossible six w ays corresp onding to the six W eyl group elemen ts a cting on their reference order ( λ 1 , λ 2 , − λ 1 − λ 3 ). If w e a sso ciate an a rro w t o eac h of these tw elv e links and w e tak e into accoun t the identification of v ertices as display ed in eq.(7.25) w e obtain the flow diagram show n in fig.4. As it is clear fro m the quoted picture, the flow s o n the w 1 w 2 w 3 w 4 w 5 w 6 1 2 4 3 5 6 5 4 Figure 4: The oriente d diagr am of the SL(3 , R ) / SO(3) flows. Th e Lie algebr a sl(3 , R ) is the maximal ly s p lit r e al se ction of the c omplex Lie alg e br a A 2 . I ts Weyl gr oup is S 3 and h as six elements identifie d by their action on the e i g envalues λ 1 , λ 2 , λ 3 of the L ax op er ator. Si x ar e ther efor e the p ossib le asymptotic states o f the universe at ±∞ and e ach p ossible m otion is an oriente d flow fr om one lower Weyl element to another higher one. The lines of t he gr aph on the right r epr es e n t p ossible orien te d flows along one-dim ensional subma nifolds of the p ar am eter sp ac e lo c ate d on the e dge s of the c ub e defi ne d by r estricting the r ange of the thr e e Euler an gles { θ 1 , θ 2 , θ 3 } to the close d interval  0 , π 2  . On the vertic es o f the cub e we find SO ( 3) gr oup elements lying in the Weyl gr oup (mo dulo the c enter Z 3 2 ), just as shown in the thr e e-dimen s ional p i c tur e o n the right. In the two-dim ensional pictur e on the left, by cho osing a s fundamental eigenvalues λ 1 = 1 , λ 2 = 2 , λ 3 = − 3 the Weyl gr oup element w i ∈ W is identifie d by the p oint in the plane that has c o or dina tes e qual to the pr o j e ctions o f w i ( λ 1 , λ 2 , λ 3 ) a long an orthonormal b asis o f v e c tors sp anning the plan e ortho gonal to (1 , 1 , 1) . edges of the cub e relate s tates of the Univ erse where there is no complete sorting of the 27 eigen v alues at past and future infinities. Indeed if w e use as r eference set the eigenv alues of eq.(7.15) then complete sorting w ould require w 2 ∈ W at −∞ , corresp onding to the decreasing ordering 2 , 1 , − 3 and w 6 at + ∞ corresp onding to the increasing ordering − 3 , 1 , 2 as we already observ ed. As it is eviden t by insp ection, the matrices lo cated on the t w elv e edges defin e one-dimensional critical surfaces. It is a a fundamental property of the Lax equation that flow s touc hing up on a critical surfa ce are completely constrained on it. Hence flo ws touc hing one link just lie on that link at all instants of time and the asymptotic states corresp ond to the ve rtices lo cated at the endpo in ts of that link. The orien t a tion of the link is also decided a prio ri. P a st infinit y is the low er of the t wo end p oin t W eyl elemen ts while future infinit y is the higher one. This is just eviden t b y comparing the gr a ph in fig.(4) with the ordering of W eyl group elemen ts as displa y ed in table 1. Besides one-dimensional critical surfaces (the links) there are also tw o -dimensional o nes (the faces) and these are obtained by studying the minors of O . F aces Let us consider the orthogonal matrix O ∈ SO(3) and let us name and parameterize its en tries as in eq.s(7.16 ,7.17). Then t here are in general exactly 6 minors that corresp ond to the conditions in volv ed in the definition of trapp ed surfaces in Section 5.2. Since w e a re dealing with sl (3) all trapp ed surfaces are a lso critical. Three o f the relev ant minors are 1 × 1 minors and three of them are 2 × 2 minors. Imp osing their v anishing one obtains equations on the three parameters θ 1 , θ 2 , θ 3 whic h would define a s man y critical surfaces, namely six. Let us en umerate these candidate tr app ed and critical surfaces Σ 1 : O 1 , 1 = 0 = cos ( θ 1 ) cos ( θ 3 ) − sin ( θ 1 ) sin ( θ 2 ) sin ( θ 3 ) , Σ 2 : O 2 , 1 = 0 = − cos ( θ 3 ) sin ( θ 1 ) − cos ( θ 1 ) sin ( θ 2 ) sin ( θ 3 ) , Σ 3 : O 3 , 1 = 0 = − cos ( θ 2 ) sin ( θ 3 ) , Σ 4 : O 1 , 1 O 2 , 2 − O 1 , 2 O 2 , 1 = 0 = cos ( θ 2 ) cos ( θ 3 ) , Σ 5 : O 1 , 1 O 3 , 2 − O 1 , 2 O 3 , 1 = 0 = sin ( θ 1 ) sin ( θ 3 ) − cos ( θ 1 ) cos ( θ 3 ) sin ( θ 2 ) , Σ 6 : O 2 , 1 O 3 , 2 − O 2 , 2 O 3 , 1 = 0 = cos ( θ 3 ) sin ( θ 1 ) sin ( θ 2 ) + cos ( θ 1 ) sin ( θ 3 ) . (7.35) It is now fairly simple to verify t ha t, while the equations for Σ 1 , Σ 3 , Σ 4 , a nd Σ 5 can b e solv ed inside the cub e 0 ≤ θ i ≤ π 2 , all solutions o f the equations for Σ 2 and Σ 6 are lo cated outside this r ange. The exis ting inside the cube critical su rfaces are sho wn in fig.s 5, 6, 7. By means of a computer programme we can ev aluate the flow s on all these critical surfaces and w e find the follow ing results for the asymptotic v a lues of the La x op erator: Σ 1 : w 2 → w 6 (surface equation θ 1 = arccos  sin( θ 2 ) sin( θ 3 ) √ cos 2 ( θ 3 )+sin 2 ( θ 2 ) sin 2 ( θ 3 )  ) , Σ 3 : w 2 → w 4 (for θ 3 = 0) , Σ 4 : w 5 → w 6 (for θ 3 = π 2 ) , Σ 3 T Σ 4 : w 5 → w 4 (for θ 2 = π 2 ) , Σ 5 : w 2 → w 3 (surface equation θ 3 = arccos  sin( θ 1 ) √ sin 2 ( θ 1 )+cos 2 ( θ 1 ) sin 2 ( θ 2 )  ) . (7.36) 28 0 Π   2 Θ 1 Π   2 Θ 2 Π   2 Θ 3 0 Π Θ 1 Θ 2 1 2 4 3 5 6 5 4 Figure 5: The pi c tur e on the left shows the critic a l surfac es Σ 1 define d by the e quation O 1 , 1 = 0 imp ose d on the SO(3) g r oup elem ent. The pictur e on the right r eminds the r e ader of the Weyl gr oup e lements lo c ate d at the vertic es of the p ar am eter sp ac e. Th e vertic es b elonging the surfac e ar e (in in cr e asing o r de r) w 2 , w 5 , w 3 , w 6 so that the flow is w 2 7→ w 6 . The fourth case listed in eq.(7.36) needs a comment. When w e set θ 2 = π 2 it happ ens tha t b oth O 3 , 1 = 0 and O 1 , 1 O 2 , 2 − O 1 , 2 O 2 , 1 = 0. Hence this plaquette o f the h yp ercub e is actually the in tersection of t w o critical surfaces. Altogether the result displa ye d in eq.(7.36) could b e predicted a priori relying on the not ion of accessible ve rtices. Giv en t he equation of a critical surfa ce, the accessible v ertices are defined a s tho se W eyl elemen ts whic h hav e at least one represen tativ e satisfying the defining condition and therefore b elong t o the surface. Once the accessible set is defined, the flo w is easily singled out. It go es from the lo w est W eyl mem b er of the set to the highest one. This task is easily carried through in the presen t case. F or the six surfaces defined in equation ( 7 .35) t he corresp onding accessible sets a re rapidly calculated and w e find t he result display ed in table 2. Expunging surfaces Σ 2 and Σ 6 whic h fall outside the cub e, w e find that the av ailable flows on critical t w o dime nsional surfaces inside the cub e are just only fo ur, namely the follow ing ones: w 2 7→ w 6 , w 2 7→ w 4 , w 5 7→ w 6 , w 2 7→ w 3 . (7.37) The o nly non v anishing inte rsection of these surfaces is the a f ore-men tioned plaquette Σ 3 T Σ 4 . W e can easily calculate the in tersection of v ertices accessible to b oth Σ 3 and Σ 4 . W e find { w 2 , w 1 , w 5 , w 4 } \ { w 5 , w 3 , w 4 , w 6 } = { w 5 , w 4 } (7.38) 29 0 Π   2 Θ 1 Π   2 Θ 2 Π   2 Θ 3 0 Π Θ 1 Θ 2 1 2 4 3 5 6 5 4 Figure 6: The pic tur e on the left shows the union of the critic al s urfac e s Σ 3 and Σ 4 , r esp e c- tively de fine d by the e quations O 3 , 1 = 0 a n d O 1 , 1 O 2 , 2 − O 1 , 2 O 2 , 1 = 0 imp ose d on the minors of the SO(3) gr oup ele m ent. The pictur e on the right r em inds the r e ader of the Weyl gr oup elements lo c ate d at the vertic es of the p ar ameter sp ac e and show s the p oss ible oriente d flows on the critic al surfac es. The vertic es b elonging to Σ 3 ar e w 2 , w 1 , w 5 , w 4 and the flow on this surfac e is w 2 7→ w 4 . The vertic es b elongin g to Σ 4 ar e inste ad w 5 , w 3 , w 4 , w 6 and the flow on this surfac e is w 5 7→ w 6 . The plaquette θ 2 = π 2 is actual ly the in terse c tion Σ 3 T Σ 4 and on this surfac es the flow go es fr om the lowest to the h ighest of the elements in the set of the vertic es ac c essible to b o th surfac e s , namel y we have w 5 7→ w 4 . where all s ets are written in ascending order. It follows that on the surface Σ 3 T Σ 4 the orien ted flo w is w 5 7→ w 4 (7.39) as indeed it is ve rified by n umerical calculation on the computer. This concludes our discussion of the SL(3 , R ) whic h has b een instrumen tal to illustrate the in volv ed mathematical structures. W e hav e seen that the top ology of the pa r ameter space H / W ( U ) is indeed complicated and cannot b e easily display ed as an h yp ercub e. Y et it is completely defined b y the trapp ed h yp ersurfaces which admit a clear definition in terms of algebraic equations. These surfaces split the parameter space H / W ( U ) into con vex hulls whic h are separated f r om eac h other. Indeed the w alls are impenetrable according to T o da evolution. Moreo v er w e can relate the initial and final states of the flows to these critical surfaces defined b y the v anishing o f the relev ant minors in the orthogo nal matrix O . Our next section is dev o ted to another maximal split case of rank r = 2 whic h will corresp ond to an en tire Tits Satake univ ersalit y class of cases. 30 0 Π   2 Θ 1 Π   2 Θ 2 Π   2 Θ 3 0 Π Θ 1 Θ 2 1 2 4 3 5 6 5 4 Figure 7: The pi c tur e on the left shows the critic a l surfac es Σ 5 define d by the e quation O 1 , 1 O 3 , 2 − O 1 , 2 O 3 , 1 = 0 imp ose d on the SO(3) gr oup element. The ide n tific ation of the cub e vertic es with Weyl gr oup elements is shown on the right. Her e the ac c essible vertic es ar e w 2 , w 1 , w 5 , w 3 and the flow on this surfac e is w 2 7→ w 3 . 8 The maximally split case Sp(4 , R ) / U(2) As w e explained in the introduction our go al is the illustration of the Lax in tegra tion formula in the non-maximally split case SO(r , r + 2s) / SO(r ) × SO(r + 2s). The Tits Satake pro jection of these manifolds is prov ided b y the maximally split coset M T S r ≡ SO(r , r + 1) SO(r) × SO(r + 1) . (8.1) In the case of rank r = 2 w e hav e M T S 2 ≡ SO(2 , 3) SO(2) × SO(3) = Sp(4 , R ) U(2) (8.2) due to the acciden ta l isomorphism b et w een t he B 2 and C 2 Lie algebras whose Dynkin diagram is displa y ed in fig. 8. F or this reason w e can rely on either formu lation in terms of 4 × 4 symplectic matrices or 5 × 5 pseudo-orthogonal mat r ices to obtain the same result. In the symplectic sp (4) in terpretatio n, the C 2 ro ot system can b e realized b y the fo llowing eigh t t wo-dimensional v ectors: ∆ C 2 =  ± ǫ i ± ǫ j , ± ǫ i  (8.3) where ǫ i ( i = 1 , 2) de notes a basis o f orthonormal unit vec tors. In the pseud o-orthog o nal so (2 , 3) in terpretation of the same a lg ebra the B 2 ro ot system is instead realized b y the follo wing eigh t v ectors: ∆ B 2 =  ± ǫ i ± ǫ j , ± 2 ǫ i  . (8.4) The t w o ro ot systems are displa yed in fig. 9. 31 Surf. Equation Access ible V ertex Flo w Σ 1 O 1 , 1 = 0       0 w 2 1 w 5 2 w 3 3 w 6       w 2 7→ w 6 Σ 2 O 2 , 1 = 0       1 w 1 2 w 3 2 w 4 3 w 6       w 1 7→ w 6 Σ 3 O 3 , 1 = 0       0 w 2 1 w 1 1 w 5 2 w 4       w 2 7→ w 4 Σ 4 − O 1 , 2 O 2 , 1 + O 1 , 1 O 2 , 2 = 0       1 w 5 2 w 3 2 w 4 3 w 6       w 5 7→ w 6 Σ 5 − O 1 , 2 O 3 , 1 + O 1 , 1 O 3 , 2 = 0       0 w 2 1 w 1 1 w 5 2 w 3       w 2 7→ w 3 Σ 6 − O 2 , 2 O 3 , 1 + O 2 , 1 O 3 , 2 = 0       0 w 2 1 w 1 2 w 4 3 w 6       w 2 7→ w 6 T a ble 2: The accessible v ertices for each of the six t w o-dimensional critical surfaces in the case SL(3 , R ) / SO(3). 8.1 The W eyl group and the generalized W eyl group of sp (4 , R ) Abstractly the W eyl group W eyl ( C 2 ) of the Lie algebra sp (4 , R ) is giv en by ( Z 2 × Z 2 ) ⋉ S 2 and its eight elemen ts w i ∈ W eyl( C 2 ) can b e described by their action on the tw o Cartan 32 Figure 8: The Dynkin diagr am of the B 2 ∼ C 2 Lie alg e br a. C 2 ✐ α 1 ❅  ✐ α 2 B 2 ✐ α 1  ❅ ✐ α 2 Α 1 Α 2 Α 1 + Α 2 2 Α 1 + Α 2 Α 2 Α 1 2 Α 1 + Α 2 Α 1 + Α 2 Figure 9: The C 2 and B 2 r o ot systems. Th e y ar e r ela te d by the exchang e of the long with the short r o ots and v i c e versa. fields h 1 , h 2 w 1 : ( h 1 , h 2 ) → ( h 1 , h 2 ) , w 2 : ( h 1 , h 2 ) → ( − h 1 , − h 2 ) , w 3 : ( h 1 , h 2 ) → ( − h 1 , h 2 ) , w 4 : ( h 1 , h 2 ) → ( h 1 , − h 2 ) , w 5 : ( h 1 , h 2 ) → ( h 2 , h 1 ) , w 6 : ( h 1 , h 2 ) → ( h 2 , − h 1 ) , w 7 : ( h 1 , h 2 ) → ( − h 2 , h 1 ) , w 8 : ( h 1 , h 2 ) → ( − h 2 , − h 1 ) . (8.5) Just as we did in the previous case study w e can in tr o duce a partial ordering of the W eyl group elemen ts whic h will gov ern t he orientation o f all dynamical flows . The ke y p oin t is the em b edding W eyl ( sp (4) ) ֒ → S 4 ≃ W eyl ( sl (4)) (8.6) 33 of the W eyl group into the symmetric group S 4 induced b y the 4 × 4 represen tation of the Lie algebra sp (4) in whic h the solv able Lie algebra is made of upper triangular matrices. The explicit construction of suc h a represen tatio n is p erformed in the next section. F or the purp ose of the considered issue, namely disco vering the structure of the generalized W eyl gr o up a nd ordering of W eyl group elemen ts, we anticipate some results. The matrices corresp onding to Cartan subalgebra elemen ts are of the follo wing form: CSA ∋       h 1 0 0 0 0 h 2 0 0 0 0 − h 2 0 0 0 0 − h 1       . (8.7) In this wa y , the action of eac h W eyl group elemen t as defined in eq.(8.5) can b e rein terpreted as a particular p erm utation of the set ( h 1 , h 2 , − h 2 , − h 1 ) a nd this interpre tation provide s the em b edding (8.6). T o ma ke it precise let us derive the structure of the generalized W eyl group. F ollo wing the definition 5.1 w e introduce as generato r s t he op erator defined in eq.(8.24 ) for the follow ing four c hoices of the θ -a ngles: { θ 1 , θ 2 , θ 3 , θ 4 } =            π 2 0 0 0 0 π 2 0 0 0 0 π 2 0 0 0 0 π 2 (8.8) whic h just corresp onds to the 4 ro o t s o f sp (4). By closing the shell of pro ducts w e obtain a group with 32-elemen ts, W ( sp (4)). This gro up has a n order four normal subgroup N( sp (4 )) with the structure of Z 2 × Z 2 whose adjoint action on an y of the Cartan matrices (8.7) is the iden tity . Explic itly N( sp (4)) is made b y the following fo ur symple ctic matrices: N 1 =       − 1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1       ; N 2 ≃       − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 − 1       , (8.9) N 3 ≃       1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 1       ; N 4 ≃       1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1       . (8.10) As exp ected the factor group W ( sp (4)) / N( sp (4)) is isomorphic to t he W eyl g roup W eyl ( sp (4)) since the adjoin t a ction of each equiv alence class pro duces the same transformation on the eigen v alues h 1 , h 2 as the abstract W eyl elemen ts listed in eq.(8.5 ). Explicitly the 8- 34 equiv alence classes of 4-elemen ts eac h are displa y ed b elow Ω 1 ≃       1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1       N( sp (4)) ; Ω 2 ≃       0 0 0 1 0 0 1 0 0 − 1 0 0 − 1 0 0 0       N( sp (4)) , (8.11) Ω 3 ≃       0 0 0 1 0 1 0 0 0 0 1 0 − 1 0 0 0       N( sp (4)) ; Ω 4 ≃       1 0 0 0 0 0 1 0 0 − 1 0 0 0 0 0 1       N( sp (4)) , (8.12) Ω 5 ≃       0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0       N( sp (4)) ; Ω 6 ≃       0 0 1 0 1 0 0 0 0 0 0 1 0 − 1 0 0       N( sp (4)) , (8.13) Ω 7 ≃       0 1 0 0 0 0 0 1 − 1 0 0 0 0 0 1 0       N( sp (4)) ; Ω 8 ≃       0 0 1 0 0 0 0 1 − 1 0 0 0 0 − 1 0 0       N( sp (4)) . (8.14 ) Considering now a Cartan Lie algebra elemen t in the fundamental represen tation of sp (4 , R ) as giv en in eq.(8.7) w e ha v e ∀ w i ∈ W ey l ( C 2 ) : Ω T i C ( { h 1 , h 2 } ) Ω i = C ( w i { h 1 , h 2 } ) . (8.15) This b eing established let us choose a s con ven tional reference set of eigenv alues the following one: h 1 = 1 ; h 2 = 2 , (8.16) then the decreasing sorting to b e exp ected at past infinity is 2 , 1 , − 1 , − 2 and cor r esp onds to the W eyl elemen t Ω 5 . If w e ta ke this as the fundamental p ermu tation, all the other eigh t p erm uta tions b elonging to the W eyl group can b e ra nk ed with the n umber of transp ositions needed t o bring them to the fundamen ta l one. This pro cedure pro vides the partia l o rdering of the W eyl group displa ye d in table 3. W e can no w study the general features of t he flows asso ciated with the maximally split coset manifold (8.2 ) and see how they follo w the general principles and connect past and future K a sner ep o c hs ordered according to table 3. T o realize this study the first essen tial step is the construction of the sp (4 , R ) Lie algebra in a basis whic h fulfils the condition that the solv able Lie algebra generating the coset is repres en ted b y upper triangular matrices. The form of the Cartan subalgebra in suc h a basis w as already anticipated in eq.(8.7), t he full construction is presen ted in the next section. 35 ℓ T W eyl group of sp (4 , R ) 0 w 5 1 w 6 2 w 1 3 w 3 3 w 4 4 w 2 5 w 7 6 w 8 T a ble 3: P artial ordering of the W eyl group of sp (4 , R ) . 8.2 Construction of the sp (4 , R ) Lie algebra The most compact w a y of pr esen ting our basis is the following. Let us b egin with the solv able Lie a lgebra S ol v (Sp(4 , R ) / U(2)). Abstractly the most g eneral elemen t of this algebra is give n b y T = h 1 H 1 + h 2 H 2 + e 1 E α 1 + e 2 E α 2 + e 3 E α 1 + α 2 + e 4 E 2 α 1 + α 2 . (8.17) If w e write the explicit form of T as a 4 × 4, upp er triangular symplectic matr ix T sy m =       h 1 e 1 e 3 − √ 2 e 4 0 h 2 √ 2 e 2 e 3 0 0 − h 2 − e 1 0 0 0 − h 1       ∈ sp (4 , R ) (8.18) whic h satisfies the condition T T sy m 0 2 1 2 − 1 2 0 2 ! + 0 2 1 2 − 1 2 0 2 ! T sy m = 0 (8.19) all the generators of the solv able algebra ar e defined in the four dimensional symple ctic represen tation. By writing the same Lie algebra elemen t (8.1 7) a s a 5 × 5 matrix T so =          h 1 + h 2 − √ 2 e 2 − √ 2 e 3 − √ 2 e 4 0 0 h 1 − h 2 − √ 2 e 1 0 √ 2 e 4 0 0 0 √ 2 e 1 √ 2 e 3 0 0 0 h 2 − h 1 √ 2 e 2 0 0 0 0 − h 1 − h 2          ∈ so (2 , 3) (8 .20) 36 whic h satisfies the condition T T so          0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0          +          0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0          T so = 0 (8.21) w e define the same generators also in the five dimensional pseudo-ort ho gonal represen ta tion. The c hoice of the in v arian t metric displa y ed in eq.(8.21) is that which guara ntees the upp er triangular structure of the solv able Lie algebra g enerators. W e shall come ba c k on this p oint in later sections. Once the generators of the solv able Lie algebra are given the full Lie algebra can b e completed b y defining the orthonor ma l generators of the K subspace as follows : K 1 = H 1 , K 2 = H 2 , K 3 = 1 √ 2  E α 1 + ( E α 1 ) T  , K 4 = 1 √ 2  E α 2 + ( E α 2 ) T  , K 5 = 1 √ 2  E α 1 + α 2 + ( E α 1 + α 2 ) T  , K 6 = 1 √ 2  E α 1 +2 α 2 + ( E α 1 +2 α 2 ) T  , (8.22) and those of the maximal compact subalgebra H = u (2 ) as follows: J 1 = 1 √ 2  E α 1 − ( E α 1 ) T  , J 2 = 1 √ 2  E α 2 − ( E α 2 ) T  , J 3 = 1 √ 2  E α 1 + α 2 − ( E α 1 + α 2 ) T  , J 4 = 1 √ 2  E α 1 +2 α 2 − ( E α 1 +2 α 2 ) T  . (8.23) In this w ay w e hav e constructed all t he relev a n t generator s in b o th represen tations. The flows are clearly an in trinsic prop erty of the algebra and will not dep end o n the represen t a tion c hosen. 8.3 P arameterization of the compact group U( 2) and critical sub- manifolds In a w a y completely analogous to the previous case-study w e can now parameterize t he compact subgroup by writing O = exp h √ 2 θ 1 J 1 i exp [ θ 2 J 2 ] exp h √ 2 θ 3 J 3 i exp [ θ 4 J 4 ] . (8.24) The square ro ot of tw o fa ctors ha ve b een intro duced in equation (8.24) in suc h a wa y as to normalize the theta angles so that the group elemen t O b ecomes a n in teger v alued matrix 37 at θ i = π 2 . Obvious ly w e hav e t w o instances o f O : the 4 × 4 symplectic O sp and the 5 × 5 pseudo-orthogonal O so . Both of them b ecome in teger v alued f or the same c hoice of the angles and when acting by similarity transformation on the Carta n subalgebra they corresp ond to W eyl group elemen ts. F or simplicit y we use t he 4 × 4 represen ta t ion and w e find O =       O 11 O 12 O 13 O 14 O 21 O 22 O 23 O 24 O 31 O 32 O 33 O 34 O 41 O 42 O 43 O 44       (8.25) where O 11 = cos θ 1 cos θ 3 cos θ 4 − sin θ 1 sin θ 3 sin ( θ 2 − θ 4 ) , O 12 = cos θ 2 cos θ 3 sin θ 1 , O 13 = cos θ 3 sin θ 1 sin θ 2 + cos θ 1 sin θ 3 , O 14 = cos θ 2 cos θ 4 sin θ 1 sin θ 3 + (sin θ 1 sin θ 2 sin θ 3 − cos θ 1 cos θ 3 ) sin θ 4 , (8.26) O 21 = − cos θ 3 cos θ 4 sin θ 1 − cos θ 1 sin θ 3 sin ( θ 2 − θ 4 ) , O 22 = cos θ 1 cos θ 2 cos θ 3 , O 23 = cos θ 1 cos θ 3 sin θ 2 − sin θ 1 sin θ 3 , O 24 = cos θ 1 cos θ 2 − θ 4 sin θ 3 + cos θ 3 sin θ 1 sin θ 4 , (8.27) O 31 = − cos θ 1 cos θ 2 − θ 4 sin θ 3 − cos θ 3 sin θ 1 sin θ 4 , O 32 = sin θ 1 sin θ 3 − cos θ 1 cos θ 3 sin θ 2 , O 32 = cos θ 1 cos θ 2 cos θ 3 , O 33 = cos θ 1 cos θ 2 cos θ 3 , O 34 = − cos θ 3 cos θ 4 sin θ 1 − cos θ 1 sin θ 3 sin ( θ 2 − θ 4 ) , (8.28) O 41 = (cos θ 1 cos θ 3 − sin θ 1 sin θ 2 sin θ 3 ) sin θ 4 − cos θ 2 cos θ 4 sin θ 1 sin θ 3 , O 42 = − cos θ 3 sin θ 1 sin θ 2 − cos θ 1 sin θ 3 , O 43 = cos θ 2 cos θ 3 sin θ 1 , O 44 = cos θ 1 cos θ 3 cos θ 4 − sin θ 1 sin θ 3 sin ( θ 2 − θ 4 ) . (8.29) V ert ices Ha ving parameteriz ed in this w a y the U(2) group elemen t with the fo ur Euler angles θ i , in a completely analog ous w ay to the case of SL(3 , R ), w e can c hec k that when all of the θ i tak e either the 0 or the π 2 v a lue then the cor r esp onding mat rix O b ecomes in teger v a lued and its similarit y action on a Cartan s ubalgebra elemen t (8.7) correspo nds to the action of some W eyl group elemen t on the eigen v alues: If ∀ i θ i =        0 or π 2 , ∃ ω ∈ W ey l ( C 2 ) / O T C ( { h 1 , h 2 } ) O = C ( ω { h 1 , h 2 } ) . (8.30 ) 38 # ve rtex W eyl gr oup elemen t multiplicity of W eyl elem. 1 { 0 , 0 , 0 , 0 } Ω 1 1 2 { 1 , 0 , 0 , 0 } Ω 5 1 3 { 0 , 1 , 0 , 0 } Ω 4 3 4 { 0 , 0 , 1 , 0 } Ω 8 3 5 { 0 , 0 , 0 , 1 } Ω 3 1 6 { 1 , 1 , 0 , 0 } Ω 6 3 7 { 1 , 0 , 1 , 0 } Ω 2 3 8 { 1 , 0 , 0 , 1 } Ω 7 1 9 { 0 , 1 , 1 , 0 } Ω 6 3 10 { 0 , 1 , 0 , 1 } Ω 2 3 11 { 0 , 0 , 1 , 1 } Ω 6 3 12 { 1 , 1 , 1 , 0 } Ω 4 3 13 { 1 , 1 , 0 , 1 } Ω 8 3 14 { 1 , 0 , 1 , 1 } Ω 4 3 15 { 0 , 1 , 1 , 1 } Ω 8 3 16 { 1 , 1 , 1 , 1 } Ω 2 3 T a ble 4: The 16 v ertices of the h yp ercubic parameter space f o r Sp(4 , R ) / U(2) and their iden tification with W eyl group elemen ts. In this w a y the parameter space U(2) / W is reduced to lie in a four dimensional hy p ercub e and, using a notatio n analogo us to that of eq.(7.18), the iden tification of the 16 v ertices of the hypercub e with W eyl group elemen ts is displa y ed in table 4. As the reader can observ e, in the c hosen num b ering the o dd- lab eled W eyl gro up elemen ts app ear only once, while the ev en-lab eled app ear three-times. Edges Using just the same strategy as in the previous case-study w e can no w construct the 64 oriente d links connecting the 16 v ertices. These are all the p ossible segmen ts o f straigh t lines in parameter space connecting tw o vertice s and b y means of a computer programme w e can ev aluate the orien ta tion of the link, namely disco v er whic h of the end-p oints (W eyl group elemen t) corresp onds to past infinity t = −∞ and whic h to future infinity t = + ∞ . As exp ected the o rien tatio n of all the links is in the direction from low er to higher W eyl elemen ts, according to the ordering of table 3. The result of these computations is displa y ed in table 5 and summarized in the flow diagram of fig.10. A four dimensional h yp ercub e cannot b e dra wn in three dimension but a standard wa y to visualize it is provide d by presen ting its stereographic pro jection. Indeed if w e shift the origin of the co ordinate system to the p oin t { 1 2 , 1 2 , 1 2 , 1 2 } then a ll t he 16 vertice s of the h yp ercub e are lo cated on the standard three-sphere, namely , as 4-comp onent ve ctors they ha v e unit norm. So w e can consider their stereographic pro jection from S 3 to R 3 and connecting them with segmen t s w e obtain the visualization of the h yp ercub e display ed in fig.11. T rapp ed h yp ersurfaces The study o f trapp ed h yp ersurfaces can now b e p erformed o nce again in complete analogy with the case of SL(3 , R ). W e just hav e to calculate all t he relev a n t minors and imp ose their v anishing. In this w ay we determine equations on the parameters that hav e to b e solv ed within the hypercubic range. If solutions within the hypercub e exist, 39 # V ertex V ertex Flow # V ertex V ertex Flow 1 { 0 , 0 , 0 , 0 } { 0 , 0 , 0 , 1 } Ω 1 7→ Ω 3 2 { 0 , 0 , 0 , 0 } { 0 , 0 , 1 , 0 } Ω 1 7→ Ω 8 3 { 0 , 0 , 0 , 0 } { 0 , 1 , 0 , 0 } Ω 1 7→ Ω 4 4 { 0 , 0 , 0 , 0 } { 1 , 0 , 0 , 0 } Ω 5 7→ Ω 1 5 { 0 , 0 , 0 , 1 } { 0 , 0 , 0 , 0 } Ω 1 7→ Ω 3 6 { 0 , 0 , 0 , 1 } { 0 , 0 , 1 , 1 } Ω 6 7→ Ω 3 7 { 0 , 0 , 0 , 1 } { 0 , 1 , 0 , 1 } Ω 3 7→ Ω 2 8 { 0 , 0 , 0 , 1 } { 1 , 0 , 0 , 1 } Ω 3 7→ Ω 7 9 { 0 , 0 , 1 , 0 } { 0 , 0 , 0 , 0 } Ω 1 7→ Ω 8 10 { 0 , 0 , 1 , 0 } { 0 , 0 , 1 , 1 } Ω 6 7→ Ω 8 11 { 0 , 0 , 1 , 0 } { 0 , 1 , 1 , 0 } Ω 6 7→ Ω 8 12 { 0 , 0 , 1 , 0 } { 1 , 0 , 1 , 0 } Ω 2 7→ Ω 8 13 { 0 , 0 , 1 , 1 } { 0 , 0 , 0 , 1 } Ω 6 7→ Ω 3 14 { 0 , 0 , 1 , 1 } { 0 , 0 , 1 , 0 } Ω 6 7→ Ω 8 15 { 0 , 0 , 1 , 1 } { 0 , 1 , 1 , 1 } Ω 6 7→ Ω 8 16 { 0 , 0 , 1 , 1 } { 1 , 0 , 1 , 1 } Ω 6 7→ Ω 4 17 { 0 , 1 , 0 , 0 } { 0 , 0 , 0 , 0 } Ω 1 7→ Ω 4 18 { 0 , 1 , 0 , 0 } { 0 , 1 , 0 , 1 } Ω 4 7→ Ω 2 19 { 0 , 1 , 0 , 0 } { 0 , 1 , 1 , 0 } Ω 6 7→ Ω 4 20 { 0 , 1 , 0 , 0 } { 1 , 1 , 0 , 0 } Ω 6 7→ Ω 4 21 { 0 , 1 , 0 , 1 } { 0 , 0 , 0 , 1 } Ω 3 7→ Ω 2 22 { 0 , 1 , 0 , 1 } { 0 , 1 , 0 , 0 } Ω 4 7→ Ω 2 23 { 0 , 1 , 0 , 1 } { 0 , 1 , 1 , 1 } Ω 2 7→ Ω 8 24 { 0 , 1 , 0 , 1 } { 1 , 1 , 0 , 1 } Ω 2 7→ Ω 8 25 { 0 , 1 , 1 , 0 } { 0 , 0 , 1 , 0 } Ω 6 7→ Ω 8 26 { 0 , 1 , 1 , 0 } { 0 , 1 , 0 , 0 } Ω 6 7→ Ω 4 27 { 0 , 1 , 1 , 0 } { 0 , 1 , 1 , 1 } Ω 6 7→ Ω 8 28 { 0 , 1 , 1 , 0 } { 1 , 1 , 1 , 0 } Ω 6 7→ Ω 4 29 { 0 , 1 , 1 , 1 } { 0 , 0 , 1 , 1 } Ω 6 7→ Ω 8 30 { 0 , 1 , 1 , 1 } { 0 , 1 , 0 , 1 } Ω 2 ] 7→ Ω 8 31 { 0 , 1 , 1 , 1 } { 0 , 1 , 1 , 0 } Ω 6 7→ Ω 8 32 { 0 , 1 , 1 , 1 } { 1 , 1 , 1 , 1 } Ω 2 7→ Ω 8 33 { 1 , 0 , 0 , 0 } { 0 , 0 , 0 , 0 } Ω 5 7→ Ω 1 34 { 1 , 0 , 0 , 0 } { 1 , 0 , 0 , 1 } Ω 5 7→ Ω 7 35 { 1 , 0 , 0 , 0 } { 1 , 0 , 1 , 0 } Ω 5 7→ Ω 2 36 { 1 , 0 , 0 , 0 } { 1 , 1 , 0 , 0 } Ω 5 7→ Ω 6 37 { 1 , 0 , 0 , 1 } { 0 , 0 , 0 , 1 } Ω 3 7→ Ω 7 38 { 1 , 0 , 0 , 1 } { 1 , 0 , 0 , 0 } Ω 5 7→ Ω 7 39 { 1 , 0 , 0 , 1 } { 1 , 0 , 1 , 1 } Ω 4 7→ Ω 7 40 { 1 , 0 , 0 , 1 } { 1 , 1 , 0 , 1 } Ω 7 7→ Ω 8 41 { 1 , 0 , 1 , 0 } { 0 , 0 , 1 , 0 } Ω 2 7→ Ω 8 42 { 1 , 0 , 1 , 0 } { 1 , 0 , 0 , 0 } Ω 5 7→ Ω 2 43 { 1 , 0 , 1 , 0 } { 1 , 0 , 1 , 1 } Ω 4 7→ Ω 2 44 { 1 , 0 , 1 , 0 } { 1 , 1 , 1 , 0 } Ω 4 7→ Ω 2 45 { 1 , 0 , 1 , 1 } { 0 , 0 , 1 , 1 } Ω 6 7→ Ω 4 46 { 1 , 0 , 1 , 1 } { 1 , 0 , 0 , 1 } Ω 4 7→ Ω 7 47 { 1 , 0 , 1 , 1 } { 1 , 0 , 1 , 0 } Ω 4 7→ Ω 2 48 { 1 , 0 , 1 , 1 } { 1 , 1 , 1 , 1 } Ω 4 7→ Ω 2 49 { 1 , 1 , 0 , 0 } { 0 , 1 , 0 , 0 } Ω 6 7→ Ω 4 50 { 1 , 1 , 0 , 0 } { 1 , 0 , 0 , 0 } Ω 5 7→ Ω 6 51 { 1 , 1 , 0 , 0 } { 1 , 1 , 0 , 1 } Ω 6 7→ Ω 8 52 { 1 , 1 , 0 , 0 } { 1 , 1 , 1 , 0 } Ω 6 7→ Ω 4 53 { 1 , 1 , 0 , 1 } { 0 , 1 , 0 , 1 } Ω 2 7→ Ω 8 54 { 1 , 1 , 0 , 1 } { 1 , 0 , 0 , 1 } Ω 7 7→ Ω 8 55 { 1 , 1 , 0 , 1 } { 1 , 1 , 0 , 0 } Ω 6 7→ Ω 8 56 { 1 , 1 , 0 , 1 } { 1 , 1 , 1 , 1 } Ω 2 7→ Ω 8 57 { 1 , 1 , 1 , 0 } { 0 , 1 , 1 , 0 } Ω 6 7→ Ω 4 58 { 1 , 1 , 1 , 0 } { 1 , 0 , 1 , 0 } Ω 4 7→ Ω 2 59 { 1 , 1 , 1 , 0 } { 1 , 1 , 0 , 0 } Ω 6 7→ Ω 4 60 { 1 , 1 , 1 , 0 } { 1 , 1 , 1 , 1 } Ω 4 7→ Ω 2 61 { 1 , 1 , 1 , 1 } { 0 , 1 , 1 , 1 } Ω 2 7→ Ω 8 62 { 1 , 1 , 1 , 1 } { 1 , 0 , 1 , 1 } Ω 4 7→ Ω 2 63 { 1 , 1 , 1 , 1 } { 1 , 1 , 0 , 1 } Ω 2 7→ Ω 8 64 { 1 , 1 , 1 , 1 } { 1 , 1 , 1 , 0 } Ω 4 7→ Ω 2 T a ble 5: A 4- dimensional hy p ercub e has 32 edges whic h amoun t to 64 edges if we consider also the p ossible orientation. Here are displa ye d the 64 orien ted links of the h yp ercubic parameter space for Sp(4 , R ) / U(2) flo ws and the corresp onding orien ted links from W eyl group elemen ts to W eyl group elemen ts. then we ha v e a tr a pp ed surface. Otherwise w e just ha v e a W eyl replica of a n already existing surface. In our case there a r e just 14 relev ant minors distributed in the following w ay : 4 of rank 1, 4 of rank 3 and 6 of ra nk 2. Explicitly we can define the f ollo wing candidate trapp ed 40 W 1 W 2 W 3 W 4 W 5 W 6 W 7 W 8 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 2 2 1 4 3 8 7 6 5 3 3 4 1 2 6 5 8 7 4 4 3 2 1 7 8 5 6 5 5 8 7 6 1 4 3 2 6 6 7 8 5 3 2 1 4 7 7 6 5 8 4 1 2 3 8 8 5 6 7 2 3 4 1 Figure 10: The oriente d phase diagr am of the Sp(4 , R ) / U(2) flows. The Lie alg ebr a sp (4 , R ) is the maximal ly split r e al se ction of the c omplex Lie algebr a C 2 ∼ B 2 . Its Weyl gr oup is ( Z 2 × Z 2 ) ⋉ S 2 and has eight ele m ents identifie d by their action on the eigenvalues h 1 , h 2 of the L ax op er ator. Eight ar e ther efor e the p o s s ible asymptotic states of the unive rse at t = ±∞ and e ach p oss i b l e motion is an oriente d flow fr om one Weyl elem ent t o another one. The orientation fol lows the or derin g of Weyl gr oup elements: it is always fr om a low e r to a hig h er on e . In this pi c tur e, cho osing a s f und a mental eigenvalues h 1 = 1 , h 2 = 2 the Weyl gr oup element Ω i ∈ W ey l ( C 2 ) is identifie d by the p o i n t in the plane that has c o or dinates Ω i ( h 1 , h 2 ) . Each link is ther ef o r e asso ciate d wi th a Weyl gr oup element whic h multiplying on the left the p ast infinity elemen t pr o duc es the futur e infinity one. By c omp arison we display b elow the gr aph the multiplic ation table of the Weyl gr oup. Note that in e ach vertex of the diagr am ther e me et just four lines . In v e rtex Ω 5 ther e ar e only o utgoing lines. This is so b e c ause Ω 5 is the lowest Weyl element and it c orr esp ond s to the univ ersal p ast infinity p oint for gen eric flows. On the c ontr ary in the vertex Ω 8 ther e only inc o m ing lines. This is so b e c ause Ω 8 is the h i g hest Weyl element and it c orr esp onds to the universal futur e i n finity fo r generic flows. T h e other vertic es have b oth inc oming and outgoing lines. surfaces: Σ 1 : O 1 , 1 = 0 , Σ 2 : O 2 , 1 = 0 , Σ 3 : O 3 , 1 = 0 , Σ 4 : O 4 , 1 = 0 , Σ 5 : O 1 , 3 ( O 2 , 1 O 3 , 2 − O 2 , 2 O 3 , 1 ) + O 1 , 2 ( O 2 , 3 O 3 , 1 − O 2 , 1 O 3 , 3 ) + O 1 , 1 ( O 2 , 2 O 3 , 3 − O 2 , 3 O 3 , 2 ) = 0 , Σ 6 : O 1 , 3 ( O 2 , 1 O 4 , 2 − O 2 , 2 O 4 , 1 ) + O 1 , 2 ( O 2 , 3 O 4 , 1 − O 2 , 1 O 4 , 3 ) + O 1 , 1 ( O 2 , 2 O 4 , 3 − O 2 , 3 O 4 , 2 ) = 0 , Σ 7 : O 1 , 3 ( O 3 , 1 O 4 , 2 − O 3 , 2 O 4 , 1 ) + O 1 , 2 ( O 3 , 3 O 4 , 1 − O 3 , 1 O 4 , 3 ) + O 1 , 1 ( O 3 , 2 O 4 , 3 − O 3 , 3 O 4 , 2 ) = 0 , (8.31) 41 1 5 4 8 3 6 2 7 6 2 6 4 8 4 8 2 Figure 11 : Ster e o gr aphic pr oj e c tion of the hyp er cubic p ar ameter sp ac e for Sp(4 , R ) / U(2) motions. Σ 8 : O 2 , 3 ( O 3 , 1 O 4 , 2 − O 3 , 2 O 4 , 1 ) + O 2 , 2 ( O 3 , 3 O 4 , 1 − O 3 , 1 O 4 , 3 ) + O 2 , 1 ( O 3 , 2 O 4 , 3 − O 3 , 3 O 4 , 2 ) = 0 , Σ 9 : O 1 , 1 O 2 , 2 − O 1 , 2 O 2 , 1 = 0 , Σ 10 : O 1 , 1 O 3 , 2 − O 1 , 2 O 3 , 1 = 0 , Σ 11 : O 1 , 1 O 4 , 2 − O 1 , 2 O 4 , 1 = 0 , Σ 12 : O 2 , 1 O 3 , 2 − O 2 , 2 O 3 , 1 = 0 , Σ 13 : O 2 , 1 O 4 , 2 − O 2 , 2 O 4 , 1 = 0 , Σ 14 : O 3 , 1 O 4 , 2 − O 3 , 2 O 4 , 1 = 0 (8.32) where t he explicit form of the equation can b e o btained by substituting the v alues of the U(2) matrix elemen ts as giv en in eq.s (8.2 5 – 8.29). A full-fledged analysis of all the trapp ed surfaces is b ey o nd the scope of the presen t pap er whic h aims at illustrating the general principles and at explaining the metho d. What w e can do without any analytic study of the trapp ed surfaces is to determine the accessible W eyl gro up elemen ts for eac h of them and in this wa y single out the corresp onding past infinity and future infinit y states. The result is sho wn in table 6. 42 Surf Accessible W eyl el. Flow Type Σ 1 { w 5 , w 6 , w 3 , w 2 , w 7 , w 8 } w 5 7→ w 8 critical Σ 2 { w 1 , w 3 , w 4 , w 2 , w 7 , w 8 } w 1 7→ w 8 sup er-critical Σ 3 { w 5 , w 6 , w 1 , w 3 , w 4 , w 2 } w 5 7→ w 2 sup er-critical Σ 4 { w 5 , w 6 , w 1 , w 4 , w 7 , w 8 } w 5 7→ w 8 critical Σ 5 { w 5 , w 6 , w 3 , w 2 , w 7 , w 8 } w 5 7→ w 8 critical Σ 6 { w 1 , w 3 , w 4 , w 2 , w 7 , w 8 } w 1 7→ w 8 sup er-critical Σ 7 { w 5 , w 6 , w 1 , w 3 , w 4 , w 2 } w 5 7→ w 2 sup er-critical Σ 8 { w 5 , w 6 , w 1 , w 4 , w 7 , w 8 } w 5 7→ w 8 critical Σ 9 { w 6 , w 3 , w 4 , w 2 , w 7 , w 8 } w 6 7→ w 8 sup er-critical Σ 10 { w 5 , w 6 , w 1 , w 3 , w 2 , w 8 } w 5 7→ w 8 critical Σ 11 { w 5 , w 6 , w 1 , w 3 , w 4 , w 2 , w 7 , w 8 } w 5 7→ w 8 trapp ed non crit. Σ 12 { w 5 , w 6 , w 1 , w 3 , w 4 , w 2 , w 7 , w 8 } w 5 7→ w 8 trapp ed non crit. Σ 13 { w 5 , w 1 , w 4 , w 2 , w 7 , w 8 } w 5 7→ w 8 critical Σ 14 { w 5 , w 6 , w 1 , w 3 , w 4 , w 7 } w 5 7→ w 7 sup er-critical T a ble 6: Accessible W eyl elemen ts on the 14 trapp ed surfaces o f Sp(4 , R ) / U(2) flows . By insp ecting t he list A Σ of accessible W eyl eleme n t s w e easily deduce the c haracter of the surface. If there are missin g W eyl elemen ts it is critical. It is sup er-critical if one of the missing elemen ts is either Ω min = w 5 and/or Ω max = w 8 . Whe n no W eyl elemen t is missing in A Σ , the surface is just only trapp ed. It w ould b e critical inside the bigger group SL(4 , R ). As it is eviden t b y insp ection o f this table the p ossible flow s realized on critical h yp er- surfaces of parameter space are j ust a very small n umber, namely the following five : 1 : w 1 7→ w 8 , 2 : w 5 7→ w 2 , 3 : w 5 7→ w 7 , 4 : w 5 7→ w 8 , 5 : w 6 7→ w 8 . (8.33) As we are going to see this is a prop ert y shared b y the en tire univ ersality class of manifolds that hav e the same Tits Satak e pro jection. The W eyl gr o up, the flow diag ram on the links and t he p ossible flow s realized on critical surfaces do not dep end on the represen tativ e inside the class but are just a prop ert y of the class. In the next section w e j ust f o cus on the detailed discussion of a few examples of flo ws f or this maximally split manif o ld. 43 8.4 Examples for sp (4 , R ) In this s ection, as we just announced, w e consider three examples of Sp(4 , R )-flo ws. One will b e in the bulk the other tw o will b e lo cated on tw o differen t critical surfaces. W e analyse these cases in detail b oth to sho w the relation b et w een the asymptotic states and the structure o f the orthogonal gro up elemen t O and to illustrate the billiard phenomenon. In the plots of the Cartan fields w e will b e able to obse rv e t he m ultiple b ouncing of the cosmic ball on the h yp erplanes or thogonal to some of the ro ots. 8.4.1 An example of flow in the bulk of parameter space: Ω 5 ⇒ Ω 8 If, as initial data w e c ho ose the following elemen t o f the compact subgroup U(2) ⊂ Sp(4 , R ): U(2) ∋ O = exp[ π 6 J 1 ] exp[ π 4 J 2 ] exp[ π 6 J 3 ] exp[ π 3 J 4 ] (8.34) = 0 B B B B B B B @ 1 16 “ 6 − √ 2 + √ 6 ” q 3 2 4 1 8 √ 3 “ 2 + √ 2 ” 1 16 “ √ 2 − 6 √ 3 + √ 6 ” 1 16 “ 3 √ 2 − 2 √ 3 − √ 6 ” 3 4 √ 2 1 8 “ − 2 + 3 √ 2 ” 1 16 “ 6 + 3 √ 2 + √ 6 ” 1 16 “ − 6 − 3 √ 2 − √ 6 ” 1 8 “ 2 − 3 √ 2 ” 3 4 √ 2 1 16 “ 3 √ 2 − 2 √ 3 − √ 6 ” 1 16 “ − √ 2 + 6 √ 3 − √ 6 ” − 1 8 √ 3 “ 2 + √ 2 ” q 3 2 4 1 16 “ 6 − √ 2 + √ 6 ” 1 C C C C C C C A w e are just in the bulk. Indeed as it is eviden t by insp ection of eq.(8.34), the 4 × 4 matrix represen ting O has no minor with v anishing determinan t. Hence a ccording to the advocated theorems w e expect asymptotic sorting of the eigen v alues. Indeed this is what ha pp ens. Implemen ting b y nu merical ev aluation the in tegration form ula on a computer w e disco v er that the asymptotic form of the La x op erato r at t = −∞ corresp onds to the W eyl group elemen t Ω 5 lim t → −∞ L ( t ) =       2 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 − 2       ⇔ Ω 5 . (8.35) Similarly at asymptotically late times the limit of the Lax op erator is that corresp onding to the W eyl group elemen t Ω 8 lim t → + ∞ L ( t ) =       − 2 0 0 0 0 − 1 0 0 0 0 1 0 0 0 0 2       ⇔ Ω 8 . (8.36) Algebraically w e ha v e Ω 8 = Ω 2 Ω 5 , so that all generic flow s in the bulk that a void to uching critical surfaces ar e a smo oth realization of the W eyl reflection Ω 2 ∈ W . The particular smo oth realization of this reflection provided by t he presen t choice of parameters is illustrated in fig.12 whic h displays the motion of the cosmic ball on the t wo dimensional billiard t a ble whose axes are the Carta n field s h 1 , h 2 . This motio n inv o lv es t wo b ounces as it b ecomes eviden t b y plotting the pro jection of the Carta n v ector − → h = ( h 1 , h 2 ) a lo ng the t w o roo t s α 1 = (1 , − 1) and α 3 = (1 , 1). These plot s are displa yed in fig.13. 44 16 18 20 22 h1 6 8 10 12 14 h2 Figure 12: Motion of the c osmic b al l on the CSA bil liar d table of Sp(4 , R ) in a generic bulk c ase. The choic e of the angles is θ 1 = π 6 , θ 2 = π 4 , θ 3 = π 6 , θ 4 = π 3 . A c c or ding to the ory this motion r e alizes the smo oth r e fle c tion Ω 2 fr om the universal primor dial Kasner er a Ω 5 to the universal r emote futur e K a sner er a Ω 8 . This motion involves just two b o unc e s on the wal l r esp e ctively ortho gonal to the r o ot α 3 = (1 , − 1) and α 1 = (1 , 1) . In this pictur e the s tr aig h t lines r e pr esent the wal ls ortho gon al to α 1 and α 3 , r esp e ctively. -3 -2 -1 1 2 3 4 5 6 7 8 -3 -2 -1 1 2 3 22 24 26 28 30 32 Figure 13: Plot o f α 1 , 3 · h pr oje c tion s for the Sp(4 , R ) generic bulk flow gener ate d by the p ar ameter choic e θ 1 = π 6 , θ 2 = π 4 , θ 3 = π 6 , θ 4 = π 3 which c onne cts the pri m or dial Kasner er a Ω 5 to the r emote futur e Kasner er a Ω 8 . Ther e a r e two b ounc es in this flow b e c ause the pr oje ctions α 1 , 3 · h have m a xima at differ ent instant of times. 8.4.2 An example of flow on the sup er-critical surface Σ 9 : Ω 6 ⇒ Ω 8 As a next example we consider a flo w confined on a sup er-critical surface. If we set θ 2 = π 2 the U(2) matrix takes the following f o rm: O = 0 B B B B @ cos ( θ 1 + θ 3 ) cos ( θ 4 ) 0 sin ( θ 1 + θ 3 ) − cos ( θ 1 + θ 3 ) sin ( θ 4 ) − cos ( θ 4 ) sin ( θ 1 + θ 3 ) 0 cos ( θ 1 + θ 3 ) sin ( θ 1 + θ 3 ) sin ( θ 4 ) − sin ( θ 1 + θ 3 ) sin ( θ 4 ) − cos ( θ 1 + θ 3 ) 0 − cos ( θ 4 ) sin ( θ 1 + θ 3 ) cos ( θ 1 + θ 3 ) sin ( θ 4 ) − sin ( θ 1 + θ 3 ) 0 cos ( θ 1 + θ 3 ) cos ( θ 4 ) 1 C C C C A (8.37) 45 whic h has tw o notable prop erties. The first is that it has v anishing some principal minor s, the second is that it actually dep ends o n t wo v ariables only , namely θ 1 + θ 3 and θ 4 . If w e consider the equation for the critical sup er-surface Σ 9 w e find cos ( θ 2 ) cos 2 ( θ 3 ) cos ( θ 4 ) = 0 (8.38) so that the hy p erplane θ 2 = π 2 where w e ha v e c hosen our gr o up elemen t is just one of the three comp onents of Σ 9 . F urthermore, b ecause of what w e just observ ed, on this h yp erplane all p oints of the following form ha ve to b e identified: ∀ φ ∈  0 , π 2  ;  θ 1 + φ , π 2 , θ 3 − φ, θ 4  ∼  θ 1 , π 2 , θ 3 , θ 4  . (8.39) Ha ving chose n initia l data on a trapp ed surface, namely Σ 9 , we do not exp ect full asymptotic sorting of the eigen v alues: actually , according to table 6 w e exp ect flo ws from Ω 6 to Ω 8 . Indeed this is what happ ens. W e v erify it in one example. F or instance, as initial data w e c ho ose the following elemen t of the compact subgroup U(2) ⊂ Sp(4 , R ), whic h lies in t he considered hy p ersurface: U(2) ∋ O = exp[ π 3 J 1 ] exp[ π 2 J 2 ] exp[ π 3 J 3 ] exp[ π 3 J 4 ] =       − 1 4 0 √ 3 2 √ 3 4 − √ 3 4 0 − 1 2 3 4 − 3 4 1 2 0 − √ 3 4 − √ 3 4 − √ 3 2 0 − 1 4       . (8.40) Implemen ting b y nu merical ev aluation the in tegration form ula on a computer w e disco v er that the asymptotic form of the La x op erato r at t = −∞ corresp onds to the W eyl group elemen t Ω 6 whic h implies no decreasing sorting of the eigen v alues lim t → −∞ L ( t ) =       2 0 0 0 0 − 1 0 0 0 0 1 0 0 0 0 − 2       ⇔ Ω 6 . (8.41) On the o ther hand the limit of the La x operator at asymptotically late times is that cor- resp onding to the W eyl group elemen t Ω 8 whic h is the same o ccurring in generic flo ws and yields increasing sorting of the eigenv alues lim t → + ∞ L ( t ) =       − 2 0 0 0 0 − 1 0 0 0 0 1 0 0 0 0 2       ⇔ Ω 8 . (8.42) Algebraically w e ha v e Ω 8 = Ω 3 Ω 6 , so that the flows o ccurring on this sup er-critical surface are smo oth realizations of the W eyl reflection Ω 3 ∈ W . The particular smo oth realization of this reflection enco ded in this flo w is illustrated in fig.1 4 whic h displa ys the motion of the cosmic ba ll on the t wo dimensional billiard ta ble This motion inv o lv es just one b ounce on the w all orthogonal to the ro o t α 4 as it b ecomes eviden t by insp ecting fig.1 5. 46 16 18 20 22 h1 -15 -10 -5 5 h2 Figure 14: Motion o f the c o smic b al l on the CSA bil liar d table o f Sp(4 , R ) in a sup er-critic al surfac e c ase. The choic e of the angles is θ 1 = π 3 , θ 2 = π 2 , θ 3 = π 3 , θ 4 = π 3 . Th i s motion r e alizes the smo oth r efle ction Ω 3 fr om the Kasner er a Ω 6 at t = −∞ to the Kasner er a Ω 8 at t = + ∞ . T he two str aight lines app e aring in the pi c tur e ar e the wal ls ortho gona l to the r o ots α 2 = (0 , 2) and α 4 = (2 , 0) , r esp e ctively. As o n e se es the c osmic b al l just b ounc es onc e the on the α 4 wal l. -3 -2 -1 1 2 3 -27.5 -25 -22.5 -20 -17.5 -15 -3 -2 -1 1 2 3 27.5 32.5 35 37.5 40 42.5 Figure 15: Plo t of α 2 , 4 · h pr oje c tions for the Sp(4 , R ) flow on a sup er-critic al surfac e gener ate d by the p ar ameter choic e θ 1 = π 3 , θ 2 = π 2 , θ 3 = π 3 , θ 4 = π 3 which c onne cts the p ast Kasner er a Ω 6 to the futur e Kasner er a Ω 8 . The r e is just one b o unc e in this flow and this o c curs on the α 4 wal l. 8.4.3 An example of flow on the sup er-critical surface Σ 2 : Ω 1 ⇒ Ω 8 In the example considered b elow the initial state is differen t from t ha t app earing in a generic bulk flo w, namely there is not decreasing sorting of the eigen v alues at past infinity whic h rather corr espo nds to the W eyl elemen t Ω 1 . Y et the end p oint a t t = + ∞ coincides with the univ ersal one Ω 8 , namely there is increasing sorting at future infinit y . W e realize this situation b y c ho osing initial da ta on one of the critical surfaces, namely the surface Σ 2 . 47 The equation for the trapp ed surface Σ 2 , as defined in (8.31) reads as follows: 0 = cos ( θ 3 ) cos ( θ 4 ) sin ( θ 1 ) + cos ( θ 1 ) sin ( θ 3 ) sin ( θ 2 − θ 4 ) (8.43) and it can b e solved within the h yp ercub e by express ing θ 1 in terms of the remaining three Euler angles as it follows : θ 1 = arccos cos ( θ 3 ) cos ( θ 4 ) p cos 2 ( θ 3 ) cos 2 ( θ 4 ) + sin 2 ( θ 3 ) sin 2 ( θ 2 − θ 4 ) ! . (8.44) On the h yp ersurface Σ 2 w e choose the particularly nice p oint n π 3 , π 6 , π 3 , π 3 o ∈ Σ 2 (8.45) whic h is easily seen to ve rify the defining equation (8.43) and whic h leads to a quite simple form of the matrix O . With these v alues o f the Euler angles w e obtain the following elemen t of the maximally compact subgroup U(2) ⊂ Sp(4 , R ): U(2) ∋ H sp = exp[ π 3 J 1 ] exp[ π 6 J 2 ] exp[ π 3 J 3 ] exp[ π 3 J 4 ] =       1 2 3 8 3 √ 3 8 √ 3 4 0 √ 3 8 − 5 8 3 4 − 3 4 5 8 √ 3 8 0 − √ 3 4 − 3 √ 3 8 3 8 1 2       (8.46) whic h indeed has v anishing O 2 , 1 matrix elemen t as it is required b y the definition of the Σ 2 trapp ed surface. Hence according to table 6 w e expect a flow from Ω 1 to Ω 8 . Before pro ceeding to the integration o f the Lax equation it is interes ting to consider the so (2 , 3 ) 5-dimensional represen tatio n of the same U(2) group elemen t. It is explicitly given b y the follo wing matrix: H so =           √ 3 16 5 16 − 3 4 √ 2 − 3 16 − 7 √ 3 16 − 19 32 11 √ 3 32 3 √ 3 2 8 − 5 √ 3 32 − 3 32 3 √ 3 2 16 15 16 √ 2 − 1 8 15 16 √ 2 3 √ 3 2 16 − 3 32 − 5 √ 3 32 3 √ 3 2 8 11 √ 3 32 − 19 32 − 7 √ 3 16 − 3 16 − 3 4 √ 2 5 16 √ 3 16           . (8.47) Since all the prop erties of the flow s are in trinsic prop erties of the g roup and cannot dep end on the c hosen represen ta t ion it fo llo ws that also the matrix (8.47 ) should b e critical namely some of its relev an t minors (those obtained by intese cting the first k -columns with k a r bitrary ro ws should v anish. Although not eviden t at first sigh t, this is indeed true. Calculating the minors w e find that there are three relev ant 2 × 2 minors whose determinan t v anishes, namely Det   √ 3 16 5 16 3 √ 3 2 16 15 16 √ 2   = 0 ; Det √ 3 16 5 16 − 3 32 − 5 √ 3 32 ! = 0 , (8.48) Det   3 √ 3 2 16 15 16 √ 2 − 3 32 − 5 √ 3 32   = 0 . (8.49) 48 Hence the criticality condition is indeed in trinsic to the choice o f the group elemen t and no t to its sp ecific represen ta tion as a matrix. Implemen ting by n umerical ev aluation the integration form ula on a computer w e disco v er that the asymptotic form of the La x op erato r at t = −∞ corresp onds to the W eyl group elemen t Ω 1 as exp ected: lim t → −∞ L ( t ) =       1 0 0 0 0 2 0 0 0 0 − 2 0 0 0 0 − 12       ⇔ Ω 1 (8.50) while t he limit at asymptotically late times is that corresp onding to the W eyl g roup elemen t Ω 8 as w e a lso expected lim t → + ∞ L ( t ) =       − 2 0 0 0 0 − 1 0 0 0 0 1 0 0 0 0 2       ⇔ Ω 8 . (8.51) Algebraically w e ha v e Ω 8 = Ω 8 Ω 1 , so that the flows o ccurring on this sup er-critical surface are smo oth realizations o f the W eyl reflection Ω 8 ∈ W . The smoot h realization of this reflection enco ded in the flow with these initial data is illustrated in fig.16 whic h display s the motio n of the cosmic ball in the h 1 , h 2 Cartan subalgebra plane. This motion in v olve s three b ounces tw o on the w all orthogonal to t he simple r o ot α 1 and one on the w all ortho gonal to α 2 . This is clearly visible b y insp ection of fig.17. 9 The c ase of th e so ( r , r + 2 s ) algeb ra W e are in terested in considering the sigma mo del on the symmetric non compact coset manifold M ( r , 2 s ) = SO(r , r + 2s) SO(r) × SO(r + 2s) . (9.1) F or r = 4 the abov e manifold is quaternionic and corr esp onds to t he family o f sp ecial geometries L (0 , P = 2 s ). 9.1 The corresp onding complex Lie algebra and ro ot system The complex Lie algebra of whic h so ( r , r + 2 s ) is a non- compact real section is just D ℓ where ℓ = r + s . (9.2) The corresp onding D ynkin diagra m is displa yed in fig 18 and the asso ciated ro ot system is realized b y the follow ing set of ve ctors in R ℓ : ∆ ≡  ± ǫ A ± ǫ B  ; card ∆ = 2  ℓ 2 − ℓ  (9.3) 49 h1 h2 Figure 16: Motion o f the c o smic b al l on the CSA bil liar d table o f Sp(4 , R ) in a sup er-critic al surfac e c ase. T h e choic e of the angles is θ 1 = π 3 , θ 2 = π 6 , θ 3 = π 3 , θ 4 = π 3 which lie on the tr a p p e d and sup er-c ri tic al surfac e Σ 2 . This motion r e alizes the smo oth r efle ction Ω 8 fr om the Kasner er a Ω 1 at t = −∞ to the Kas ner er a Ω 8 at t = + ∞ . The p e culiar knot app e aring in this pictur e implies the e xistenc e of two b ounc es on the same r o ot wal l. The two str aight lines displaye d i n the figur e ar e the wal ls ortho gonal to the two simple r o ots α 1 = (1 , − 1) and α 2 = (0 , 2) . The b al l b ounc es twic e on the α 1 wal l. -1.5 -1 -0.5 0.5 1 1.5 -15.2 -14.8 -14.6 -14.4 -14.2 -14 -13.8 -3 -2 -1 1 2 3 45 50 55 Figure 17: Plo t of α 1 , 2 · h pr oje c tions for the Sp(4 , R ) flow on a sup er-critic al surfac e gener ate d by the p ar ameter choic e θ 1 = π 3 , θ 2 = π 6 , θ 3 = π 3 , θ 4 = π 3 which c onne c ts the p a st Kasner er a Ω 1 to the futur e Ka sner er a Ω 8 . The two b ounc e s ar e cle arly visible i n the m axima and minima of the fi rst gr ap h. where ǫ A denotes an orthono r mal basis of unit v ectors. The set of p ositive ro ots is then easily defined as follows: α > 0 ⇒ α ∈ ∆ + ≡  ǫ A ± ǫ B  ( A < B ) . (9.4) A standard basis of simple r o ots represen ting the D ynkin diagram 18 is giv en b y α 1 = ǫ 1 − ǫ 2 , 50 Figure 18: The Dynkin diagr am of the D ℓ Lie algebr a. D ℓ ✐ α 1 ✐ α 2 ✐ α 3 . . . ✐ α ℓ − 3 ✐ α ℓ − 2   ❅ ❅ ✐ ✐ α ℓ − 1 α ℓ α 2 = ǫ 2 − ǫ 3 , . . . . . . . . . , α ℓ − 1 = ǫ ℓ − 1 − ǫ ℓ , α ℓ = ǫ ℓ − 1 + ǫ ℓ . (9.5) The maximally split real for m of the D ℓ Lie algebra is so ( ℓ, ℓ ) and it is explicitly realized b y the following 2 ℓ × 2 ℓ matrices. Let e A,B denote the 2 ℓ × 2 ℓ matrix whose en tries are all zero except the en try A, B wh ic h is equal to one. Then the Cartan generators H A and the p ositiv e ro ot step op erato rs E α are represen ted as follo ws: H A = e A,A − e A + ℓ,A + ℓ , E ǫ A − ǫ B = e B ,A − e A + ℓ,B + ℓ , E ǫ A + ǫ B = e A + ℓ,B − e B + ℓ,A . (9.6) The solv able algebra of the maximally split coset M ( ℓ, 0) = SO( ℓ, ℓ ) SO( ℓ ) × SO( ℓ ) (9.7) has therefore a v ery sim ple form in terms of matrices. F ollowing the general constructiv e principles S ol v ( ℓ,ℓ ) is just the algebraic span of all the matrices (9 .6 ) so that S ol v ( ℓ,ℓ ) ∋ M ⇔ M = T B 0 − T T ! ; ( T = upp er triangular , B = − B T an t isymmetric. (9.8) The matrices o f the form (9.8) clearly f o rm a subalgebra of the so ( ℓ , ℓ ) algebra whic h, in this represen tation, is defined as the set o f matrices Λ fulfilling the following condition: Λ T 0 1 l 1 l 0 ! + 0 1 l 1 l 0 ! Λ = 0 . (9.9) 9.2 The real form so ( r, r + 2 s ) of the D r + s Lie algebra The main p o in t in order to apply to the coset manifold (9.1) the g eneral in tegration a lgorithm of the Lax equation devised f or the case SL(2 ℓ ) / SO(2 ℓ ) consists of in tr o ducing a con v enien t 51 basis of generators of the Lie algebra so ( r , r + 2 s ) where, in the fundamental represen tation, all elemen ts of the solv able Lie alg ebra asso ciated with the coset under study turn out to b e given b y upp er triangular matrices. With some ingen uity such a basis can be found b y defining the so ( r, r + 2 s ) Lie algebra as the set of matrices Λ t satisfying t he f ollo wing constrain t: Λ T t η t + η t Λ t = 0 (9.10) where the symmetric in v ariant metric η t with r + 2 s p ositiv e eigen v a lues (+1) a nd r negativ e ones ( − 1) is giv en b y the follow ing matrix. η t =     0 0  r 0 1 2 s 0  r 0 0     . (9.11) In the ab ov e equation the sym b ol  r denotes the completely anti-diagonal r × r matrix whic h follo ws:  r =             0 0 . . . . . . 0 1 0 0 . . . . . . 1 0 0 0 . . . 1 0 0 . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . 0 1 0 0 . . . . . . 0             | {z } r                                r . ( 9.12) Ob viously there is a simple ortho gonal transformatio n wh ic h maps the metric η t in to the standard blo ck diag onal metric η b written b elow η b =     1 r 0 0 0 1 2 s 0 0 0 − 1 r     . (9.13) Indeed w e can write Ω T η b Ω = η t (9.14) where the explicit form of the ma t r ix Ω is the follo wing: Ω =     0 1 2 s 0 1 √ 2 1 r 0 1 √ 2  r 1 √ 2 1 r 0 − 1 √ 2  r     . (9.15) Corresp ondingly the orthog o nal transformation Ω maps the Lie alg ebra and group elemen ts of so ( r , r + 2 s ) from the standard basis where the in v ariant metric is η b to the basis where it is η t Λ t = Ω T Λ b Ω . (9.16) 52 In the t - basis the general for m of an elemen t of the solv able Lie algebra whic h generates the coset manifold (9.1) has the following a pp earance: S ol v  SO(r , r + 2s) SO(r) × SO(r + 2s)  ∋ Λ t =     T X B 0 0 X T  r 0 0 −  r T T  r     (9.17) where T =             T 1 , 1 T 1 , 2 . . . . . . T 1 ,r − 1 T 1 ,r 0 T 2 , 2 . . . . . . T 2 ,r − 1 T 2 ,r 0 0 T 3 , 3 . . . . . . T 3 ,r . . . . . . . . . . . . . . . . . . 0 0 0 . . . T r − 1 ,r − 1 T r − 1 ,r 0 0 0 . . . . . . T r,r             upp er triangular r × r , B = − B T an t isymmetric r × r , X = arbitrary r × 2 s (9.18) while an elemen t of the maximal compact subalgebra has instead the following app earance: so ( r ) ⊕ so ( r + 2 s ) ∋ Λ t =     Z Y C  r − Y T Q − Y T  r  r C  r Y −  r Z T  r     (9.19) where Z = − Z T an t isymmetric r × r , C = − C T an t isymmetric r × r , Q = − Q T an t isymmetric 2 s × 2 s . Y = arbitrary r × 2 s (9.20) Ha ving clarified t he structure o f the matrices represen ting Lie algebra elemen ts in this basis w ell adapted to the Tits Satak e pro jection, w e can now discuss a ba sis of generators also w ell adapted to the same pro jection. T o t his effect, let us denote b y I ij the r × r matrices whose only non v anishing en try is the ij -th one whic h is equal to 1 I ij =               0 0 . . . . . . . . . 0 0 0 . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . 0 0 . . . 1 . . . 0 } i-t h row 0 0 . . . . . . . . . 0 0 0 . . . . . . |{z} j-th column . . . 0               . (9.21) 53 Using this notatio n the r non- compact Cartan generators are giv en by H i =     I ii 0 0 0 0 0 0 0 −  r I ii  r     ; ( i = 1 , . . . , r ) . (9.22) Next w e in tro duce the coset generators asso ciated with the lo ng ro ots of t yp e: α = ǫ i − ǫ j . α = ǫ i − ǫ j i < j = 1 , . . . , r ⇒ K ij − = 1 √ 2  E α + E − α  = 1 √ 2     I ij + I j i 0 0 0 0 0 0 0 −  r ( I ij + I j i )  r     (9.23) and the coset generators asso ciated with the long ro ots of type α = ǫ i + ǫ j : α = ǫ i + ǫ j i < j = 1 , . . . , r ⇒ K ij + = 1 √ 2  E α + E − α  = 1 √ 2     0 0 ( I ij − I j i )  r 0 0 0  r ( I j i − I ij ) 0 0     . (9.24) The short ro ots, after the Tits-Sata ke pro jection, are just r , namely ǫ i . Eac h of them, ho wev er, a pp ears with m ultiplicit y 2 s , due to the pain t group. W e in tro duce a 2 s -tuple of coset g enerators asso ciated to eac h of the short roo t s in suc h a w ay that suc h 2 s -t uple transforms in the fundamen tal represen tation of G paint = so (2 s ). T o this effect let us define the rectangular r × 2 s matrices J im analogous to the square matrices I ij , namely J im =              0 0 . . . . . . . . . . . . . . . 0 0 0 . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 . . . 1 . . . . . . . . . 0 } i-th row 0 0 . . . . . . . . . . . . . . . 0 0 0 . . . . . . |{z} m -th column . . . . . . . . . 0              . (9.25) Then w e in tro duce the following coset generators: α = ǫ i i = 1 , . . . , r m = 1 , . . . , 2 s ⇒ K i m = 1 √ 2     0 J im 0 J T im 0 −J T im  r 0 −  r J im 0     . (9.26) The remaining generators of the so ( r, r + 2 s ) a lgebra are all compact and span the subalgebra so ( r ) ⊕ so ( r + 2 s ) ⊂ so ( r , r + 2 s ). According to the no menclature of eq.(9.19) w e in t r o duce 54 four sets of generators. The first set is asso ciated with the long ro ots of t yp e α = ǫ i − ǫ j and is defined as follows : Z ij = 1 √ 2  E α − E − α  = 1 √ 2     I ij − I j i 0 0 0 0 0 0 0 −  r ( I ij − I j i )  r     . (9.27) The second set is asso ciated with the long ro ot s of t yp e α = ǫ i + ǫ j and is defined as follow s: C ij = 1 √ 2  E α − E − α  = 1 √ 2     0 0 ( I ij − I j i )  r 0 0 0 −  r ( I j i − I ij ) 0 0     . (9.28) The third group of compact generators spans the compact coset SO(r + 2s) SO(r) × SO(2s) (9.29) and it is give n b y Y i m = 1 √ 2     0 J im 0 −J T im 0 −J T im  r 0  r J im 0     . (9.30) The fourth set of compact generator s spans the pain t group Lie alg ebra so (2 s ) and is giv en b y Q mn =     0 0 0 0 Q mn − Q nm 0 0 0 0     (9.31) where Q mn denotes the analogue of the I ij in 2 s rather than in r dimensions. By p erforming the c hange of basis to t he blo c k dia g onal fo rm of the matrices w e can v erify tha t C ij − Z ij generate the so ( r ) subalgebra while C ij + Z ij together with Q mn and Y im generate the subalgebra so ( r + 2 s ). The full set of generators is ordered in the following wa y: T Λ =        H i |{z} r , K ij − |{z} 1 2 r ( r − 1) , K ij + |{z} 1 2 r ( r − 1) , K i m |{z} 2 rs , Z ij |{z} 1 2 r ( r − 1) , C ij |{z} 1 2 r ( r − 1) , Y i m |{z} 2 rs , Q mn | {z } s (2 s − 1)        (9.32) and satisfy the trace relation: T r ( T Λ T Σ ) = g ΛΣ , g ΛΣ = 2 diag   + , + , . . . , + | {z } r ( r +2 s ) , − , − , . . . , − | {z } r 2 − r +2 rs +2 s 2 − s   . (9.33) 55 In this w ay w e ha v e obtained the needed and detailed construction of the embedding (6 .1 ) whic h is necessary to apply the in tegration algo rithm. In the next section w e make a detailed study of the case r = 2 , s = 1. 10 A case study for the Tits Satak e pro jection: SO(2 , 4) The simplest example of not maximally split manifold inside t he series defined by eq.(9.1 ) corresp onds to the c hoice: r = 2 , s = 1, namely M 2 , 2 ≡ SO(2 , 4) SO(2) × SO(4) . (10.1) The Tits Satak e pro jection yields the manifold studied at length in section 8 Π T S : SO(2 , 4) SO(2) × SO(4) 7→ SO(2 , 3) SO(2) × SO(3) ∼ Sp(4 , R ) U(2) (10.2) and the pain t group is the simplest p ossible group G paint = SO(2) . (10.3) This ma nif o ld will b e the targ et of our case study in order to illustrate t he b earing of t he Tits Satak e pro jection and the features of the Tits Sa tak e unive rsalit y classes. F ollo wing the discussion of section 9 w e can organize the ro ots in a w ell adapted wa y for the Tits Sata k e pro jection and in tro duce a basis where the solv a ble Lie algebra of the coset is represen ted b y upp er triangular matrices. The ro ot system asso ciated with so (2 , 4) is a ctually that of D 3 ∼ A 3 described by the Dynkin diagram whic h follows: D 3 ✐ α 1 ✐ α 2 ✐ α 3 (10.4) There are 6 p ositiv e ro ots that are vec tors in R 3 and can b e organized as it follows : α 1 , 1 = ǫ 2 − ǫ 3 Π T S − → ǫ 2 ≡ α 1 , α 1 , 2 = ǫ 2 + ǫ 3 Π T S − → ǫ 2 ≡ α 1 , α 2 = ǫ 1 − ǫ 2 Π T S − → ǫ 1 − ǫ 2 ≡ α 2 , α 3 , 1 = ǫ 1 − ǫ 3 Π T S − → ǫ 1 ≡ α 1 + α 2 , α 3 , 2 = ǫ 1 + ǫ 3 Π T S − → ǫ 1 ≡ α 1 + α 2 , α 4 = ǫ 1 + ǫ 2 Π T S − → ǫ 1 + ǫ 2 ≡ 2 α 1 + α 2 . (10.5) 56 In the ab ov e formulae the last three columns describ e the Tits-Satak e pro jection of the ro o t system whic h, in this case, is simply giv en b y the geometrical pro jection of the three–v ectors on t o the plane { 12 } . In this w ay the corresp ondence with the sp (4 , R ) ro ot system b ecomes explicit (compare with fig.9). W e hav e 2 s = 2 preimages of eac h of the short ro ots α 1 and α 1 + α 2 while the long ro ots α 2 and 2 α 1 + α 2 ha ve a single preimage. In complete analog y with eq.(8.20) we can define the appropriate basis fo r the realizatio n of the considered Lie algebra by giving the explicit expression o f the most general elemen t of the solv able Lie algebra S ol v (SO(2 , 4) / SO(2) × SO(4)). Abstractly this is giv en b y T = h 1 H 1 + h 2 H 2 + e 1 , 1 E α 1 , 1 + e 1 , 2 E α 1 , 2 + e 2 E α 2 + e 3 , 1 E α 3 , 1 + e 3 , 2 E α 3 , 2 + e 4 E α 4 . (10.6) W e define the form of a ll Cartan and step generators b y writing the same Lie algebra elemen t (10.6) as a 6 × 6 matrix T =             h 1 + h 2 − √ 2 e 2 − √ 2 e 3 , 1 − √ 2 e 3 , 2 − √ 2 e 4 0 0 h 1 − h 2 − √ 2 e 1 , 1 − √ 2 e 1 , 2 0 √ 2 e 4 0 0 0 0 √ 2 e 1 , 1 √ 2 e 3 , 1 0 0 0 0 √ 2 e 1 , 2 √ 2 e 3 , 2 0 0 0 0 h 2 − h 1 √ 2 e 2 0 0 0 0 0 − h 1 − h 2             (10.7) whic h satisfies the condition (9 .1 0) with the metric η t defined in eq.(9.11). Then in full analogy with eq.s (8.22,8.23) we can construct a basis for the subspace K and for the subalgebra H by writing K 1 = H 1 , K 2 = H 2 , K 3 = 1 √ 2  E α 1 , 1 + ( E α 1 , 1 ) T  , K 4 = 1 √ 2  E α 1 , 2 + ( E α 1 , 2 ) T  , K 5 = 1 √ 2  E α 2 + ( E α 2 ) T  , K 6 = 1 √ 2  E α 3 , 1 + ( E α 3 , 1 ) T  , K 7 = 1 √ 2  E α 3 , 2 + ( E α 3 , 2 ) T  , K 8 = 1 √ 2  E α 4 + ( E α 4 ) T  (10.8) and J 1 = 1 √ 2  E α 1 , 1 − ( E α 1 , 1 ) T  , J 2 = 1 √ 2  E α 1 , 2 − ( E α 1 , 2 ) T  , J 3 = 1 √ 2  E α 2 − ( E α 2 ) T  , J 4 = 1 √ 2  E α 3 , 1 − ( E α 3 , 1 ) T  , J 5 = 1 √ 2  E α 3 , 2 − ( E α 3 , 2 ) T  J 6 = 1 √ 2  E α 4 − ( E α 4 ) T  . (10.9) 57 In this w ay w e hav e constructed 8 + 6 = 14 g enerators. One is still missing to complete a 15- dimensional basis for the Lie algebra so (2 , 4). The missing it em is Q , namely the generator of the pain t group SO(2) Q =             0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0             . (10.10) Naiv ely one might think that the six generators J i defined in (10 .9) close the Lie a lge- bra of so (4 ) , while Q generates the factor so (2) in the denominator group of our manifold SO(2 , 4) SO(2) × SO(4) . F ro m the discus sion of the previous section 9 w e know tha t this is not the case. Indeed the pain t gro up is inside the factor SO ( r + 2s) so that the listed J i constitute a tangen t basis for the coset manifold e P = SO(2) × SO(4) SO(2) paint (10.11) whic h is the unive rsal co v ering o f the true parameter space fo r the integration of our Lax equation. The actual P is obtained from e P b y mo dding out the generalized W eyl group a s stated in equation (5 .8). Ha ving established these notations w e can just pro ceed to the construction of the initial data in the usual w a y . The Cartan subalgebra elemen t is given b y the following 6 × 6 matrix: C =             h 1 + h 2 0 0 0 0 0 0 h 1 − h 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − h 1 + h 2 0 0 0 0 0 0 − h 1 − h 2             (10.12) while the orthogona l ma t r ix O ∈ P c an b e defined in complete ana logy to eq.(8.24): O ( θ 1 , . . . , θ 6 ) = exp h √ 2 θ 1 J 1 i exp h √ 2 θ 2 J 2 i exp [ θ 3 J 3 ] exp h √ 2 θ 4 J 4 i exp h √ 2 θ 5 J 5 i exp [ θ 6 J 6 ] =             O 11 O 12 O 13 O 14 O 15 O 16 O 21 O 22 O 23 O 24 O 25 O 26 O 31 O 32 O 33 O 34 O 35 O 36 O 41 O 42 O 43 O 44 O 45 O 46 O 51 O 52 O 53 O 54 O 55 O 56 O 61 O 62 O 63 O 64 O 65 O 66             . (10.13) 58 W e do not write the explicit functional form of the 36 en tries b ecause it tak es to o m uc h space yet it is clear tha t they are uniquely defined b y the ab ov e equation and b y the explicit form of the generators. W e just go ov er to discuss the W eyl group. 10.1 The generalized W eyl group for SO(2 , 4) Applying the pro cedure of definition 5.1, w e introduce six generators f o r the generalized W eyl group corresp o nding to the reflections with respect to the 6 ro ots. These can b e represen ted as the rotation matrices γ i = O   0 , . . . , 0 , | {z } i − 1 π 2 , 0 , . . . , 0 | {z } 6 − i   ; i = 1 , . . . , 6 (10.14) whic h are in teger v alued. Considering all pro ducts and a ll relations among these generator s w e obtain the finite group W ( so (2 , 4)) whic h ha s 32 elemen ts. The group W ( so (2 , 4)) has a normal subgroup Z 2 × Z 2 ∼ N ( so (2 , 4)) ⊂ W ( so (2 , 4) ) (10.15) giv en b y the follo wing four diagonal matrices: n 1 =             1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1             ; n 2 =             − 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 − 1             , n 3 =             1 0 0 0 0 0 0 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1             ; n 4 =             − 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 − 1             . (10.16) The normal subgroup N ( so (2 , 4)) when acting b y similarit y transformation on a Cartan subalgebra elemen t of the form (10.12) leav es it inv ariant ∀ n ∈ N ( so ( 2 , 4) ) : n T C n = C . (10.17) The order 8 factor group o bt a ined b y mo dding W ( so (2 , 4)) with resp ect to N ( so (2 , 4)) is isomorphic to the W eyl g roup of the Tits Satake subalgebra W eyl ( sp (4)) and has the same action on the eigen v alues h 1 , h 2 W ( so (2 , 4 )) N ( so (2 , 4)) ≃ W eyl ( sp (4) ) . (10.18) 59 A represen tative for eac h of the eigh t equiv alence classes can b e easily written. W e find Λ 1 =             1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1             ; Λ 2 =             0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0             , Λ 3 =             0 0 0 0 0 1 0 − 1 0 0 0 0 0 0 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 − 1 0 1 0 0 0 0 0             ; Λ 4 =             1 0 0 0 0 0 0 0 0 0 − 1 0 0 0 1 0 0 0 0 0 0 − 1 0 0 0 − 1 0 0 0 0 0 0 0 0 0 1             , Λ 5 =             0 1 0 0 0 0 − 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 − 1 0 0 0 0 1 0             ; Λ 6 =             0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0             , Λ 7 =             0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0             ; Λ 8 =             0 0 0 0 1 0 0 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 1 0 0 − 1 0 0 0 0 0 0 1 0 0 0 0             . (10.19) Assem bling the inf o rmation presen ted ab ov e w e come to a stronger conclusion. Not o nly the factor group is isomorphic to the W eyl group of the Tits Sata k e pro jection but ev en the generalized W eyl group is isomorphic. Indeed we hav e found W ( so (2 , 4)) ∼ W ( sp (4) ) . (10.20) W e hav e not prov ed so far that this is tr ue in g eneral but it is an attractive conjecture to p ostulate that W ( U ) ∼ W ( U TS ) . (10.21) W e lea v e the pro of o f suc h a conjecture t o future publications. 60 10.2 V ertices, edges and trapp ed surfaces By means of a computer prog ramme w e can no w study the ve rtices, the links, the critical surfaces and the accessible ve rtices o n each critical surface. W e summarize the results. V ert ices Our parameter space is now a 6 dimensional h yp ercub e tha t has 64 v ertices and 192 edges. On each of the 64 v ertices w e find one o f the 8 W eyl elemen ts whic h ob viously reapp ears sev eral times. Eac h of the o dd elemen ts Λ 1 , 3 , 5 , 7 app ears 4 t imes, while each of the ev en elemen ts Λ 2 , 4 , 6 , 8 app ears 1 2 times so that w e ha v e 4 × 4 + 4 × 12 = 64. The 64 v ertices with their W eyl elemen t corresp ondence are listed in ta ble 7. Edges The one dimensional links connecting t he 64 v ertices are 192 and each of them represen ts a flow from one low er W eyl elemen t to a higher one. A priori we might exp ect that the lines connecting the 8 W eyl elemen ts could now b e more n umerous than in the case of the Tits Satak e pro jected manifold. Ho wev er calculating all these links on a computer w e find that the indep enden t lines are just 16 and the same 16 app earing in the Tits Satak e pro jection. This is made eviden t by the flow diagram displa y ed in fig. 19 L 1 L 2 L 3 L 4 L 5 L 6 L 7 L 8 Figure 19 : The oriente d phase d iagr am of the SO(2 , 4) / SO(2) × SO(4 ) flows. The Lie algebr a so (2 , 4) is not maximal ly split and its T its Satake sub algebr a is sp (4 , R ) ∼ so (2 , 3) ⊂ so (2 , 4) . The r e l e v ant Weyl gr oup is that of the T i ts Satake sub algebr a and the flow diagr am for the inte gr ation of the L ax e quation on this sp ac e just c oincide s with that of sp (4 , R ) . At fixe d value of the Cartan fields if h 1 , h 2 , − h 1 , − h 2 ar e the eigenv a l ues of the sp (4 , R ) L ax op er ator, those of the so (2 , 4) L ax op er ator ar e h 1 + h 2 , h 1 − h 2 , 0 , 0 , − h 1 − h 2 , − h 1 + h 2 . Using { h 1 + h 2 , h 1 − h 2 } as c o o r d inates to ide ntify the Weyl ele m ent we o b tain the pr esente d flow gr aph. T rapp ed surfaces and accessible vertices The trapp ed h yp ersurfaces in parameter space are obta ined b y equating to zero the minors obta ined by in tersecting the first k columns 61 of O with an equal n um b er of arbitrar ily c hosen rows . In this wa y w e generate a to tal of 62 trapp ed surfaces. They are en umerated a s follo ws: Order of the minor Number 5 6 4 15 3 20 2 15 1 6 62 (10.22) W e can no w calculate the set of accessible W eyl elemen ts fo r each of these 62 surfaces and within t he accessible set w e can single out the low est and the hig hest W eyl elemen ts whic h will corresp ond to the initial and final end p oin ts of the flows confined on that surfa ce. The result is displa y ed in table 8. Inspection of this list reve als that the av ailable flo ws, although rep eated on man y differen t surfaces are just a small set of fiv e p ossibilities, exactly the same fiv e p ossible flow s app earing in the the case of the Tits Satake pro jection that w ere sho wn in eq.(8.33) This concludes our discussion. As w e ha v e seen the vertice s and the p ossible flows on critical links o r trapp ed hy p ersurfaces do not dep end on the c hosen represen tativ e within a Tits Satake univ ersalit y class rather they dep end only on the class. In other w o rds the study of the maximally split Tits Sata ke pro jection already provides us with a complete picture of all possible flow s. It is only the detailed structure of b ouncing whic h v aries from one represen tativ e to the other. 10.3 Examples of flo ws for SO(2 , 4) W e come no w to the analysis of t w o explicit examples of flo ws aiming at illustrating three asp ects: a The em b edding of the Tits Satake flows within t he flows of the bigger coset manifold. b The role of the extra parameters not contained in the Tits Satak e pro jection. c The instability of sup er-critical and in g eneral of trapp ed surfaces. T o this effect w e shall reconsider t he case analyzed in section 8.4.3 of a flow on the critical surface Σ 2 for the Tits Satake pro jection of SO(2 , 4), namely Sp(4 , R ) ∼ SO(2 , 3). The unp erturb ed sup er-critical flow The c hoice of the Euler angles is that of eq.(8.46) whic h, in the fiv e dimensional represen tation SO(2 , 3) pro duces the matrix of eq.(8.47) . This latter has three v anishing minors, a s shown in eq.(8.49). It is quite easy t o em b ed this case and the corresp onding flow into the non maximally split represen tative SO(2 , 4) of the same univ ersality class. It suffices to choose the six θ i angles as follow s: { θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6 } = n π 3 , 0 , π 6 , π 3 , 0 , π 3 o (10.23) 62 1) ; { 0 , 0 , 0 , 0 , 0 , 0 } = Λ 1 2) ; { 1 , 0 , 0 , 0 , 0 , 0 } = Λ 5 3) ; { 0 , 1 , 0 , 0 , 0 , 0 } = Λ 5 4) ; { 0 , 0 , 1 , 0 , 0 , 0 } = Λ 4 5) ; { 0 , 0 , 0 , 1 , 0 , 0 } = Λ 8 6) ; { 0 , 0 , 0 , 0 , 1 , 0 } = Λ 8 7) ; { 0 , 0 , 0 , 0 , 0 , 1 } = Λ 3 8) ; { 1 , 1 , 0 , 0 , 0 , 0 } = Λ 1 9) ; { 1 , 0 , 1 , 0 , 0 , 0 } = Λ 6 10) ; { 1 , 0 , 0 , 1 , 0 , 0 } = Λ 2 11) ; { 1 , 0 , 0 , 0 , 1 , 0 } = Λ 2 12) ; { 1 , 0 , 0 , 0 , 0 , 1 } = Λ 7 13) ; { 0 , 1 , 1 , 0 , 0 , 0 } = Λ 6 14) ; { 0 , 1 , 0 , 1 , 0 , 0 } = Λ 2 15) ; { 0 , 1 , 0 , 0 , 1 , 0 } = Λ 2 16) ; { 0 , 1 , 0 , 0 , 0 , 1 } = Λ 7 17) ; { 0 , 0 , 1 , 1 , 0 , 0 } = Λ 6 18) ; { 0 , 0 , 1 , 0 , 1 , 0 } = Λ 6 19) ; { 0 , 0 , 1 , 0 , 0 , 1 } = Λ 2 20) ; { 0 , 0 , 0 , 1 , 1 , 0 } = Λ 1 21) ; { 0 , 0 , 0 , 1 , 0 , 1 } = Λ 6 22) ; { 0 , 0 , 0 , 0 , 1 , 1 } = Λ 6 23) ; { 1 , 1 , 1 , 0 , 0 , 0 } = Λ 4 24) ; { 1 , 1 , 0 , 1 , 0 , 0 } = Λ 8 25) ; { 1 , 1 , 0 , 0 , 1 , 0 } = Λ 8 26) ; { 1 , 1 , 0 , 0 , 0 , 1 } = Λ 3 27) ; { 1 , 0 , 1 , 1 , 0 , 0 } = Λ 4 28) ; { 1 , 0 , 1 , 0 , 1 , 0 } = Λ 4 29) ; { 1 , 0 , 1 , 0 , 0 , 1 } = Λ 8 30) ; { 1 , 0 , 0 , 1 , 1 , 0 } = Λ 5 31) ; { 1 , 0 , 0 , 1 , 0 , 1 } = Λ 4 32) ; { 1 , 0 , 0 , 0 , 1 , 1 } = Λ 4 33) ; { 0 , 1 , 1 , 1 , 0 , 0 } = Λ 4 34) ; { 0 , 1 , 1 , 0 , 1 , 0 } = Λ 4 35) ; { 0 , 1 , 1 , 0 , 0 , 1 } = Λ 8 36) ; { 0 , 1 , 0 , 1 , 1 , 0 } = Λ 5 37) ; { 0 , 1 , 0 , 1 , 0 , 1 } = Λ 4 38) ; { 0 , 1 , 0 , 0 , 1 , 1 } = Λ 4 39) ; { 0 , 0 , 1 , 1 , 1 , 0 } = Λ 4 40) ; { 0 , 0 , 1 , 1 , 0 , 1 } = Λ 8 41) ; { 0 , 0 , 1 , 0 , 1 , 1 } = Λ 8 42) ; { 0 , 0 , 0 , 1 , 1 , 1 } = Λ 3 43) ; { 1 , 1 , 1 , 1 , 0 , 0 } = Λ 6 44) ; { 1 , 1 , 1 , 0 , 1 , 0 } = Λ 6 45) ; { 1 , 1 , 1 , 0 , 0 , 1 } = Λ 2 46) ; { 1 , 1 , 0 , 1 , 1 , 0 } = Λ 1 47) ; { 1 , 1 , 0 , 1 , 0 , 1 } = Λ 6 48) ; { 1 , 1 , 0 , 0 , 1 , 1 } = Λ 6 49) ; { 1 , 0 , 1 , 1 , 1 , 0 } = Λ 6 50) ; { 1 , 0 , 1 , 1 , 0 , 1 } = Λ 2 51) ; { 1 , 0 , 1 , 0 , 1 , 1 } = Λ 2 52) ; { 1 , 0 , 0 , 1 , 1 , 1 } = Λ 7 53) ; { 0 , 1 , 1 , 1 , 1 , 0 } = Λ 6 54) ; { 0 , 1 , 1 , 1 , 0 , 1 } = Λ 2 55) ; { 0 , 1 , 1 , 0 , 1 , 1 } = Λ 2 56) ; { 0 , 1 , 0 , 1 , 1 , 1 } = Λ 7 57) ; { 0 , 0 , 1 , 1 , 1 , 1 } = Λ 2 58) ; { 1 , 1 , 1 , 1 , 1 , 0 } = Λ 4 59) ; { 1 , 1 , 1 , 1 , 0 , 1 } = Λ 8 60) ; { 1 , 1 , 1 , 0 , 1 , 1 } = Λ 8 61) ; { 1 , 1 , 0 , 1 , 1 , 1 } = Λ 3 62) ; { 1 , 0 , 1 , 1 , 1 , 1 } = Λ 8 63) ; { 0 , 1 , 1 , 1 , 1 , 1 } = Λ 8 64) ; { 1 , 1 , 1 , 1 , 1 , 1 } = Λ 2 T a ble 7: V ertices/W eyl group corresp ondence fo r the case SO(2 , 4). 63 Σ 1 { w 6 , w 8 } Σ 2 { w 5 , w 8 } Σ 3 { w 5 , w 8 } Σ 4 { w 5 , w 8 } Σ 5 { w 5 , w 8 } Σ 6 { w 5 , w 7 } Σ 7 { w 5 , w 8 } Σ 8 { w 5 , w 8 } Σ 9 { w 5 , w 8 } Σ 10 { w 5 , w 8 } Σ 11 { w 5 , w 8 } Σ 12 { w 5 , w 8 } Σ 13 { w 1 , w 8 } Σ 14 { w 5 , w 8 } Σ 15 { w 5 , w 8 } Σ 16 { w 5 , w 8 } Σ 17 { w 5 , w 8 } Σ 18 { w 5 , w 2 } Σ 19 { w 5 , w 8 } Σ 20 { w 5 , w 8 } Σ 21 { w 5 , w 8 } Σ 22 { w 5 , w 8 } Σ 23 { w 5 , w 8 } Σ 24 { w 5 , w 8 } Σ 25 { w 5 , w 8 } Σ 26 { w 5 , w 8 } Σ 27 { w 1 , w 8 } Σ 28 { w 5 , w 8 } Σ 29 { w 5 , w 8 } Σ 30 { w 5 , w 8 } Σ 31 { w 5 , w 8 } Σ 32 { w 5 , w 8 } Σ 33 { w 5 , w 8 } Σ 34 { w 5 , w 2 } Σ 35 { w 5 , w 8 } Σ 36 { w 5 , w 8 } Σ 37 { w 5 , w 8 } Σ 38 { w 5 , w 8 } Σ 39 { w 5 , w 8 } Σ 40 { w 5 , w 8 } Σ 41 { w 5 , w 8 } Σ 42 { w 5 , w 8 } Σ 43 { w 5 , w 8 } Σ 44 { w 5 , w 8 } Σ 45 { w 1 , w 8 } Σ 46 { w 5 , w 8 } Σ 47 { w 5 , w 8 } Σ 48 { w 5 , w 8 } Σ 49 { w 5 , w 8 } Σ 50 { w 5 , w 2 } Σ 51 { w 5 , w 8 } Σ 52 { w 5 , w 8 } Σ 53 { w 5 , w 8 } Σ 54 { w 5 , w 8 } Σ 55 { w 5 , w 8 } Σ 56 { w 5 , w 8 } Σ 57 { w 6 , w 8 } Σ 58 { w 5 , w 8 } Σ 59 { w 5 , w 8 } Σ 60 { w 5 , w 8 } Σ 61 { w 5 , w 8 } Σ 62 { w 5 , w 7 } T a ble 8: Initial and final endp oin t s of flo ws confined on tr app ed surfaces for the case SO(2 , 4). 64 since, according to eq.(10.5) and (1 0.9), the a ngles θ 2 and θ 5 corresp ond to the second cop y of the compact generators resp ectiv ely asso ciated with the first and the third of the sp (4) ro ots. The result of this choice is the follo wing matrix in SO(2) × SO(4) ⊂ SO(2 , 4): O unp =              √ 3 16 5 16 − 3 4 √ 2 0 − 3 16 − 7 √ 3 16 − 19 32 11 √ 3 32 3 √ 3 2 8 0 − 5 √ 3 32 − 3 32 3 √ 3 2 16 15 16 √ 2 − 1 8 0 15 16 √ 2 3 √ 3 2 16 0 0 0 1 0 0 − 3 32 − 5 √ 3 32 3 √ 3 2 8 0 11 √ 3 32 − 19 32 − 7 √ 3 16 − 3 16 − 3 4 √ 2 0 5 16 √ 3 16              . (10.24) As one sees, by deleting the 4th row and the 4th column one retriev es the mat rix of eq.(8.47) . Indeed the matrix (10 .2 4) is manifestly inside the Tits Sata k e subgroup SO( 2 , 3) ⊂ SO(2 , 4). If w e use O unp as initial data for o ur inte gration algo rithm implemen ted on a computer w e find that the asymptotic limits are Λ 1 at past infinit y and Λ 8 at future infinit y just as in the original case discussed in section 8.4 .3. Consider fig.20. It displays the plot of the Cartan fields pro jected a lo ng the ro ot α 1 and clearly demonstrates that there are just tw o b ounces on the w a ll or t hogonal to this ro ot. Bo th of them o ccur in a narrow time range around t = 0. A t v ery early and v ery late times there are no more b ounces and the result is the tra jectory of the cosmic ball displa y ed in fig.21. The t wo asymptotic lines (incoming and outgoing) are the Kasner ep o c hs Λ 1 and Λ 8 , resp ectiv ely . -2 -1 1 2 -7.6 -7.5 -7.4 -7.3 -7.2 -7.1 -6.9 -6 -4 -2 2 -8 -6 -4 -2 Figure 20: Plo t of the α 1 · h p r oje ction for the SO(2 , 4) flow gener ate d by the p ar ameter choic e { θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6 } =  π 3 , 0 , π 6 , π 3 , 0 , π 3  . This is actual ly a flow in the Ti ts Satake submanifold and c orr esp onds to a sup er-critic al surfac e. This sup er-critic al flow c onne cts the primor dial Kasner er a Λ 1 to the r emote futur e Kasne r er a Λ 8 . The p lot on the left and on the right ar e the sa m e. The on l y differ enc e is that on the right we have an enl a r gement of the time r e gion ar ound t = 0 , while on the left we c onside r a time r ang e c ove ring a much wider p ortion of the e arly ep o c h s. 65 3 4 5 6 7 h1 11 12 13 14 h2 Figure 21: T r aje ctory of the c os m ic b al l in the SO(2 , 4) flow gene r ate d b y the p ar ame ter choic e { θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6 } =  π 3 , 0 , π 6 , π 3 , 0 , π 3  . This flow is inside the Tits Satake submanifold and c o rr e sp onds to a sup er-critic al surfac e. It c onne cts the primor dial Kasner er a Λ 1 to the r emote futur e K a s ner er a Λ 8 . P erturbing the sup er-critical flow wit h pain t ed wa lls In order to illustrate b oth the nature o f the Tits Sata ke pro jection and the instabilit y of sup er-critical surfaces w e consider no w a small p erturbation of the initial da t a used in the previous example. Keeping all the other angles unc hanged w e shift from zero the angle θ 2 b y a very small amount. Explicitly w e c ho ose θ 1 = π 3 , θ 2 = arcsin  1 100  , θ 3 = π 6 , θ 4 = π 3 , θ 5 = 0 , θ 6 = π 3 . (10.25) As explained θ 2 is asso ciated with the second cop y of the r o ot α 1 . Hence introducing this small angle is equiv alen t to creating a new α 1 w all just pa inted with a differen t color. This new wall is very v ery small and therefore it will pro duce v ery little effects at finite times. Y et it is sufficien t to remo ve us from the sup er-critical surface and t his necessarily changes asymptotics. Instead of Λ 1 w e exp ect no w Λ 5 at past infinit y . It is interes ting to ana lyze in detail ho w this happ ens. If w e name O per t the mat rix corresp onding to the choice of angles (10.25) w e can appre- ciate the p erturbation of initial data b y writing O per t in the follow ing w a y: O per t = O unp + ǫ 1             0 0 0 0 0 0 0 0 0 − 6 0 0 0 0 0 6 √ 6 0 0 − 15 2 − 9 √ 3 2 3 √ 6 0 − 9 √ 3 2 − 15 2 0 0 0 − 6 0 0 0 0 0 0 0 0             66 + ǫ 2              0 0 0 0 0 0 − 5 16 − 3 √ 3 16 √ 3 2 4 0 − 3 √ 3 16 − 5 16 5 √ 3 2 8 9 8 √ 2 − 3 4 0 9 8 √ 2 5 √ 3 2 8 0 0 0 2 0 0 − 5 16 − 3 √ 3 16 √ 3 2 4 0 − 3 √ 3 16 − 5 16 0 0 0 0 0 0              , ǫ 1 ≃ 1 . 2 × 10 − 3 , ǫ 2 ≃ 1 . 0 × 10 − 4 . (10.26) Then we can implemen t the in tegration algorithm on our computer and calculate the asymp- totic v alues of the Lax op erator . Not withstanding the smallness of the p erturbatio n, the past infinit y regime j umps from Λ 1 to Λ 5 as exp ected, while at futur e infinit y it remains Λ 8 whic h is already the highest p ossible W eyl elemen t. W e can a ppreciate the mec hanism whic h r e- alizes this effect b y lo oking at fig.s 22 and 23. There is an extra b ounce on the α 1 w all as -1.5 -1 -0.5 0.5 1 1.5 -2.2 -2.1 -1.9 -1.8 -1.7 -1.6 -1.5 -6 -4 -2 2 -3 -2 -1 1 2 Figure 22: Plo t of the α 1 · h p r oje ction for the SO(2 , 4) flow gener ate d by the p ar ameter choic e { θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6 } =  π 3 , arcsin  1 100  , π 6 , π 3 , 0 , π 3  . This flow is a p erturb ation of a sup er-c ritic a l flow. The s h ift fr om Λ 1 to Λ 5 at p ast infinity o c curs via an extr a b ump on the α 1 wal l which o c curs at very e a rl y times. This bump is not vis i b le in the pl o t on the right which is in the r ange n e ar t = 0 but it is evident in the plo t on the left which go e s further b ack i n time. This pictur e is to b e c omp ar e d with fi g .20. w e exp ected whic h corrects the tra jectory and directs the cosmic ball to Λ 5 rather than Λ 1 when w e go back in time. Since the p erturbat io n is small t his b ounce o ccurs a t v ery early times so that for most of the time the flo w is almost on the critical surface. The smaller t he p erturbation, the earlier the o ccurrence of the primev al b ounce. It should also b e noted that w e w ould ha v e obtained exactly the same effe ct if w e had perturb ed the θ 1 angle instead of the θ 2 . Indeed they are asso ciated with the same ro ot . This is the meaning of the Tits Satak e pro jection whic h captures all the essen t ia l features of the dynamical pro cesses for the en tire univ ersality class. 67 7 8 9 10 h1 10 11 12 h2 2 4 6 8 10 h1 -2 2 4 6 8 10 12 h2 Figure 23: Comp arison o f the tr aje ctories of the c osmic b al l in the SO( 2 , 4) flow gene r a te d by the p ar ameter choic e { θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6 } =  π 3 , 0 , π 6 , π 3 , 0 , π 3  and i n its p erturb ation by a smal l θ 2 = arcsin  1 100  . The thin line is the unp erturb e d flow, the fatter line is the p erturb e d one. The first plot c overs a time r ange ar ound t = 0 w h i le the se c ond plo t extends much e arli e r in time. The additional b ounc e r esp onsible for the c hanging of asymptotic is visible in the se c ond plot. P art I I I P ers p ectiv es 11 Summary of results In this pap er w e hav e made a few steps forw ard in dev eloping the general programme of sup ergra vit y billiards as a paradigm for sup erstring cosmology . O ur results are b oth of ph ysical and mathematical natur e. On the phys ical side, which for us means sup ergravit y/sup erstring theory , the essen tial p oin ts are the following ones: 68 1) W e hav e sho wn t hat all s up ergrav it y billiards are completely in tegr a ble, irresp ectiv ely whether they are defined on a maximally split coset manifold U / H as it happ ens in the c ase of maximal su sy or a non maximally split U / H, as it happ ens in all lo w er sup ersymmetry cases. W e ha ve pro vided the explicit in tegration algorithm whic h just dep ends on the triangula r em b edding of the solv able Lie algebra S ol v (U / H) into that of S ol v (SL(N) / SO ( N) ) . 2) W e ha v e discov ered a new principle of time orien tation of the cosmic flow which relies on the natural ordering of the W eyl group elemen ts (or of the p erm uta t ions) according to their length ℓ T in terms of transp ositions. Cosmic ev olutio n is alw ays in the direction o f increasing ℓ T whic h play s the role of an entrop y . There is a fascinating similarity , in this con text b etw een the laws of cosmic ev olution a nd those of blac k hole thermo dynamics. 3) W e hav e clarified the meaning of Tits Satake univ ersality classes, in tro duced in [18], at least from the v an tag e p oint of cosmic billiards. The a symptotic states, the type of a v aila ble flows and the critical surfaces in para meter space ar e prop erties of the class and do not dep end on the represen tativ e manifold in the class. On the mathematical side the highlig h ts of our pap er are the follow ing ones: 1) W e hav e introduced the notion of generalized W eyl group for a non compact symmetric space U / H and shown that the f a ctor gro up with resp ect to its nor ma l subgroup is just the W eyl group of the Tits Satak e subalgebra U T S ⊂ U . Moreo v er, w e hav e demonstrated that not only the factor gro up is isomorphic to the W eyl gro up of the Tits Sata ke pro jection but ev en the g eneralized W eyl gro up is also isomor phic W ( U ) ∼ W ( U TS ). A t least this is true in the considered examples and w e mak e the conjecture that it is true in general. 2) W e ha ve established a remark able conjecture enco ded in prop erty 5.1 o f the main text: the constraints on minors of the diago nalizing orthog o nal matrix for the La x op erator comm ute with the T o da flow . 3) W e ha v e pro p osed a v ery simple efficien t metho d of calculating the T o da flow asymptotics at t = ±∞ for the Lax op erato r of a σ -mo del with target space an y non compact- symmetric coset space U / H. Our algorithm requires only the know ledge of the corre- sp onding W eyl group W eyl( U ) as w ell as that of the small gro up H. 4) W e hav e p o sed the question ho w the equations cutting out a lgebraic lo ci in compact group or coset manifolds and defined in terms of v a nishing minors in the defining represen tation can b e lifted to the abstract group lev el and extended to all irreducible represen tations. 12 Op en problems and directi o ns to b e pursue d The results w e ha v e obtained are just steps ahead in a programme to b e further dev elop ed. They ha v e solved some standing pro blems and op ened new dir ections of inv estigations which seem to us quite exciting. W e just men tion, as conclusion, the milestones w e would lik e to 69 attain in the near f uture, ev aluated fr o m the view-p oint whic h w as generated b y our presen t results: 1) Construction of all the triangular em b eddings for all the solv able Lie algebras of all sup ergra vit y mo dels and corresp onding construction o f the in tegra tion alg orithm for all sp ecial homogeneous geometries. 2) Oxidation and ph ysical interpre tation of the T o da flo ws w e hav e just shown ho w to construct within sup ergra vity mo dels as those coming fro m string compactific ations on manifolds of restricted holono my . 3) Extension of the in tegra t io n algorithm t o sigma mo dels with a p o t en tial emerging from flux compactifications and gauged superg r avities in higher dimensions. 4) Study of the in tegration algorithm in affine and hyperb olic Ka ˇ c-Mo o dy extensions of the symmetric space U / H, as they emerge from stepping do wn to D = 2 a nd D = 1 dimensions. 5) Comparison b et we en the law of increasing ℓ T and the second principle of blac k hole mec hanics in searc h of an adequate formulation of cosmological thermo dynamics and of a p ossible mapping b etw een the a ttractor mec hanism in black hole phy sics and T o da flo ws in cosmic billiards. 6) More in depth study of the top ology of parameter space H / W ( U ) a nd in particular its partition in complex h ulls t hat admit the trapp ed surfaces as w alls. 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