JHEP05(2008)079 May 7, 2008 May 22, 2008 vity Models. February 19, 2008 eduction to three Accepted: Published: Received: that the bosonic sector of the sponding non-linear realisation. c manifolds. We conjecture that ed Kac-Moody algebras for suitable eit Leuven, [email protected] , Published by Institute of Physics Publishing for SISSA Space-Time Symmetries, Extended Supersymmetry, Supergra We consider all theories with eight supersymmetries whose r [email protected] [email protected] Department of Mathematics, King’sStrand, College London, London, WC2R 2LSE-mail: U.K. Instituut voor Theoretische Fysica, Katholieke Universit Celestijnenlaan 200D B-3001 LeuvenE-mail: Belgium Fabio Riccioni and Peter West Antoine Van Proeyen Abstract: dimensions gives rise to scalarsthese that theories parametrise are symmetri non-linear realisationschoices of of very-extend real forms. Wesupersymmetric show theory for is the precisely most reproduced interesting by cases the corre Keywords: Real forms of verytheories extended with Kac-Moody eight algebras supersymmetries and JHEP05(2008)079 d 6 1 17 14 19 14 22 27 +++ 2 G [1], and eralised to 2 − +++ D ity theory, the D h is a six dimensional +++ 4 F te dimensional semi-simple n-linear realisation of [4]. Indeed, it was proposed eir dimensional reductions. It rsymmetry, but the non-linear f an algebra originally denoted rees with that of the theory to 3 metries [1]. Also in all the other − tion of +++ D A , then the very extended algebra is G algebras [5] and their low level content +++ G – 1 – , is at low levels the bosonic sector of maximal +++ G +++ 8 E = 2 supergravity in five dimensions, that is invariant even N P , i.e. ), 11 is just one of a class of special algebras called very extende E 1) , P 1) , 11 , (0 E (8 dimensions is associated with L (4 L L D and . It was proposed that the bosonic string effective action gen conjecture [1] concerned the eleven dimensional supergrav and and 11 5) 24) +++ E +4(4) − − G 2 P +++ +++ +++ 7( 8( D E E in the 26-dimensional case and which was later identified wit 0) theory, which again has eight supersymmetries [6]. , 27 4.3 4.2 theories with eight supersymmetries 4.1 K = (1 dimensions is associated with the non-linear realisation o similarly gravity in D denoted by under eight supersymmetries, while the non-linear realisa by supergravity theories invariant under thirty two supersym cases the low levelwhich content it of is the non-linear associated realisation with ag [1, 2, 4, 6]. In particular, the no was found in [6].realisation Clearly associated not all with these theories possess supe to consider non-linear realisations for all N has the bosonic field content of algebras [3]. SuchLie algebras algebra. can be constructed If from the each latter fini algebra is denoted by 4. Field content of real forms of 1. Introduction The original 2. Review of theories with3. eight supersymmetries Relationship between Kac-Moody algebras and 3 Contents 1. Introduction ten dimensional IIA and IIBwas supergravity later theories realised [2] that and th 5. Discussion A. The calculation of the form fields JHEP05(2008)079 8) − m odd [7] and the e theories arises n = 1 supergravity the very extended N for ) of this algebra with rs of compact genera- is associated with the , but while the bosonic 1) ra used to construct the m 2 − in their maximally non- s are linked under dimen- 8) with local sub-algebra n m,m ( D mmetries that exist in six gebras have come up in the the first three nodes of the +++ m ing local sub-algebra of the ase for the ten dimensional 15]. Different real forms of − ubgroup. The coefficients of imally non-compact. +++ gebra used in the non-linear 8+ O( s that arise has been studied. ong have been the subject of ly, if one uses a different real D l be the group element in the e local sub-algebra. Clearly, a orm being considered and the rmations of the algebra which m B which for this real form is the ber of non-compact generators, cal equations. Hence, different , certain supergravity theories which is associated to the very field content. Furthermore, the f cosmological billiards [16 – 18]. 2 n a general group element which re physically inequivalent. , 1, vector fields then the associated ≥ = 1 supergravity coupled to 2( n N even and = 1 supergravity coupled to one vector, n , for n N for and related conjectures an important stepping – 2 – 2 n 11 E +++ 8+ D ), ten-dimensional m SO( we find the finite dimensional algebra ⊗ . However, if one adds ) +++ m m + 2 dimensions and also with ten-dimensional D +++ 8 8) [7]. m B 8) Yang-Mills multiplets. The different field content of thes − − , as well as ten-dimensional m m +++ 8 SO(2 D ⊗ This is apparent in the above examples as the algebra Different real forms of a complex algebra have different numbe In all the above cases the real form of the very extended algeb The way the cosets and the corresponding non-split real form In the original understanding of the Different real forms of finite dimensional semi-simple Lie al = 1 supergravity theory which was found to be associated with bosonic string in tors and so differentrealisation compact is sub-algebras. the maximal Thefields compact local in sub-algebra the sub-al of non-linear realisation the aredepends real found on f by space-time writing and dow removethe parts generators of that it remain using areform, the the local the fields s local of sub-algebra thenon is theory. linear different Clear realisation and and asdynamics consequently a the is result corresponding so just wil thatis which used is to invariant define under thedifferent rigid non-linear choice transfo realisation of and local also sub-algebrareal under also forms th lead affects to the different dynami non-linear realisations which a required real form of these very extended algebras is not max from the fact thatDynkin the diagram real of forms used are different. Deleting coupled to 2( string corresponds to the maximally non-compact real form S extended algebra corresponding to the addition of one Yang-Mills multiplet, non-linear realisations is the one whichcalled has the the maximal num maximally non-compactnon-linear real realisation form. is The theone correspond maximal invariant compact under sub-algebra theN Cartan involution. This is also the c algebra local sub-algebra SO( very extended Kac-Moody algebra is sional reduction or oxidationsemi-simple have Lie algebras been have also studied occurred in in the [8, context 9, o 12 – Yang-Mills multiplets corresponds to the real form SO(8 context of supergravity theorieshave in scalar the fields past. thatcompact In belong form. particular to cosets Thedimensions and of supergravity less groups theories and that with themuch study. spaces are eight In to not supersy particular which in their [8 – scalars 11] bel the geometry of the space SO(8) JHEP05(2008)079 , of H factor 2 and one A group +++ . Hence if one G H w of theories with tries whose scalars where the Dynkin diagram. This G and its real form one will is the internal symmetry ⊗ ort cut. The three dimen- esentations of the internal -linear realisation of a very classified in [12]. We con- 2 a is found by decomposing ead to precisely the correct G s of the very extended Kac- ymmetries and in particular as eleven dimensional super- A s. In section 3 we derive the heories exactly coincides with rs of the algebra that turn up rsymmetries. In section 4 we +++ ended Kac-Moody symmetry. ty are always found to belong ra is taken to be the smallest ree dimensional theory contains ear realisation of G and a subgroup Moody algebra associated with a ensions, however the field content which tells us which real form of from the scalar coset and thus the G G G by dimensionally reducing the theory in as this is inherited from the real form of G +++ – 3 – G of the higher dimensional theory. The reduction dimensions starting from a decomposition of the +++ D G non-linear realisation. Section 5 contains a discussion . +++ associated with different real forms of various Lie algebras G +++ -forms in G p one can find the group +++ G +++ G that turns up in the three dimensional theory. Indeed the sub G into the algebras that result in deleting the affine node of its The paper is organised as follows. In section 2 we give a revie In this paper we consider the theories with eight supersymme Having chosen the very extended Kac-Moody algebra arose. +++ is the node usually labelled three. The resulting algebra is gravity, are non-linear realisations.such algebra The that Kac-Moody contains all algeb in the the generators non-linear and realisation. commutato Ifgiven one theory suspects is that a the very Kac- sional extended theory algebra arising then one fromG can a adopt very a extended sh Kac-Moody algebr stone was the result [19] that the theories under study, such is associated with three dimensional gravity and the factor conjectured very extended algebra group in three dimensions. Atonly lowest gravity levels the and resulting scalars th to which a for coset theories space with or supergravi non-linear realisation for the group suspects that a theory hasextended algebra a formulation at low levels as a non eight supersymmetries in six,Kac-Moody five, algebras four and three dimension parametrise symmetric manifolds.jecture These that theories all have theseWe been identify theories this have Kac-Moody an symmetriesfield and underlying content verify for very-ext that some they of l the theories of most interest. can test if this agrees with the theory in question or not. question to three dimensions and read off the group and conjecture their relationshow that with the theories bosonic field withthe content eight of one these supe supersymmetric obtained t from the also provides the real form of the algebra adjoint representation of automatically recover the correct scalar cosetin in all three other dim dimensions is uniquely predicted by the non-lin symmetry group for all of our results. An appendix explains how one obtains the repr the algebra G the scalar coset is the maximal compact subgroup of 2. Review of theories with eightIn supersymmetries this paper wetheir are field interested content in in theories order with to eight compare supers it with the prediction JHEP05(2008)079 . ). a ave aαi ψ ,B a µ rality is (2) sym- e the spinors USp . mposed, so that the s, theories with eight and a gravitino (5) V 2 inimal because a spinor ) a n 1 − ajorana condition can be a ( sed on a single irreducible )) (2.1) nal gravity multiplet gives s, they are usually called 2-form is self-dual and the ss as is readily apparent if iplets both give rise to the lso a hyper-multiplet which A and a symplectic Majorana- (6) V ctor multiplet, which has the theory to five dimensions has sider this and its dimensional e chiral. The six-dimensional en by multiplet is given by a 2-form he gravitino is a ymplectic Majorana condition ns are often called symplectic orm is dual to a vector in five a n eed in six dimensions a Dirac mmetric theories in six dimen- s the bosonic content ( ( A a rbitrary )) (2.2) A ); (5) V n (6) T ( , a 2-form n a ( , ϕ µ 2 index. In six dimensions there are also ) is real, while the dimensional reduction to e , , φ ) (5) V ϕ n (6) T ( = 1 n a – 4 – i ( 2 , A a 1 a (+) a A ,B ; a 0) theories to denote the fact that the corresponding 2 the number of tensor and vector multiplets respectively, µ ) , a in brackets denote the number of such fields. The scalars e 1 − ( [10]. ( a (6) V (6) V n ) 1) , A , n = (1 a ) where the scalar (6) T (6) T µ n e n N and , ϕ ( and a + 1. In the case in which the five-dimensional theory does not h SO( SO( A (6) T (6) T (6) V , together with a symplectic Majorana-Weyl spinor whose chi n n n φ (2) doublet of spinors, leading again to 8 real components if + (6) T USp n = is real. The vector multiplet consists of a vector (5) V φ n and a scalar 2 We now consider the five dimensional case. As in six dimension In six dimensions theories with eight supersymmetries are m If we denote by a = 2 theories. The dimensional reduction of the six-dimensio 1 (+) a supersymmetries are minimalN in five dimensions. Nonetheles Moody algebras. Such theories exist in dimensions six and le one considers the representationsspinor of has Clifford 8 algebras.corresponding complex Ind irreducible components, spinor andimposed has a on 8 Weyl a real condition components. can be A i M in five dimensions the gravitybosonic multiplet field together content with ( one ve plectic Majorana-Weyl spinor, hence the supergravity multiplet consists of the vierbein supercharge is a Weyl spinor and therefore these theories ar The 2-form has a field strength which is anti-self dual while t five dimensions of thevector six multiplet dimensional in five tensor dimensions,dimensions. and using vector The the five mult fact dimensional supergravity that multiplet a ha 2-f with eight real components issions irreducible. are Minimal usually supersy called are Weyl. Spinors satisfyingMajorana this spinors. type of In Majorana fivespinor, dimensions which conditio has no 8 condition real components. cancan As be be in impo six imposed dimensions, on a s a doublet of spinors in five dimensions. where a six-dimensional origin, still equation (2.2) holds with a As a result the dimensionalthe reduction field of content the six dimensional tensor and vector multiplets. TheA field content of the tensor opposite to the onescalar of the gravitino. The field strength of the Weyl spinor of theconsists same of chirality two of fermions the andreductions gravitino. four further scalars, There in but this is we paper. will a not con where the numbers then the bosonic content of the six dimensional theory is giv parametrise the coset JHEP05(2008)079 rise (2.9) (2.8) ained (2.16) . t which consists = 4 supergravity = 3) (2.12) = 6) (2.13) = 9) (2.14) = 15) (2.15) = 27) D (4) V (4) V (4) V (4) V (4) V at a supersymmetric n n n n n the manifold that the ( = 2, . N ng time ago in [8], and the n which the scalar manifold to = 26) (2.7) = 8) (2.5) = 14) (2.6) = 5) (2.4) educible spinor representation ametrised by the scalars have = 2 Yang-Mills theory, has the sional theory has no six or five- 3 , in [21] the following possible )) ries with eight supersymmetries (3) ( (1) (2.11) (6) ( (5) V (5) V (5) V (5) V (3)] ( (1)] (2.10) N (4) V (1)] n n n n U U U n U U ( + 2. The field content is SO(2) ( / / / U are real. The dimensional reduction )( ) × × 2 / 2 1) × (6) V ) , φ 6 , R 1) n iφ (12) . , (3) 4 ∗ (4) V + 1) (2.3) ± /E n U (6) ( , [ /F 1 and ) SU(1 Sp(6 SO (6) T φ 26) ) 1 ( SU(3) ( /S SO(3) ( 26) n [SU(1 ⊗ (5) V − , φ [SU( / (5) V / − – 5 – ) 3) 7( SO(1 / ) ) /USp = ,n n , = 2 five-dimensional massless theories describing 6( E (4) V 1) × E ,C , R , (6) n (4) V N ( ∗ SO( n 1) a SO(2)] (4) V SO(1 SU(3 , 1) n 1 SU × SL(3 SL(3 ), where , A − 2 a − 1) SU( iφ (5) V ,B (5) V − n a ± n vector multiplets was achieved in [20]. The scalars paramet µ 1 e (4) V ( SO( n (5) V , φ SO( a n A [SO( / = 2 Yang-Mills multiplets where, in case the theory can be obt 2) , N 1 − = 2 theories. The gravity multiplet has a bosonic field conten (4) V n ), while the vector multiplet, giving rise to the (4) V N a n ,B a SO( µ In the case in which the theory has a five-dimensional origin, The complete classification of the In four dimensions the minimal amount of supersymmetries th e scalars parametrise is called very special [22]. K¨ahler Equation (2.9) also holds in thedimensional case origin. in which the The four-dimen been special spaces widely K¨ahler that studied are innon-compact par the symmetric literature spaces were [9, given: 11]. In particular coupled to from a reduction from 6 dimensions, of the above six dimensional theory to four dimensions leads manifolds which are calledis very a special symmetric real. spacecorresponding a For cosets the complete are cases classification i was derived lo bosonic field content ( supergravity coupled to of ( theory can have is four,has corresponding four to real components the in fact four that dimensions.are an called Therefore irr theo JHEP05(2008)079 , P m = 2, (2.25) three- and q q , for integer = 4 for of freedom, while m . rom 6 dimensions, +1 q [12, 18]. = 4 , and thus the bosonic D = 16) (2.22) = 28) (2.23) r manifolds of theories = 10) (2.21) = 7) (2.24) = 2) ˜ q φ ree dimensions an irre- q D (3) H (3) H (3) H (3) H (3) H he parent six dimensional n n n n n of semi-simple Lie algebras s scalar manifold have been = 1, ver the complex numbers as lassification, i.e. the Dynkin uction of the six dimensional ious real forms of a given Lie th eight supersymmetries are re. The parameters esponding manifold is called = 16 q a. A real form of a Lie algebra scalars l origin it is called very special +8 q is the dimension of the irreducible D )] (2.20) +1 (1)] (2.19) (2)] (2.18) q (3) H = 2 for )) (2.17) U n D SO(4) ( 8 and SU(2)] ( SU(2)] ( SU(2)] ( SU(2)] ( (3) H , / × +1 USp n q 7 × × × × SO( , D 7 × (4 + 1 dimensions with positive signature, and 6 2(2) × ˜ ) φ E (6) – 6 – G q 0 integers. In the case [ , , , . Here 0, / a SU(2) (3) H , µ ≥ +1 1 n [SU(6) q e × = 5 24) USp / ( [SO(12) − [ (2 ) − D P q / / [SO(4) 8( P / 5) (3) H = = 4 hyper-multiplet in three dimensions consists of four 6(2) ) E − n 4(4) E USp = q [ 7( (3) H N F / E 3 and (6) V ,n 2) [SU( n , = 16 for / 2) ≥ − (3) H +1 , SO(4 n q q = 1 for (3) H (2 D n +1 4, + 3. The scalars parametrise quaternionic manifolds. If the +1 and q , ) for USp q D SU( (6) V = 3 = n q, P ( q + L (6) T is the number of hyper-multiplets, and in case of reduction f n = 4. In three dimensions, gravity is not a propagating degree (6) T (3) H n n N = 8 for = We refer the reader to reference [23] for a review on the scala Finally, we perform the reduction to three dimensions. In th theories with eight supersymmetries +1 q (3) H with eight supersymmetries. Thelabelled theories by with a homogeneou n dimensional theory hasspecial a quaternionic, four-dimensional while origin, isquaternionic. it the The also corr quaternionic has symmetric a spaces five-dimensiona are there is an extra possibility which will not be considered he are related to thetheory number by of tensor and vector multiplets in t representation of the Clifford algebra in field content is where real scalars and fourtheory Majorana to spinors. three dimensions The gives dimensional gravity red coupled to the real takes the values vectors are dual to scalars. The D 3. Relationship between Kac-Moody algebras and A central role in ouralgebra considerations defined will over be the playedwas complex by originally numbers. the carried var The out when classification this the is algebras a are complete takendiagram, field. to does be Consequently, not the o specify end a preferred result real of form the of c the algebr ducible spinor has two realcalled components, and thus theories wi JHEP05(2008)079 1 α < A E ) into a (3.1) (3.2) (3.3) cp , α ) which a G and as a rators in α ,C ( − . ) algebra. The C and this is just )= G a , R K one can construct are compact while ,H is an automorphism cp K a G . For example, the H . rs possible third real form is any positive root, esses a unique real form he scalar product of the . K rm ct that the only non-zero olutive automorphisms of eferences are [24, 25]. ration. In particular, the -algebra α pact SL(2 1) plex algebra SL(2 = 1). These are also called e scalar product (the Killing ⊂ K − ke the generic form ticular choice of constantven by in taking the generators ( , whereupon the algebra now ] 2 . The negative rather than the ˆ θ ˆ P P ⊂ nts are real. Given such a choice i will have positive definite scalar nd this can lead to very different , ] )=1and( − ˆ P P α P , where [ a − , = P iH [ , E P ˆ α P ). All real forms of the finite dimensional , = E . We denote these eigenspaces by , ˆ P θ a , R ˆ K⊕ ⊂ H ⊂ P ] and to its complexification as = – 7 – ] ˆ P cp . P , cp θ G , K G ) and [ K α [ the elements of the Cartan sub-algebra. The compact is given by ( − a , α E ) which squares to one ( , H − − E ⊂ K 1 eigenvalues of ∈ G α ] ⊂ K − ] E K and we can divide the generators of the compact real form K , α θ , K = ( we define new generators [ A, B E K [ α P ∀ ˆ is an automorphism it preserves the structure of the algebra V ) θ ), are non-compact. This follows from the fact that all the gene B α ( − θ P ) E A ( + θ α E ( ) = i 1) is included as it is isomorphic to SL(2 , Given an involution By considering all involutions of the unique compact real fo Any finite dimensional complex semi-simple Lie algebra poss = AB ( α θ ˆ is just a choice of generators for which the structure consta we can then take the algebra to be defined over the real numbers ‘Cartan involutions’. nature of the generators follows in an obvious way from the fa scalar product between the corresponding generator and U all other real formsreal forms of are the in complexthe one Lie compact to real algebra one algebra. under( correspondence conside By with an all involution those we inv mean a map which 0. We refer to this compact algebra as in which all thegenerators, generators defined are by compact. the Killingusually Compact form, preferred means is positive that negative definite the t nature Killing is form just usually a adopted by result mathematicians. of It the is par gi semi-simple Lie algebras were found by Cartan in 1914. Some r representations of these groups havephysical different properties effects a depending onSU(1 which real form is adopted. The Dynkin diagram, which is a single dot, corresponds to the com has two real forms; the compact SU(2) algebra and the non-com those which possess +1 and respectively. Since result the algebra when written in terms of this split must ta From the generators takes the generic form Thus we find a new real form of the algebra in which the generato the generators products. Clearly, the new real form has maximal compact sub the original algebra areform) compact and and so as have a negative result definit of the change all the generators the part of the algebra invariant under JHEP05(2008)079 ) n lt K E ion and (3.4) which on all of the , where c 8(8) I a σ θ E ˆ H . The roots . Clearly, it } P α H y map → − − pace of roots into . ) as its maximally a θ involution )= ˆ G H α ( n,n pact generators and so rators of the algebra it θ SU(2). ; ts is the character of the artan sub-algebra and it f the authors one has so ators of all the real forms and α ⊗ ) while for the real form of . The character . { s ∆ of the algebra. In fact α q . Clearly, the generators of aximal compact sub-algebra 7 rank of K a ˆ = U E on for this real form has been can be split between compact a ed. , the non-compact part of the compact real form or split real as the roots belong to the dual s H . It also follows form the above ˆ SO( θ ifferent real forms. For example, H dim i . For example, the complex Lie G H c ⊗ → − − θ ∈ P become non-compact. By abuse of ) is the dimension of → − we can write the corresponding real of G p = a G − and ∆ θ Y a θ α G a H H ˆ } U H α , of and ,H . Using this involution we find a real form α θ ˆ a and α V and r )= . Let us denote the Cartan sub-algebra ele- ˆ U H α α i α P ∈ K ( → U – 8 – − = − . The number of compact generators is dim( θ I ; X E a = G α α ˆ H ˆ { V α = = ↔ − both the Cartan generators of the complex Lie algebra cp , U c and α α a G )=0if ˆ the maximally non-compact form is denoted by α V E H − 8 1; ∆ = E . The real rank E ± X,Y α ( + V − ∩ P α is an automorphism it preserves the Killing form and as a resu )) is negative definite. In fact one can define a Cartan involut 120) and we will use this notation for all the real forms of the the maximal compact sub-algebra is E ) as its unique compact real form and SO( . θ H Y n 2 )= ( = 24) = − . α − ˆ K P U 8( X,θ X,Y , H E α and non-compact generators − 2dim has SO(2 by E K . For the compact real form the involution is just the identit )) = ( ) the maximal compact sub-algebra is SO( n θ P , but as we will see it is more illuminating. We can divide the s − Y G remain compact generators while G− D ∗ ( p,q α α H ,θ E ) V As each real form corresponds to an involution Rather than consider the action of the involution on the gene An important real form can be constructed by considering the As the involution As we have discussed the Cartan sub-algebra denoted by = X ( = dim 8 α θ ˆ form as generators and so we may write and the number of non-compact generators is dim is more convenient tothis consider is its equivalent to action its on action the on space the Cartan of subalgebra root ( space those that have eigenvalues real form is theσ number of non-compact minus the number of com is a linear operator that takes takes its maximal value for the split case where it equals the discussion that ( ments in non-compact real form. For to be an involutionfar for mainly which dealt with this thecalled is split ‘the true. case, Cartan and involution’, In as the the no Cartan past other involuti one papers was o consider E V the compact real form transform as its maximal compact subgroupreal is form SO(16). (8 The = number 248 in bracke generators The with generators algebras. Taking different non-trivial involutionsfor we find SO( d notation we are denoting with algebra and the Cartan generators in this particular real form. The m is just that invariantreal under form the of Cartan the involution.turns algebra out Clearly found that it in hasone this the can way maximal construct. contains number It all ofform. is non-compact of therefore gener called the Using the C the maximally non- above notation we can write it as JHEP05(2008)079 . ) θ θ is − α P θ − (3.5) . E is the ) = 0. ( . This ∗ P , but a β 1. The P − = 1 and ( θ na H θ ± g H 2 ra, can be t that P θ rresponding = )= α by an arrow. α E c the dual of the b ( , then θ c ith the fact that ∆ follows from the its and and e roots have the same ∈ where a H c is the group whose Lie ) β of the associated group α ∆ r ( eneral be divided up into g g form on the dual space to und in reference [24]. Let θ act real form all the nodes lgebra whose roots are the onto the subspace ∈ artan matrix is constructed kin diagram of the complex ng the fact that sub-algebra element is com- rtan sub-algebra generators E are white as all the elements o the non-compact elements. s-Satake diagram we find the on the simple roots that are α α α c 1 under the action of oots with respect to and . θ on all of ± , we can then split the roots in P ) if θ P ) α ) = H β . Clearly, if α E ( H θ we find that ,θ E ) P ( β α θ ( + = θ ) = ( α α 2 θ N E β E P , we can consider the restriction of the roots ( c α,β θ α and the rest which are negative, denoted Σ H c – 9 – N is in the maximal compact sub-group, + = c g ]= and the rest of the nodes are white. The map β α,β a α N , E β α corresponds to a black dot. This follows from the action + where E is such that α a ). In this case we can connect nodes r ) = c b g a α )) is a projection α c on the roots we can deduce the action on the generators by α ( na α ( θ ( g θ θ c θ P g − = α i.e. )= ( g a preserves the Cartan matrix. The latter follows from the fac 1 2 if the label c , and so the determination of the real form of a given Lie algeb α θ a θ ( θ and so there are roots that give neither H )= P θ α ( θ with some of the nodes coloured black. A node is black if its co ) is just the projection or restriction of the root )= on the roots. a P α ( . Given an ordering in Σ, or equivalently C θ . The action of the remaining generators must be consistent w H θ P s ( G P θ H ∆ . We denote this restricted root space by Σ. We can think of Σ as P The action of Given a Tits-Satake diagram we know the action of Having the action of An exception to the black and white dots occurs when two simpl Given a real form of a complex Lie algebra, any element ∈ H α combination of roots. defined by simple root is in ∆ encoded on the so calledalgebra Tits-Satake diagram. This is the Dyn are associated with the underlying algebra and so can not in g eigenspaces of space in to be just the eigenvalue components corresponding to the Ca Σ into those that are positive denoted Σ is the Iwasawa decompositionus a be description a of little whicheigenvectors more of can precise. the be full fo Given Cartan the sub-algebra generators of the real a In fact algebra consists of the generators which have the positive r black and one can deduce its action on all the simple roots usi taking action on the dual space of roots. We also take on the corresponding simple root, indeed the action of maximal commuting non-compact subalgebra, that is an automorphism, so preserves thefrom Killing the form and scalar that product thethe on C roots, the that roots is the inducedaction Cartan from of sub-algebra. the Hence Killing from the Tit projection, i.e. that the action of The black nodes are justpact the ones and whose obviously corresponding the Cartan whiteClearly, nodes for the are maximal those non-compact correspondingof real t the form all Cartan the sub-algebra nodesare are black. non-compact while for the comp can be expressed as is an automorphism, acting on [ assignment of the constants if JHEP05(2008)079 y he = 1 ) it is N and all p,q na g = 10, nerators whose D is just the Borel by adding to the . r is generated by all λ g r − na g +++ g ll the Cartan generators to G case for y extended node, which ) while for SO( iven a semi-simple finite ar realisation. So far, all λ n ch e way the symmetries are Clearly, the dimension and he generators appearing in oice of a local sub-algebra. n the non-linear realisation he maximally non-compact corresponding supergravity sed that all the Kac-Moody n algebra is carried out with l supergravity, IIA and IIB the positive root generators ut for more than one vector g theory was associated with ear realisations based on the n of the Kac-Moody algebra ators, and their commutation re the maximal non-compact SO( l algebra appear in sation and so only at the lowest l form to another. For example, ⊗ n this paper always chosen to be effective bosonic string actions [1] is a symmetry of the low energy ) n 11 the dimensions of the two cosets are . This latter algebra was not a Kac- E 11 q G 6= ) do all the Cartan subalgebra generators n,p n,n , the conjectured Kac-Moody algebra contains – 10 – sends a restricted root 11 = 2 θ G q + p that arose in the non-linear realisation of the supergravit is just the Cartan sub-algebra while 11 na g G = 1 supergravity one must use symmetry algebras that are not t one constructs the very extended algebra first the affine node and then the over extended node, which is . The action of + N G G is just that generated by the Lie algebra consisting of all ge ). Clearly, even if q r ) the maximal compact sub-algebra is SO( g SO( n,n ⊗ ) The construction of non-linear realisations based on a give For the case of a maximally non-compact form of the algebra, a Important for the original understanding that p . r different and so istwo the algebras. physics resulting from In the particular, two non-lin only for SO( appear in the coset and so correspond to fields in the non-line algebras considered insupergravity the and context their of formulations theform in of eleven lower real dimensiona algebras dimensions andcan we so be the chosen group to element be which appears that i of the Borel subgroup. This is also the respect to a particularThe real choice form of of a localrealised. given subalgebra The algebra affects local and subalgebra the thethe is field maximal ch for compact content the subalgebra cases and of consideredthe th a i properties the of real this form local beingfor sub-algebra used. change SO( from one rea simple roots can beg generated from multiple commutators of t supergravity theory coupled tomultiplet no coupled or to one vector multiplet, b sub-group. The important pointreal to form note that is all that Cartan it subalgebra elements is of only the for origina t SO( The group positive root generators ofin the the usual complex sense. algebra, that As is such, all this is the decomposition for whi restricted root is in Σ are non-compact and so maximally non-compact real form [7]. theory under study. However, unlike effective actions of stringtheories theory as non-linear was realisations the of formulation an of algebra the Moody algebra, but it wasthe conjectured smallest that Kac-Moody the algebra correspondin thatrelations, contained of all the the gener algebra many more generators.contains As many a more fields result than thelevels the original non-linear do non-linear the realisatio reali two coincidealgebras found [1, by 19]. considering maximal As supergravitiesand [1], noted gravity above [4] it was indimensional reali this algebra way were very extended algebras [5]. G Dynkin diagram of connected to the affine node by a single line, and finally the ver JHEP05(2008)079 = 16) . The 2 A (6) V n ⊗ will have all 1). In three G , has rank three that is just the by deleting the (4 +++ L into H G +++ +++ five tensor multiplets G ith eight supersymme- colouring some of the G +++ . In fact the scalars are G us resence of gravity in the dimensions, it being just G c-Moody algebra, but also am of s ddition to gravity a set of erlying very extended Kac- redicts actually agrees with onding very extended Kac- sions. Then we can find the x dimensions has an exten- al theory. tion, or coset, of a particular = 9) and sixteen ( on of on of interest. If this precise D , the formulation of the three SU(2). This is the coset space . The Kac-Moody algebra now (6) T ⊗ on is found by carrying out the lies in G n has an underlying very extended if the scalars in three dimensions 7 +++ with the real form in which nodes E G D 24) +++ − G +++ 8( into the algebra that results by deleting E part of the Dynkin diagram which coincide with a local sub-algebra of the three dimensional theory. Thus not G where the subscript indicates that this is the G G will be the maximal compact sub-algebra. We +++ – 11 – 1). In three dimensions we find that the scalars 24) H G , − and taking into account the local sub-algebra as (8 8( 2 L E dimensions is A D ⊗ G is the internal symmetry of the three dimensional theory. that has the corresponding real form. As explained above the = 8) vector multiplets is associated with G in the Dynkin diagram. The first test of the conjectured very into (6) V D +++ n G +++ G and the local sub-algebra . If a theory is a non-linear realisation of G G = 5) and eight ( Clearly if we suspect that a theory in dimension The scalars in three dimensions will be a non-linear realisa Let us explain how this works with some examples. The theory w The six dimensional theory with eight supersymmetries with (6) T n In particular, the fielddecomposition content of of the non-linear realisati the only dynamical degrees of freedom of the three dimension local sub-algebra of the full non-linear realisation which second factor is thethree-dimensional algebra theory. SL(3) and it corresponds to the p Kac-Moody algebra we can reducecoset it to to three which space-time the dimen moody scalars algebra belong for and the deduce theory that in the corresp described above. Clearly,scalars at which the are lowest a level non-linear we realisation will of find in a extended Kac-Moody symmetry is tothe see if content the of field the contenttest it theory p is under not consideration true in thenMoody the the algebra dimensi conjecture is that not the true. theory has an und real form of are a non-linear realisation constructed from the algebra determines uniquely all the field content of the theory in more than affine node which is usually labeled the node three. This break the node usually called dimensional theory is given by carrying out the decompositi can then conjecture a a consequence of the decomposition of is connected to the over extended node by a single line [5]. Th with those found in the internal symmetry white dots except for some black dots in the real form of annodes algebra black. can This be means encoded that in the its corresponding Dynkin Dynkin diagram diagr by only can we deduce fromits three real dimensions form. the very extended Ka tries which in six dimensions has nine tensor multiplets ( vector multiplets is associated with belong to the non-linear realisation real form which has the maximal compact sub-algebra ( labelled 7, 8, 9 and 11 are black as in figure 1. in equation (2.23).sion Thus such we that conjecture it that is this the theory non-linear realisation in of si JHEP05(2008)079 5) P we − r +++ 7( E . The = 3) and where the +++ 6(2) (6) T 5) E 4) with local ), that when n , − 7( , P jecture that this E + 4 1). 1). (0 , 1). , , P L (8 (4 (2 L L 11 L e 2. @ y h eight supersymmetries imal compact sub-algebra @ r multiplets ( ealisation @ which has maximal compact on-linear realisation of set of SO( ar realisation of @ @ @ 6(2) even, the conjectured Kac-Moody E @ 10 @ P i i y 1), when reduced to three dimensions , 9 8 (2 L – 12 – Dynkin diagram corresponding to Dynkin diagram corresponding to Dynkin diagram corresponding to vector multiplets, associated with 5) 24) − − P +++ 7( +++ +++ 6(2) 8( E E E = 4), associated with The The The (6) V n SO(4), as in equation (2.20). For iiiiiii ⊗ SU(2). This is the coset space in equation (2.21). As a result 1234567 , and the corresponding Dynkin diagram is shown in figure 4. Fo ⊗ +4) Figure 2: Figure 3: Figure 1: +4(4) P 2 P +++ iiiiiiyiy 123456789 D SU(2). This is the coset space in equation (2.22). Thus we con ⊗ iiiiiiyyyi 1 2 3 4 5 6 7 8 910 Alternatively, the six dimensional theory with three tenso Other two examples concern the six dimensional theories wit subscript indices that thisSO(12) is the real form which has the max theory in six dimensions has an extension such that it is the n dimensions we find that the scalars belong to the non-linear r reduced to three dimensions have scalars parametrising a co subgroup SO( with the real form in which nodes 7, 9 and 10 are black as in figur algebra is four vector multiplets ( has scalars which belong to the coset constructed from corresponding Dynkin diagram is shown in figure 3. with one tensor multiplet and conjecture that this theory is associated with the non-line subgroup SU(6) JHEP05(2008)079 s, ly to + 6 + 5 + 6 2 2 y y 1 P P 2 − @ P @ @

@ odd). quation (2.18), even).

@ P + 5 P + 4 ) ( 1 ) ( 2 P − 2 , P P ctor multiplets and is , P non-linear realisations. (0 est dimension in which ry corresponds to pure lifted to any dimension (0 L whose Dynkin diagram is L 24) and (2.25) have already +++ 2(2) theory corresponding to the +3 pergravity theories. We will G +++ P A y y y y y and , whose Dynkin diagram is shown in non-linear realisations and show that even) explicitly, corresponding to the +++ 4(4) +4(4) P F whose Dynkin diagram is shown in figure 1 2 − +++ ) ( P +++ G +2 B , P +++ P – 13 – C (0 L + 7 1 Dynkin diagram is shown in figure 21. The theory 1) cases above, are special because the corresponding + 7 − 2 , 2 P P (0 Dynkin diagram corresponding to ) theory, corresponding to the three-dimensional coset of Dynkin diagram corresponding to L i i +++ 1) and 4(4) , , P F non-linear realisations respectively, but our results app 2 (4 non-linear realisation can not be uplifted to six dimension +4(4) 1 +4(4) − L ( 2 − 2 P +++ P +++ 0) and L , D +4(4) B 1), +++ 2(2) 2 , P +++ (0 G 1). The L (8 D , The The L (1 ) theory, corresponding to the three-dimensional coset of e L and , P 5) 3 − − +++ ( 7( Figure 4: Figure 5: L E iiiiii iiiiii , non-linear realisation has two tensor multiplets and two ve 1234567 1234567 24) The theories giving rise to the coset spaces in equations (2. The In the next section we will analyse the − +++ +++ 8( 4(4) odd, the conjectured Kac-Moody algebra is all the other cases as well. figure 5. been conjectured in [6] to be associated to the These cases, like the Lie algebra is maximallyF non-compact. The six-dimensional associated with as it is evidentsupergravity from in five the dimensions. Dynkin diagram of figure 22. This theo 19. As itabove is three. evident Finally, from the the diagram, this theory can not be up shown in figure 20.this theory The can diagram live makes is it four. manifest that the high corresponding to the has a conjectured Kac-Moody symmetry equation (2.18), has a conjectured Kac-Moody symmetry E consider the cases their field content exactly agrees with the corresponding su JHEP05(2008)079 as cor- 24) +++ 8 1) must − , E +++ 8( (8 E L 1) theory. The 1) theory. The 1) theory. The , , , (8 (8 (8 as shown in figure L L L as illustrated in the 5 A 11 11 11 1) as this corresponds 24) , ⊗ y y y ities also have − 5 n-linear realisation to six +++ 8( y algebra for fact that the two real forms D ras by computing their low E SO(9 d content of the theory it is a onding non-linear realisations. es the maximally non-compact @ +++ – 14 – G @ . . @ 1). , 25) 26) , which has a Dynkin diagram in which all of its nodes are − − theory is obtained by taking the decomposition of 7( 6( E E 1) theory we are using the real form +++ 8(8) , E Dynkin diagram corresponding to the 6-dimensional Dynkin diagram corresponding to the 4-dimensional Dynkin diagram corresponding to the 5-dimensional 1) (8 , L 24) 24) 24) (8 − − − L +++ +++ +++ 8( 8( 8( E E E and The The The 24) six dimensional − iiiiiiyyyi iiiiiiyyyi iiiiiiyyyi 1 2 3 4 5 6 7 8 910 1 2 3 4 5 6 7 8 910 1 2 3 4 5 6 7 8 910 +++ 8( E The Figure 6: Figure 8: Figure 7: internal symmetry group is SO(9 internal symmetry group is internal symmetry group is 4. Field content of real forms of In this section welevel will field test content the andassociated conjectured seeing with. Kac-Moody if algeb it agrees with the4.1 actual fiel At first sight it would appear that this conjectured Kac-Mood be wrong as the ten and eleven dimensional maximal supergrav their corresponding non-linear realisation.real form, The denoted former by us white, while for the Dynkin diagram of figure 1.are As different we leads will see to in different the field following, contents the for the corresp 6. The latterdimensional factor gravity. is The the internal algebra symmetry SL(6) is and the it real leads form in the no responding to deleting node six in figure 1 leaving the algebr JHEP05(2008)079 l s 9 of 24) 1). , − 0. It +++ 8( > E it arises c which are using this f SO(9 m r g algebra pre- ct real form g . As such the na P 24) g H − riate node in the +++ = 8( ) (4.1) E ree forms of g 45 is in ( g techniques similar to c 4 n of 1) and we find in addition H . If we adopt an ordering ...a s case the 2-forms in the , c his consideration does not eted node, labelled c, is a 1 nal theory associated with a oup element can be written H to zero and the remaining deleted node, i.e. r the local sub-algebra which ent me set of fields that occurs in ter group element are just the of SO(9 ng more 4-forms than scalars is 36 all generators that have positive , , A 9) to the SO(9) sub-algebra leads ) , 16 ( 3 a dimensions, arising from the non-linear 2 a 1 of SO(1 D a 1), that is the adjoint. These are duals to we must first carry out the decomposition part of the Dynkin diagram. The maximal , 45 , A is a local transformation that belongs to the 5 ) h D – 15 – +++ 10 1). These fields are the duals of the 1-forms. The case being studied here, the form fields that arise ( , G 2 a 1 a where +++ 8 E , A ) gh 1) with local subgroup SO(9). non-linear realisation every field appears together with it , 16 → ( that arise in six dimensions form the spinor representation 1) is SO(9) and so the scalars in six dimensions belong to the a , g . The resulting set of generators is independent of which rea +++ 24) , A . The non-linear realisation consists of group elements − G ) representation of SO(9 +++ 1 +++ 8( is the first element then a restricted root will be positive if ( G b E c . The 2-forms belong to the 10-dimensional representation o 45 +++ a H h representation of SO(9 G 16 and so the group element can be brought to the form non-linear realisation are the same as for the maximal compa 16 r g 9) must satisfy (anti)self-duality conditions. The rank th 24) , na − into the algebra that remains after the deletion of an approp . The dynamics will set the field strength of the g +++ 8( c such that g E 36 and can for example be read off from table 5 of reference [27]. 1), i.e. the P , = ⊕ +++ The 1-forms of In view of the last remark, the form fields can be computed usin Thus we find that in six dimensions the non-linear realisatio As discussed above to find the theory, in say H of SO(1 g G 9 +++ 8(8) in the for cisely predicts SO(9 from a generator which has positive level with respect to the E those of reference [26] and in the follows that the theory will containlevel fields with corresponding respect to to thethe deleted split node. case or Inapply indeed fact, for for this level any is zero other the generators. real sa form. We note that t will be dual to the scalars. realisation of a very extended algebra local transformation. The parametersfields that appear of in the the theory. lat white node In and the so cases its corresponding studied Cartan in sub-algebra this elem paper the del where the numbers in brackets denote the representations of 4-forms are in the restricted roots contain a component that is the eigenvalue compact subgroup of SO(9 non-linear realisation of SO(9 the nine scalars mentioned above. In the actual six dimensio to the positions of the black dots in the dual. As in10 six dimensions 2-forms are dual to 2-forms, in thi We recall that in any of form we take for the 9 scalars. The apparent contradiction arising from havi Dynkin diagram of resolved by remembering that theat dynamics the is lowest level invariant unde is SO(9). Decomposing the to subject to transformation as compact subalgebra. As noted in section three the general gr belong to the JHEP05(2008)079 . 6 of 27 27 has ...a 1 6 25) a ⊕ − E A 7( E 1728 . The SL(5) als. The 2- 4 non-linear re- representation representation F 5 which is five breaks into the 25) − (1). As such there 56 912 25) 7( U e ,..., − and we have in addi- E 2 ⊗ =9+16+1 =26 and ars in five dimensions 7( with local sub-algebra , and so we expect this 6 and we have in addition = 16) vector multiplets E 25) E (5) V ontent. The find that we algebra precisely predicts algebra precisely predicts Dynkin diagram in figure 25) − ) (4.2) 26) n 26) (6) V = 1 of we must delete node five in − pace-filling 6-forms 7( − at belong to the also predicts the number of − n 24) 24) 78 7( 24) E 6( 6( ( − 3) we see that this is precisely − − . 24) E 3 E E +++ d as 24) 8( +++ a 8( 133 +++ − 8( , a, b ) (4.3) 2 1) so predicting the presence of belong to the − b E E a +++ , 8( E of a 3 1 +++ 8( with local subgroup a a E h 2 E 133 a ( 1 2 26) , A a 351 a , which is the adjoint, and are dual to ) − 1 A a 6( 25) 27 E ( − 1) theory is given in equation (2.2). Finally 2 , A , . The 2-forms are dual to the 1-forms. There a 7( ) 1 = 9) and sixteen ( E a (8 26) 56 L – 16 – ( − (6) T a , A 6( n ) SL(5) as shown in figure 7. The internal symmetry E , A 27 ⊗ ) ( which are dual to the 26 scalars once one takes account theory predicted by 1 1) theory in four dimensions. The a ( algebra leads to 1-forms and 2-forms that are in the representation of SO(9 , 6 26) b a − (8 E , A h 24) 6( L ) which belong to the representation. − 144 1 E 4 ( +++ 8( . The 2-forms in the first bracket are dual to the scalars while b 6 a we delete the node four of the 10 E ...a representation of 1 h E a ⊕ A 133 ) of SL(4). This leads to the diagram of figure 8. The real form as the distribution of the black dots shows. This real form of 126 1 ⊗ ⊕ 26) ⊕ to be in the five dimensional − 25) 5 6( 78 − ( 320 E 7( ...a 1 ⊕ E . They lead to 28 vector fields together with their magnetic du a . ) A (1). In this non-linear realisation the 1-forms belong to th representations respectively of 25) 25) four dimensions 27 − U − 7( 1) we have nine tensor multiplets ( 27 7( To find the In Hence in four dimensions the non-linear realisation of Hence in five dimensions the non-linear realisation of ⊕ , ⊗ as its maximal compact subgroup. As a result there are 26 scal that arises is the one which has maximal compact sub-algebra E E 7 6 (8 4 27 and dimensional gravity. The is therefore F E figure 1 to find the algebras and they belong to the non-linear realisation of are 54 scalars which belong to the non-linear realisation of factor leads in the non-linear realisation to the field the 54 scalars in the sense discussed above. Indeed, the forms belong to the ( E of where the numbers in brackets denote the representations of L 1 to leave the fields strengths of the latter will vanish in the dynamics and from equation (2.1) inhave section precise two agreement. we The can non-linear read realisation off of the field c are 78 3-forms in the adjoint of where the numbers in brackets denote the representations of 5-forms tion the 45 scalars mentionedthe above. correct Examining field equation content of (2. the of the above comments. the 26 scalars mentioned above. This is precisely as require the actual content of the five dimensional alisation also predicts that the deformations forms we also have 4-forms to be in the 144 gauged supergravities for this theory and the number of s of theory to have 351 gauged extensions. We also have 5-forms th JHEP05(2008)079 ) g 2 , 4 SU(2) ociated it must . Delet- ⊗ 1) , ve results in 1) theory. The , ) representation (8 2 L , 1) is isomorphic to 4 belong to the ( , 11 5) ns is six. Clearly, this hat the deleted dot be gram of figure 10. The − y 1). This six-dimensional non-linear realisation we +++ 7( , sing. As such we can not have some other real form y extended algebra and its 5) E three dimensions (4 eted node must all be white ection. As such the gravity linear realisation, it leads to e Dynkin diagram of figures − nsidered in this paper and it ultiplets and as argued above L of +++ 7( . E 5) 1) with local sub-algebra SO(5). SL(3), as shown in figure 9. The − , +++ 7( ⊗ theory in ) representation of SO(5 E 1) is SO(5), while SU(2) is compact. 1 , which has the maximal compact sub- 24) , 24) − 6 − 8( 24) +++ 8( E − E ) as this form contains all the Cartan sub- 8( D E – 17 – SU(2). We note that SO(5 ⊗ has 1-forms that belong to the ( 1) , 5) theory predicted by the − . that is SL( +++ 7( 1) theory only exists in six dimensions and less. The gravity 1 , 24) E − − (8 D 8( SU(2) and are the fields dual to the 1-forms. The 4-forms belon L A E ⊗ 1) @ Dynkin diagram corresponding to the 3-dimensional dimensional theory under study. To actually lead to gravity , 1) , 24) D (4 − part of the Dynkin diagram and, as the name suggests, it is ass +++ 8( L six dimensional 1 E − SU(2). This is the correct coset, but this, unlike all the abo SU(2). The 2-forms belong to the ( D ⊗ and ⊗ A The 7 5) 1) E , − iiiiiiyyyi 1 2 3 4 5 6 7 8 910 +++ 7( E (4). The maximal compact subalgebra of SO(5 Finally we consider the formulation of the It is also clear why the To find the ∗ Figure 9: internal symmetry group is ing node three in figure 1 we find the algebra representation of SO(5 scalars belong to non-linear realisation of algebra higher dimensions, was guaranteed byreal the form way were in guessed. which the ver with gravity in the line is the contain the real form of which are all either self dual or anti-self dual. The 3-forms of SO(5 algebra elements as non-compact elementsthe and diagonal so, components in of the thethen non- metric. some of Put the anotherhave diagonal way, if a components we gravity of line thewhite metric that in will contains view be a of mis line, black the which dot. considerations must begin at We from thenodes also the beginning and node demand of labelled looking t this one, atapplies and s figure to the all del 1 the oneis real see amusing forms that to of verify the the the1–5. very maximal upper extended dimensio dimensions algebras is co indeed six for4.2 th Thus we find 5 scalars which belong to the coset SO(5 The non-linear realisation of Let us now turn our attention to the theory associated with supersymmetric theory has 5 tensorshould multiplets and be 8 associated vector m with the very extended algebra must delete node sixinternal as symmetry in algebra figure is 2.SU SO(5 This leads to the Dynkin dia JHEP05(2008)079 or- (4.5) 1) as , tations ) repre- (4 2 L , ) (4.4) 20 ( 3 1) theory. The 1) theory. The , , , ⊕ 1 (4 (4 ( ) L L 2 SU(2) and we have ⊕ , ) 4 ⊗ 1 (6) where the maximal ities. The space-filling , 1) ∗ , 15 ( SU 4 ) representation. . we must delete node five in multiplets. 1 ⊗ SO(5 ...a appropriate to five dimensions and so we expect this number , ly the content of the 1 ) fields are dual to the scalars 1) theory in five dimensions as d by the non-linear realisation ) 6 a , 1 5) ( ) are set to zero. , 21 − 35 (4 3 5 ⊕ 10 10 ( +++ , 7( , A L ⊕ 3 in six dimensions contains the forms ) ) belong to the ( 1 y y a E 1 2 2 @ 5) 5 , , a 1 − 6 4 105 a ( ...a ( +++ 7( 1 3 a a ⊕ E 2 , A A ) + 1 = 14 and examining equation (2.2). The a ) 1 3 @ a (6). The 14 scalars belong to the corresponding , (6) V five dimensions 15 6 – 18 – ( n ( 2 (6) representations the form belongs to. This is in , A in a + ∗ ) USp ⊕ 1 1 a ) 5) (4). , (6) T ∗ 3 SU − 6 n , ( +++ 7( , A 2 ) SU = a E 10 1 ( ⊗ a 15 . ( (5) V (6). ⊕ a n ∗ ) , A 15 A ) 1 , SU 2 ⊕ , SU(2). The corresponding field strengths are dual to mass def ) representations. The former decomposes into the represen 64 4 Dynkin diagram corresponding to the 6-dimensional Dynkin diagram corresponding to the 5-dimensional ( 3 ⊗ a , 105 5) 5) 1 − − 1) ⊕ +++ +++ 7( 7( , , A ) ) of the local sub-group SO(5). The ( E E 1 1 ( , 384 b a ) and ( The The 10 iiiiiiyiy iiiiiiyiy h 1 123456789 123456789 , 15 ) and ( Finally we also predict that the 5-forms To summarise, the non-linear realisation of To find the field content of 1 , 5 given in equation (2.1) with 5 tensor multiplets and 8 vector where the numbers in bracketsin denote addition the the representations of five scalars mentioned above. This is precise precise agreement with the field content of the actual can be seen by noticing that mations, and so we6-forms expect belong to the the same ( number of gauged supergrav Figure 10: Figure 11: sentation of SO(5 The numbers in brackets denote the to the ( internal symmetry group is SU(2) internal symmetry group is while the fields strengths of the latter as well as the ( non-linear realisation also predicts 4-forms in the ( of gauged supergravities.belong The to space-filling the forms predicte figure 2, as shown in figure 11. The resulting algebra is SL(5) compact subgroup of the latter factor is coset. The forms fields in the non-linear realisation of are JHEP05(2008)079 r 133 (4.6) isation are gure 2 and g to the . four dimen- 1) theory. The 1) theory. The , , (12) with local ∗ (4 (4 ) . The remaining , and the maximal L L 66 SO 5) − ⊕ 7( E pergravities, as they are field content of the three 462 ⊗ e into SU(6) representations ⊕ 3) ear realisation. SU(2). One can compute the ion. ⊗ 2079 10 10 ( 4 y y ...a 1 ). What is different to the above cases is case. The Dynkin diagram of the three- a P even, that was conjectured in section 3 to , A ) SO( P SU(2). There are 64 scalars describing the non- are dual to the scalars and the field strengths ⊗ – 19 – 352 (12) representations. The 1-forms do account for ⊗ ( ∗ 6) 3 non-linear realisation appropriate to 15 , a Dynkin diagram in figure 4 we find the diagram of 2 5) ⊕ a SO 1 − @ a +++ 7( 15 +4(4) E 2 P , A . +++ ) even 5) (12). D ) theory with three dimensional ∗ − with local subgroup SO(12) 66 non-linear realisation. We refer to the appendix for a prope P @ ( 7( . The 2 , P , SO E 5) a ) 1 , and are related to the scalars by duality. 15 (0 (12). The 30 scalars belong to the coset of − a 7 Dynkin diagram corresponding to the 4-dimensional Dynkin diagram corresponding to the 3-dimensional ∗ L 7( +4(4) , P ⊕ E 2 P +++ E 5) 5) (1). The form fields of this four dimensional non-linear real (0 non-linear realisation is obtained deleting node three in fi , A SO − − ) D 15 L U +++ +++ 7( 7( 5) 1) supersymmetric theory. In particular, the 1-forms belon ⊗ ⊕ 32 E E ⊗ , − ( +++ a 7( (4 and 35 E A L The The iiiiiiyiy iiiiiiyiy 123456789123456789 ⊕ 1 +4(4) 2 P in shown in figure 12 where node four of figure 2 has been deleted +++ → D Deleting node six of the The Dynkin diagram of the Finally, we consider the 66 explanation of the computations carried out in this subsect We now consider the figure 14. The resulting algebra is SO(6 correspond to the Figure 12: Figure 13: internal symmetry group is internal symmetry group is sions algebra is SL(4) subgroup SU(6) the 16 vectors and their duals, while the 66 2-forms decompos as The numbers in brackets denote the dimensional of the remaining fields are set to zero. We expect 352 gauged su associated to the number of 3-forms predicted in the non-lin compact subgroup of the latter is SO(12) leads to the diagram of figure 13. The remaining algebra is SL( linear realisation of field content in thisdimensional case, finding precise agreement with the that is the adjoint of 4.3 JHEP05(2008)079 . ). ⊗ 1 D ); 1 are , P ⊕ with 2 1 2 µ 15 SO( , a rank G occurs in 1 ⊕ ⊗ and 6 ⊕ ) is compact α . The reader 6) into those 15 6) 4 2 , , P µ 35 ⊕ can be thought G . The latter we φ ) as well as ( 6 35 ) ), i.e. the adjoint of 6) Dynkin diagram. 1 ⊕ ) le root , associated with node − 1 2 = 0 we just have the → ), while the local sub- P 1 − p as SO( joint representation of ion techniques of refer- ( 20 1 2 ) is graded by the level c , that is P G , P b ( note that P el one representation that of the Dynkin diagram of ⊕ a 1 66 m , P h 1 i e representation of , 6 o ) (4.7) respectively all in the vector ations of SO(6 first node in 1 SO( ng to space-time, the adjoint 1 5 ine of nodes one to five. The , cal subgroup which is SO(6) . In fact the → a ra after the deletion of node tors in all the above cases. ton ⊗ b ), the deleted node attaches to 1 4 at level one. ( a a D 3 ) and ( h 6) , then one finds the representation 32 a ⊕ , 1 , that is ( 2 2 +7 in the SO(6 ) , a φ G 1 8 2 P a 6) respectively. To find the fields with 66 , , λ = SL( A 2 1 µ ( G ). The 32 dimensional spinor representation and ⊕ P 3 , a scalar ) a 2 1 – 20 – ) algebra for the space-time part of the remaining a 1 and so one finds the vector fields. As a result, ), that is ( 15 λ 1 D 2 a P − , = 1, there is always an obvious solution for any such A . That is the 32 dimensional spinor representation of 1 c is the fundamental weight of 7 , D ( i 1 SO( λ m a 1) with trivial local subgroup. µ ⊕ = ⊗ , 3-forms are the duals of the vectors which are also in the ⊗ A ) , that is i 2 vector of SO( 1 5 6) a P , , µ 1 4 P a µ where of the Dynkin diagram of A ( into representations of SO(6 6). As a result we must carry out a further decomposition of = 2, we find the following representations of SO(6 j j , λ c and the anti-symmetric part of ) breaks into SL(6) representations as ). The 1 and so m ⊗ 1 . In all the cases above i P , +4(4) 15 j − µ 2 P +++ λ 66 D D ). Clearly, any scalar fields that might arise in ( the ( P and the deleted node attaches to the node labelled i.e 2 SO( G ⊗ 6) which is valued as a 6), ) are completely removed by the last part of the local subgrou ⊗ , , At the next level, The fields at the next levels can be found using the decomposit In the case under study in this sub-section we find that the lev 1 P associated with node six. This is the number of times the simp and the node labelled G c 1 and similarly for which are labelled in terms of their fundamental weights. We the 495 and 66 dimensionalspace-time representation indices of of SO(6 SL(6) we must decompose the represent of SL(6). This can be achieved by also deleting node The resulting fields are subjectSO(6) to transformations of the lo vector representation. These are the only representations i adjoint representation of SO(6 G recognise as leading to the fields that the deletion does not lead to an SL( in the cases above, one finds that the vector fields belong to th with highest weight SO( the node labelled SO(6 of as belonging to the coset SO(1 ence [26]. In fact, at the next level, representation of SO( can verify that this is indeed the representation for the vec arises in the decomposition is the root being considered in the decomposition. At level zer m the highest fundamental weight which is associated with the These correspond in the non-linear realisation to the gravi in question decomposes into SL(6) representations as reduction of a very extended algebra. If the resulting algeb is and thus coincides withSO(6 its maximal compact subgroup. The ad this group to SL(6)representation to of find the SL(6) usual beingdecomposition representations associated of belongi with the gravity l algebra, but rather SO(6 two anti-symmetric tensor algebra removes the JHEP05(2008)079 3 ). P P ≥ 6) is + 6 + 5 , c 2 2 y y P P m = 0, one ng of the and 5. @ c ) theory ( @ m , P of SO(6

@ i (0

@ µ L is the number of i = 1, we find the remaining algebra + 4 q ) (4.8) i t a six dimensional c m 2 P P n part of the internal orresponding to the ( al theory as it arises m ). In fact as P 3 a P 2 = hese are the fields of six a 1 µ a SO( line to contain nodes 1,2,3,4 of space-time indices on the ng for forms, which excludes ) theory. , A ) which corresponds to 2-forms ight ) er indices. If we focus on 2-form vector multiplets. This is indeed 2 y y , P P ( + 1) with local sub-group SO( + P 1 (0 a L P , P A , ); ) actually have a block of six indices and 15 1 1 @ ( + 1). At level two, λ + 1 scalars. As a result, the field content it is the number of times the root of the node 2 l , P , φ + 1 tensor multiplets. ). Consequently, in this model we find gravity, , P ) 1 = 0 and so we are really only interested in the 1 1 l P – 21 – , ( 2 a ). The last two correspond to gravity and the first 1 where 105 (+) a 1 l + 7 , A 1 2 P + 2 ); i c 1 ) and ( ( ) m 2 ) a 1 2 and for form fields of rank two or less we have the fields 1 − ) and ( + = ( a 1 P ≤ , 2 m )( Dynkin diagram corresponding to the 6-dimensional c + 1). It is straightforward to verify that at level zero, 2 , A = 2 we must have ), which is compact. The gravity line connects nodes 1, 2, 3, 4 ) 35 + m c 1 + 6 P P ( , P ( j m b ), ( +4(4) , q a ) 2 ) P +++ 1 h j 6). As such we have only one 2-form which occurs at the beginni = 1, we find the representation ( 15 + D , SO(1 c − P 2 )( ⊗ vector multiplets and one tensor multiplet. The non-abelia m 2 (6 then as The j + P + 7 occurs in the way the highest weight state occurs and 2 P a + 1 scalars which belong to the coset SO(1 P ( 1 2 P , a iiiiii 1234567 P in the vector representation of SO(1 1 +7 in figure 4. This leads to the Dynkin diagram in figure 15. The A 2 P 2 Hence up to level In fact deleting node six is not the only way in which one can ge Upon dimensional reduction to five dimensions the theories c a part of SO(6 1 + 2 self or anti-self dual rank 2-forms and a 5 representations ( A 495 multiplet. The apparent scalars in the ( so are not forms in the required sense. finds ( At level one, to the where the numbers in brackets denote the representations of Figure 14: labeled given by is then SL(6) A we can not getdimensional more supergravity rank plus one three tensor or multiplet less and forms at higher levels. T blocks of six indices.the Among possibility this of having set blocks offields of states six we indices are as well searchi as oth symmetry group is SO( It is straightforward to seegenerators, that or having fields, done for this a the representations number with highest we P even) with what we expect from the bosonic field content of the theory. One can also deleteand node five and then take the gravity corresponds to is supergravity coupled to Dynkin diagrams of figure 14 and 15 give the same five dimension JHEP05(2008)079 P P even) + 6 + 5 + 6 + 5 nsional 2 2 2 2 y y y y P P P P P @ @ ) ( ) theory ( ) theory ( @ @ , P , P , P

@ @ (0 (0 (0

@ @ L L L symmetry group is + 4 + 4 1). The gravity line 2 2 P P , It is straightforward to +1 recover the correct field 1) and f figure 16. This is very , P be derived from the non- e supersymmetric theories (4 s in ten dimensions and the l theory arises from deleting ons give rise to scalars that L tended Kac-Moody symmetry. eting node four and gives the cture that all the theories with s SO( + 7. 1), y y y y , 2 P (8 L ) theories in five, four and three dimensions – 22 – , P @ @ (0 L + 7 + 7 2 2 P P i i Dynkin diagram corresponding to the 5-dimensional Dynkin diagram corresponding to the 6-dimensional [2] giving rise to a unique nine dimensional theory upon dime +4(4) +4(4) +++ 2 2 8 P P +++ +++ E D D + 1 tensor multiplets and no vector multiplets. The internal The The P 1). The gravity line connects nodes 1, 2, 3, 4 and , iiiiii iiiiii 12345671234567 +1 P cases, and we haveis found precisely that reproduced the by bosonic the field non-linear content realisations. of thes connects nodes 1, 2, 3 and 4. from deleting node fivesimilar as to is the evident situation fromway of they the the fit Dynkin into well known diagram IIA o and IIB theorie reduction. The four-dimensionalDynkin theory diagram corresponds of to figure 17, del node and three finally and the corresponds three-dimensiona also to confirm the that Dynkin bycontent diagram deleting of of nodes figure the five, 18. bosonic four sector and of three we the also eight supersymmetries that uponparametrise symmetric reduction manifolds to have an three underlyingIn very dimensi ex particular thelinear bosonic realisation. sector We of have worked any out of in these detail the theories can respectively. 5. Discussion In this paper we have given substantial evidence to the conje Figure 16: Figure 15: even) with even). The non-abelian part of the internal symmetry group i SO( JHEP05(2008)079 P P re 1), , non- re 20 + 6 + 5 + 6 + 5 (1 2 2 2 2 L y y y y P P P P +4 ), corre- 1 @ @ 2 − ing to the ) theory ( ) theory ( , P to the case P +++ @ @ 2 B , P , P

@ @ − ( (0 (0

@ @ L L L re 3, and + 4 + 4 2 2 P P re 22. The low-level 1), corresponding to , re 19, s can also be done for ), corresponding to the (2 bras that we conjecture L ticular the other theories ic spaces in three dimen- , P nded Kac-Moody algebras. 3 rators of the very-extended n derived in [6], where it was sional theories whose scalars − s. These are presented using ( r realisation at low levels. L y y y y 1). , SU(1 odd, corresponding to the ⊗ P 2) 4). , , +2 +4 – 23 – ) for P P , P + 7 + 7 (0 L 2 2 P P i i @ non-linear realisation whose Dynkin diagram is shown in figu Dynkin diagram corresponding to the 3-dimensional Dynkin diagram corresponding to the 4-dimensional non-linear realisation whose Dynkin diagram is shown in figu @ +++ 4(4) F )), and minimal five-dimensional supergravity, correspond +3 +4(4) +4(4) 2 2 +++ P P P +++ +++ , P = 2 4-dimensional supergravity without matter corresponds 2 A D D − N ( L The The non-linear realisation whose Dynkin diagram is shown in figu 1 of iiiiii iiiiii non-linear realisation whose Dynkin diagram is shown in figu 1234567 1234567 non-linear realisation whose Dynkin diagram is shown in figu +++ 6(2) − Crucial to our analysis are different real forms of very-exte The analysis of the field content of the very extended algebra E +2 = +++ +++ 2(2) P linear realisation whose Dynkin diagram is shown in figure 5, the Figure 18: Figure 17: corresponding to the 21. One cansions also that consider can the not be theories uplifted associated to to six symmetr dimensions, namely even). The internal symmetry group is SO( even). The internal symmetry group is SO( We explain the realto forms describe of the the variousTits-Satake very theories diagrams. extended with Kac-Moody Given eight alge thisalgebra supersymmetrie diagram and we the field derive content the of gene the corresponding non-linea the other cases whosethat symmetries live we in have sixparametrise conjectured. dimensions symmetric and In spaces that par are give rise to three-dimen C sponding to the fields associated to all these Kac-Moody symmetries have bee (in particular P G JHEP05(2008)079 + 5 imal P +4 are + 6 @ P P non-linear i + 4 + 5 ( P ). P +++ ( G le dualisations of , P 1). , 3 ( + 5. − (1 + 3 ( he non-linear realisa- ( + 4 P L L ( P P ( in exotic supersymmetry leven dimensions. All the at all the massive deforma- ( a potential, for theories with ebra [29], and thus it applies least one set of ten or eleven hat it automatically encodes though the ). All nodes from 6 to ( sponding supergravity. d to on-shell propagating de- upergravity theory. This lifts re thus relevant in the context non-linear realisations, turning these theories are the infinitely ( , P 2 e dimensions. Although 2-forms @ ( m to those considered before, it is − ( ( +++ L 1 form whose field strength is dual to G ( non-linear realisation describes demo- − ( ( D +++ non-linear realisation describes the bosonic ( G ( – 24 – ( +++ 4(4) ( F ( Dynkin diagram corresponding to Dynkin diagram corresponding to ( ( +2 +++ 4(4) ( +++ P F C ( non-linear realisation describes the bosonic sector of min The The @ +++ 2(2) Dynkin diagram corresponding to G +6 are connected by arrows, as well as nodes 5 and iiiiiii 1234567 +3 P 1) theory. +++ P , A (1 Figure 21: Figure 19: , there is an infinite preferred set that describes all possib L The iiiiiy yi 1 2 3 4 5 6 +++ 8 E In [28] it was shown that amongst the infinitely many fields in t In the democratic formulation that arises in the Supersymmetric theories with eight supersymmetries conta iiiyiyy yyy 1234567 black. Nodes 4 and Figure 20: also shown that the five-dimensional supergravity and the tion of cratically all the fieldseither and description. the corresponding duals means t sector of the realisations discussed in this paperstill have the different case real that for allmany the dual propagating descriptions degrees of of the freedom propagating of fields of therepresentations, corre like for instancecan tensor be dualised multiplets to in vectorsnon-trivial in fiv five vacua dimensions this in is theof no absence of longer gauged true. supergravities. These The multiplets fact a that the the on-shell degrees ofthe freedom infinite of set the ofinfinitely eleven-dimensional many dualities s remaining that fields in occurantisymmetric eleven in indices, dimensions and two have at dimensions thereforegrees they to of freedom. do e not This isto correspon actually all true the for cases any Kac-Moody discussed alg in this paper as well. Therefore al on a mass deformation correspondsthe to mass having and a thus is non-vanishing. In [27, 30] it was shown th JHEP05(2008)079 rmations re 23. As y with 18 non-linear re- +++ G @ non-linear realisation. A the fermions are included @ @ y i ld mean that the number of ies with 16 supersymmetries, +++ being used, this would mean y theory with more than 16 8 esentation to which each form wever, we expect these defor- fferent for different real forms than maximal supersymmetry @ on for a suitable real form of a E would possess the same massive G @ igate in this direction and in par- @ @ @ 24) i y − +++ 8( 9 8 E Dynkin diagram. Dynkin diagram. Dynkin diagram. SO(9), and we conjecture that it is associ- theory, that is associated to the maximally / 14) 20) − − 20) +++ 2(2) +++ +++ +++ 8(8) 6( 4( − G – 25 – 1)-form, but are associated with symmetries that F E E 4( non-linear realisations of [7]. If this is true also − F The D The The +++ m D and i i i i i 1 2 3 4 5 Figure 22: 1) theory associated to Figure 23: Figure 24: , +++ m (8 B L . In particular, the scalars of the three-dimensional theor iiiiyyi i i i y y non-linear realisation whose Dynkin diagram is shown in figu 1234567 1234567 dimensions would give in all cases the number of massive defo +++ 20) D G − +++ 4( F 1 forms in The fact that different real forms can be accounted for in the − similar analysis was carriedcorresponding out to in the [31] for the case of theor tions of gauged maximal supergravities are encoded in the in the case ofD theories with eight supersymmetries, this wou of the supersymmetric theory. Moreover,belongs given does that the not repr dependthat for on instance the the particular real form of ticular examine if theaffects fact this that result. the It localthe can subalgebras deformations happen do are that not di arise for from theories a with ( less supersymmetric theories. It would be interesting to invest only transform the fermionicmations sector to of be the accounted theoriesin for [32]. the by non-linearly the Ho realised very theory extended [29]. algebra once ated to the deformations in a given dimension as the supersymmetries parametrise the coset alisation also leadssupersymmetries to can the be conjecture describedvery as that extended a any non-linear supergravit realisati JHEP05(2008)079 di- 2 D R 1), and , non-linear hat are not 2 avity theory 5) = (2 R − +++ 7( N E 8 g 3 R st dimension in which the EU Marie Curie, re- hows that this real form non-linear realisation, one man Potential Programme ly, the supergravity theory 001. The work of F.R. and 3 ersymmetries, or indeed no ank the physics department her thank the Galileo Galilei non-linear realisation, whose 5) ory is called R c sector, so that there should mental forces and symmetries − in [34]. The reader can check he one of the lar real forms of very-extended with the labelling of the nodes as e dimensions. The supergravity +++ tive action generalised to 7( larly apparent when one writes the 14) he maximally non-compact form 8 E − E NFN for partial support. The work +++ 6( non-linear realisation whose Dynkin E 1 and 5) R 7 − E +++ 7( , E 6 2 1 E 1) theory, supergravity with 24 supersymmetry R R , , – 26 – 5 (4 D L , 2 4 R A dimensions is associated with the maximally non-compact 3 D R 7 1 g R 3 R 2 R 1) theory considered in this paper and one giving the supergr [4]. These are examples of real forms which lead to theories t , 3 (4 The Dynkin diagrams for − L +++ D [1], and gravity in A 2 − 4 5 6 1 +++ D g g g ggggggggggggg gggggggggggg 1234561234567 123123412345 R D 3 1 R R exists in six dimensions and below. The six-dimensional the realisation derived in section 4.2. The bosonic string effec it was originally conjecturedthat in the bosonic [33] field and content later of this constructed theory coincides with t corresponding Dynkin diagramthis is theory shown exists in iswith figure 4, as 24 24. can supersymmetries be corresponds The read to highe from the the diagram. Final Figure 25: with 24 supersymmetries. Just like the giving the supersymmetric. More generally itKac-Moody is algebras possible lead that to particu supersymmetry theories at with all. less than eight sup Acknowledgments We thank Duncan Steele forin discussions. Leuven P.W. for would like their to hospitalityP.W. th is where supported this by work a wassearch PPARC rolling begun training grant network in PP/C5071745/1 grant 2 HPRN-CT-2000-00122. and Institute We for furt Theoretical Physics for hospitalityof and the A.V.P. is I supported in partunder by contract the MRTN-CT-2004-005104 European ‘Constituents, Community’s funda Hu in appendix A. Theforms pattern next for to the forms the of nodes low to rank which is they particu are associated. it is evident from thetheory diagram, this with theory 20 only supersymmetries lives in is thre associated to the diagram is shown in figure 2. This last example in particular s can lead to separatebe theories two that different only ways differ of in embedding the the fermioni fermions in the form of mensions is associated with theof non-linear realisation of t JHEP05(2008)079 , is D +++ (A.1) (A.2) G cus on indices required in terms 1 − D A +++ +++ corresponding . G G c α one must carry out +++ can be taken to be 1 G − he generators known. D here the deletion of the +++ A 235.05 an in part by the ntent of such Lorentzian for suitable choices of G 1 ry extended algebra hrough the ‘Interuniversity -P. nt representation of certain are finite dimensional semi- − t the emphasis there was on of ne node whose deletion leads esentation of and algebras, which includes very sibly one affine algebra, which d content in terms of familiar D 2 a given dimension is found by a is not . 2 A α G he case of simply laced algebras. 1 ly antisymmetrised G i these generators are associated to G λ this appendix is completely general , corresponding the the remaining ⊗ ci and 1 and the simple root A = 0 in that paper. We will eventually G 1 2 +++ +++ Del i ∗ G G G G ν X m . However, our methods are quite general of 1 − − i − j x α D µ where = A cj – 27 – c 2 A is , although when α G 1 . The simple roots corresponds to the decomposition of 1 i c G ⊗ X G 1 . In this appendix we will show how to algebraically cal- − G +++ = G that can arise in the decomposition of = +++ ν 2 G in terms of representations of G index structure. This work is carried out by the authors of th +++ Del 1 , the simple roots of − G 1 +++ and D G G multiplet, that is the brane charges, while here we want to fo . The latter simple root can be written as 1 A c 1 G of l i β . In the non-linear realisation of +++ is a subalgebra of G 1 is a vector orthogonal to the root space of − D x A In this appendix we will restrict our attention to the cases w For no indefinite Kac-Moody algebra is a complete listing of t Let us label the deleted node by into representations of simple Lie algebras. We willThe also discussion restrict our is attention the tothe same t content as of that the given in reference [26], bu Dynkin diagram after the deletion. node in the Dynkin diagram of arising in consider in detail the case in which of the representations of the remaining algebra the adjoint representation. In essence on takes were paper in collaboration with Duncan Steele. However, there is aextended class algebras, of whose such Dynkin algebras diagramto called contains Dynkin Lorentzian diagrams at that least are o thoseare for finite more algebras amenable with pos toalgebras analysis. in terms Indeed, of one these can remaining analyse algebras the [3]. co Given a ve culate the representations of the generators with complete the field content ofdeleting the a non-linear particular realisation node it leads and to decomposing in the adjoint repr fields with the same of the universe’, in part by the FWO - Vlaanderen, project G.0 Federal Office for Scientific,Attraction Poles Technical Programme and — Cultural Belgian Affairs Science Policy’ t P6/11 A. The calculation of the formIn fields this paper weKac-Moody have required algebras the decomposition of the adjoi and apply to any semi-simple algebra a further decomposition to thisrepresentations. latter Nonetheless, the algebra analysis to carried find outand in the applies fiel to any the simple roots to the deleted node where JHEP05(2008)079 2 2 1 n ν G G can (A.3) (A.6) (A.4) (A.5) (A.8) (A.7) i, j, . . . and and 1 +++ 1 occurs in ν G G c is a positive α respectively. as 2 respectively. The . G 2 2 +++ i G G . α Λ G are the Dynkin indices i 2 occurs provided and i ν − i m q of 1 nding to node and λ 1 + algebra. Taking the scalar i G i ey generators. As a result, α may be split into by the same indices G 1 p . 2 . 1 X to denote the Dynkin indices, i Λ ν G x i G j est weight must occur as one of − ) . r p − P n k 2 i , where i x ν i − β c , λ c i α µ i vary. As such, a necessary condition i i n =2= m m λ q ( m )= 2 c i i i the fundamental weights of = j = p α X i i +++ j P 2 ,µ X α i λ G G − 1 i X implies that [26] ν − Λ 1 ( m ν j 2 c − c and ν µ i ) c m m i k X – 28 – µ m , − algebra, and using = j , λ + , to contain the highest weight representation of ) 2 β i = 2 j j with highest weight ν j that can occur. µ i β ( G i n 2 j i c ,µ λ q +++ i i q we find that [26] n +++ G m p j µ G i k ( G X j i being considered. All generators in the algebra X λ i by the level. The level is just the number of times the root are positive integers and this places a bound on the possible = q X the level and we will classify the result of the decompositio − i X α 2 k c with highest weight n i 1 + G m ν 1 X m c c ⊗ α G belonging to the weight spaces of c m and 1 is that ν c m G = j are positive or negative integers depending if the root m 1 q = , c G i α q m Λ ’s that appear as the roots of the 1 G and i is determined by the requirement that n 2 denote the coefficients of the simple roots of the is the Cartan matrix of , i x i ab n m A Using the above expressions, we may write a general root We will call the integer Repeating this procedure for the If a representation of is just the number of times the Chevalley generator correspo occurs in a particular root c c Here respectively and the nodes of these two algebras are labeled for simplicity, although the ranges are different. The vecto value of where or negative root. Also we define the above quantities as product of both sides of this equation with In these equations α which are the parts of highest weights, or Dynkin indices into representations of with Dynkin indices We note that these two vectors belong to the weight spaces of where that must be positive integersthe or possible zero, Λ occurs then this high we find that the representation of for the adjoint representation of the multiple commutator which leads to the root of interest. be constructed from them multiple commutators of the Chevall Taking the scalar product with JHEP05(2008)079 , , 5 by D +++ D 8 (A.9) which − (A.11) (A.10) E i ation of 11 attaches β i E bra which n D . , or i 11 we mean P . An identical E 3 1 +++ ) and belongs to − ν E G ebra, as shown in D = = 1 i and . ν the possible represen- . lgebra µ 1. As a result we find i D 4 g here. It becomes q − ) − i − E is correlated as equations j i ation with highest weight 11 , µ . 2 as these conditions are not P relations. However, almost i 5 , λ D roposed in [3], the notion of E i q G ave glossed over some subtle ed to the end of the Dynkin E j ysis of Lorentzian algebras in i λ lways have the representation at ( n dimensions is found by deleting j P +++ − p and i D in [36] and in general in [35]. and the deleted node, G we relabel the nodes of p under 1 . 1 )= 10 G D D j i,j λ − X E 2 1 using its definition, that is the multiple 11 ,µ λ 1 E )+ − − j D . We find that +++ this again places a constraint on the allowed 1 µ 1 ,µ D i ( ), corresponds in the non-linear realisation to G λ c µ . ( D – 29 – c m = ,... j although one does not discover the multiplicity of q 2 2 m i ) in the adjoint representation of − ν =1+ q − 2 ) , j 2 0 ,ν +++ x i,j , 1 X ,µ paper [1], but was spelt out explicitly together with the is in this case , or SL( G ν i 1 + 2 µ − 11 ( 2 c i G D E q m = 1, equation (A.5) becomes A 2 i c x X m = 2 subalgebra, that is to the node labeled α and decomposing into representations of the remaining alge 1 that can occur in the decomposition of the adjoint represent − 2 D D respectively. Once we delete node G A . The algebra 1 1 ⊗ A . The algebra − 2 1 D with highest weights ( to have a sensible labelling from the view point of the subalg ⊗ G G µ 2 2 ). This corresponds to a generator which is a vector under SL( can only take the values 2 D ⊗ 1 G A = 2 1 − 1 , λ . Not every solution will correspond to a root in α ⊗ − 1 ν n 1 The level one solution discussed just above is the represent Squaring equation (A.3) gives [26] Let us first analyse equation (A.6) for the case we are studyin Our task is to analyse equations (A.6), (A.8) and (A.9) to find At level one, that is We now specialise to the cases concerning the very extended a D − and G A D → +++ 4 µ ( the fundamental representation with highest weight (A.6) and (A.8) contain the same level We note that the occurrence of the highest weights in representations. tations of G Since as strong as the construction of the algebra all solutions are in fact present in commutator of the Chevalley generators subject to the Serre the representations using thesepoints equations. that In are the described above interms we more of h detail algebras in that [35, occurlevel 26]. after was the inherent The deletion in anal of the a first node was p discussion applies to equation (A.7).of Hence at level one we a lies in the negative root lattice. One obvious solution is th constraints on the representations in the context of whose Dynkin diagram in given inthe figure 1. node labelled The theory in that the gravity sector. Indiagram these of cases the the deleted node is attach to the first node of this algebra which we label by one. By is A n figure 25. Consequently, we have that JHEP05(2008)079 is = D ion 2 = 1, , i α k (A.14) (A.13) (A.15) (A.12) +1 − = 0 and D D i . At level s q blocks of K , corresponds D k s i f q they must have dition for = . On the other i c vanishing. c q vector indices. As j m m associated with the q into representations c is m +1 D c e decomposition of the D A totally anti-symmetrised m dices. However, equation D l ultiple commutators of the K D f fundamental weights of an lies that the solution always . to the generators j .11) are all the representations on of . are given by l requirement that there are no al Dynkin index and so the Chevalley generators n 1 ) indices that the corresponding = 1 in equation (A.13). For the b − − . a D k D indices and in particular the repre- i − j K ) A i 1 sD . )= D ≤ j with Dynkin indices − ≤ q − 1 D ,µ )+ j − i 1 A ( , i , j i D − ) ) q − i are just j D A j − − 1 µ D D ≤ D ( D i − D X ( ( = 0 and ( -form we mean an object that has just one set of i k by deleting the end node i.e. node – 30 – j i D Chevalley generators , with all the Dynkin indices , which is equivalent to the condition q k s + 1 k A − c ( D i − − ) sj m j X D D = µ )= totally anti-symmetrised indices. As such we find that = A c ,µ = j i j k c of equation (A.12) into equation (A.11) we find that the m n − ,µ − c i m D into µ completely antisymmetrised indices. This is a representat m µ D ( ( D k A = 0. As such in equation (A.12) we require = 1 and s S , they belong to. By a D D indices where the last term corresponds to the possibility o = − 11 k sD E blocks each with indices, i.e. i )+ i we find that the effect is to lead to a generator that has q D . The actual problem may have more or less solutions as the con − ) with fundamental weight 1 algebra D indices that is a set of D − ( 1 1 i D Substituting the value of We are interested in forms in terms of their the multiple commutator contains q ,...D − − A i 2 c D D , deleted node. As these are related by the action of 1 such, the number of indices on a generator that arises at leve m P It follows that among the solutionsthat that occur occur to in equation the (A decomposition of the adjoint representati of SL( antisymmetrised indices. We recall that having a non-trivi generators carry. TheChevalley generators generators. are The generators constructed of from the m To analyse this equation it is useful to consider the SL( latter is automatically solved and that the root coefficients different in the latter case to the problem being studied here hand, as the generators are representations of of they contain do not(A.11) add or is subtract the from sameadjoint the equation representations total as number of of one in would find if one analysed th to having with all other Dynkin indicesblocks vanishing, of with the additiona equation (A.11) becomes The fact that theexists. right-hand Here side we is have non-negative usedA indeed the imp formula for the scalar product o which is automatically solved by taking sentations of A exceptional case of space filling forms we have one block of indices and so JHEP05(2008)079 , k we = 1 8 are 6 -form , (A.19) (A.18) (A.20) (A.16) (A.17) p k = 1 all 7 , 2 p = 6 for a 2 ,n n = 4, we find that =1 or + Λ E j = 0 and so 4 k and examining the k 2 8 one has 2 p , ≤ 8 the internal symmetry 2 λ 1 + e fundamental weights D , i 1 2 − 1 k es for a form or rank + Λ − = 3 1 2 7 5. To illustrate how this 4 − λ 3 2 k or the case of forms, using 7/2 − =1 or = 2 or Λ 1 s which for = = p . = 1, with all the other Dynkin ,...,n 2 2 6 2 3 2 2 3 2 5 λ 14 ,...,n α j 2 ,...,n . p 3 α 2 − − 3 highest weights it is quicker to first ≤ − ) n 2 5 − + Λ 6 n n + Λ 30 λ =1 or D , i 4/3 5/4 = 2 1 n singlet. Finally, for 3 − k n 5 − and so Λ λ = p 6 2 ,...,n 11 = 1, or = , j − 2 (9 3 E 1 E D 2 4 )+ p 2 1 20 12 λ , for which = 1 − 5/4 4/5 , i = − 10/3 6 λ ) i E − − nn ) , i, j and so ,...,n – 31 – ) n n , i =0+Λ 1 2 3 + 1) ) (1 − 3 5)) 6 3 10 3)( − n 2 12 λ 2 ,...,n i 6/5 − (9 = 1 ) = 2, we find that 15/2 − D k 3 − α n and that n ( n ( = (9 ( i E k 3 4 = = 1 D 2 − ) A    2 2 n 2 2 = = 2 6 4 (( α 3) − λ , i, j = 6/5 = 5, that is 2 2 − 10/3 ) (9 or Λ , i n j α ) ) ) in j D 4 3 + ) n n = 1 or it is straightforward to find solutions to this equation once )(( n − ). While for space-filling forms, i.e. 1 j 2 1 − − j n 1 2 − k = − − ( λ (9 (9 i 4/3 4/5 3/2 ) n (9 2 . , λ ( i n ,... 2 i 2 E λ        5 6 8 7 4 ( A − +++ j 8 E E E A D = 3, we find that = . (( p E i 0 k p ij , ) . Taking and so Λ 1 4 i,j − = 2 2 A ) P 2 n , E α Table giving the square length of the fundamental weights of = + Λ A 2 3 4 ( − k < D Rather than solving equation (A.8) for the In fact in many cases it suffices to know the length squared of th Since = 2 groups occurring in indices zero. For Table 1: α above table we conclude that Λ solve equation (A.9). Forequations the (A.10) groups and we (A.14)with are we considering find and that f equation (A.9) becom the other Dynkin indices zero. all the other Dynkin indices zero or we have an and and which are given ingoes table let 1, us study where the the case labelling of is as in figure 2 where Λ generator of know the possible scalar products of the fundamental weight given by [26] JHEP05(2008)079 D er ) ) − 4 2 and find λ 11 λ 8 ) . The and it + 3 E + λ 2 = 2 we already λ 3 4 4 ( , λ k 2 λ λ 7 ( 5 ν (2 7-forms λ + Λ , 45 2 3 70 10 λ ) − ) , 4 3 1 ) λ ation (A.8). For ) 5 λ ) = λ 4 4 λ + λ + λ 2 3 (2 ( 2 (2 α λ = 1 have multiplicity λ ( . The 3-forms always ( d in table 2 [27]. Once 6-forms 10 k 8 of all the generators with 15 126 E . Indeed these generators 40 tern for all the groups is 320 G 1 λ as usual, while for ) ) 6 ) 5 1 4 λ with highest weight = 5 and λ λ λ ) + 5 c + . However the former case has an + 5 λ 4 4 ( m 3 λ = 1 we find that each form belongs λ ( λ λ ( ( , and the 4-forms always contain the k n table 5 of [27], where the corresponding . Equation (A.8) rules out and so we have 1, 5-forms 27 6 D 2 λ 24 . In this case from equation (A.16) we − 144 and 1728 8 . The exception is the case of four forms, 11 + 1 E ) D λ ) ) 1 λ 5 we have ) ) 6 4 − λ 2 3 λ 6 + Λ λ λ ( 6 λ λ 11 ( ( − ( ( E E – 32 – = and . The 5-forms always contain the representation 4-forms 10 45 , and taking 351 133 = 3 or dimensions belong to the representation of 8645 6 2 D in dimension from 7 to 3, and the corresponding highest 4 = 1 we have 1 and α E − λ ) D 9 D k 8 ) have multiplicity zero. ) ) 7 λ λ ) ) , 2 7 1 ( ) +++ 5 6 8 λ 1 1 4 λ λ ( λ λ ( ( λ E λ λ ( ( ( , 2 5 3-forms 16 78 912 248 λ 3875 147250 and 2 4 . The three-dimensional case in exceptional because togeth = 5 and so have multiplicity zero. Hence for the space-filling forms we λ 2 ) D ) 4 7 . Hence for ) ) λ , 6 D − ) 4 5 λ 2 5 λ 3 λ ( λ λ ( 10 λ λ . It turns out the latter and the singlet in ( ( (0) ( , 2 1 λ 1 2 + Λ λ 5 2-forms 10 27 λ 133 k 3875 , 1 ) + ) ) ) ) and 1 . λ 1 1 1 1 2 6 λ λ λ λ λ 7 ( k λ ( ( ( ( − , λ = 1 for which we have the solutions + Table giving the representations of the symmetry group 1-forms 16 27 56 10 248 c 1 = , which follows the patters, one also gets a singlet of λ m 5 8 6 7 4 7 2 G E E E A D λ α We now must check that the above solutions do indeed solve equ Let us now consider the case of These results and those for all the other cases are summarise For the case of space-filling forms for . It turns out that 1 3 5 4 7 = 0 and has multiplicity zero. and D 6 1 2 5 λ completely antisymmetric indices of turns out that Table 2: weight. The representations are the conjugatesfields of were the listed. ones i example, for the case of possible solutions are λ have zero. For space-filling branes we find can have 1 2 are the level one generators, and this is the representation the results areapparent. listed The in 1-form terms generators always of have their highest weight highest weights a pat α to only one fundamental representation of that is discussed. The 2-form generators in with with highest weight contain the representation with highest weight representation with highest weight JHEP05(2008)079 +4 ]. 2 P +++ B D Nucl. ]. 09 6) and , , . There is =2 JHEP D , − N Phys. Lett. = SO(6 11 e highest weight , λ 1 hest weight of the hep-th/0104081 G (1986) 763. hep-th/0309198 Maxwell-Einstein d in this paper. How- tations of the forms in 3 x using the programme nsider the case of ds to =5 od that can be carried out t one might otherwise miss ions can also be applied to D Maxwell-Einstein supergravity orms arising in the non-linear licities of each representation. ]. forms, that always contain the (2001) 4443 [ =2 (2004) 2493 [ ]. 18 N 21 More on (1984) 89; 6) into SL(6) to find the recognisable , The symmetry of M-theories Very-extended Kac-Moody algebras and their . B 245 Exceptional supergravity theories and the magic n hep-th/0401196 A class of Lorentzian Kac-Moody algebras ]. Class. and Quant. Grav. (1984) 244; – 33 – , . It is amusing to draw the Dynkin diagrams of hep-th/0107209 D The geometry of − 10 and the one with highest weight 2 , and the 6-forms always contain the representation λ B 242 ]. D Nucl. Phys. Kac-Moody symmetries of IIB supergravity D Potentials and symmetries of general gauged (2004) 019 [ , − − Coset symmetries in dimensionally reduced bosonic string Class. and Quant. Grav. for arbitrary Kac-Moody symmetries of ten-dimensional non-maximal 11 , 10 ]. 05 λ Class. and Quant. Grav. λ (2001) 117 [ (1983) 72; , + hep-th/0205068 +++ + n D D D − JHEP Nucl. Phys. − , 9 , B 615 10 B 133 λ λ hep-th/0107181 and M-theory (2002) 403 [ 11 hep-th/0304206 E ). As explained above the lowest level representation has th Phys. Lett. Nucl. Phys. P , B 645 , (2001) 421 [ and place the forms against the node corresponding to the hig ) and the reader will readily find the higher level results use 1 D (2003) 020 [ theory interpretation at low levels supergravity theories square supergravity: symmetric spaces and kinks and Jordan algebras supergravity: Yang-Mills models Phys. 517 The reader can apply the above technology to find the represen One can calculate the representations found in this appendi = SO( − , λ 5 2 11 [5] F. Englert, L. Houart, A. Taormina and P. West, [9] B. de Wit and A. Van Proeyen, [6] A. Kleinschmidt, I. Schnakenburg and P. West, [7] I. Schnakenburg and P. West, [8] M. G. G¨unaydin, Sierra and P.K. Townsend, [4] N.D. Lambert and P.C. West, [1] P.C. West, [2] I. Schnakenburg and P.C. West, [3] M.R. Gaberdiel, D.I. Olive and P.C. West, µ with highest weight of figure 4 discussed in section 4.3, which deleting node 6 lea also an additional patternrepresentation involving with the highest spacetime-filling weight with highest weight representations to which it belongs. the other cases required in this paper. For example one can co E G ( SimpLie [30]. This has theHowever, advantage we that think it gives it theby is multip hand. useful By to doing givesuch such a as calculations purely one the algebraic can above meth cases spot pattern where features one of tha wants to highest computerealisation weights. the of representations groups These of like the calculat f representations of the forms in six dimensions. ever, in this case one must further decompose SO(6 References JHEP05(2008)079 , (2003) , Phys. ]. , 05 coset (2007) 063 Nucl. Phys. . ]. vector , s =3 07 ]. JHEP , D =2 (2004) 052 Class. and Quant. N JHEP , s supergravity-matter 08 , cible symmetric spaces ns for six-dimensional (2000) 007 , Academic Press, New hep-th/0406018 (2003) 303 08 =2 hep-th/0110263 JHEP hep-th/0304109 N supergravities , , supergravity from hep-th/9112027 (1985) 445. (1990) 57. JHEP B 658 =4 2 , =5 D (2004) 018 [ (1985) 385. supergravity: superHiggs effect, flat D 06 B 241 (2003) 023 [ =2 Cosmological billiards and oxidation (1992) 307 [ ]. N B 250 07 Lagrangians of Nucl. Phys. Higher-dimensional origin of No-scale JHEP , 149 , Special geometry: K¨ahler an intrinsic formulation -model isometries in very special geometry – 34 – (1986) 71. JHEP ]. σ Phys. Lett. , , ]. ]. theories Nucl. Phys. Classification of manifolds K¨ahler in Unified Maxwell-Einstein and Yang-Mills-Einstein origin of all maximal supergravities , B 276 Class. and Quant. Grav. Broken Special geometry, cubic polynomials and homogeneous 11 =6 hep-th/0312251 , E . D (1985) 569; Hyperbolic billiards of pure The hep-th/9207091 ]. B 255 Nucl. Phys. Commun. Math. Phys. hep-th/0606173 , (1962) 1. (2004) 548 [ Vector multiplets coupled to The group theory of oxidation. II: cosets of non-split group hep-th/0212024 , ]. ]. ]. ]; Special geometries, from real to quaternionic 13 52 The group theory of oxidation Selfduality for interacting fields: covariant field equatio origin of brane charges and U-duality multiplets Tits-Satake projections of homogeneous special geometrie Differential geometry, Lie groups and symmetric spaces (1992) 94 [ hep-th/9909099 space-time supersymmetry Hidden superconformal symmetry in M-theory , 11 (2007) 27 [ On root systems and an infinitesimal classification of irredu E Nucl. Phys. =2 , (2003) 348 [ 24 B 293 N hep-th/0304233 hep-th/0406150 arXiv:0705.0752 hep-th/0005270 hep-th/0210178 [ York U.S.A. (1978). Osaka J. Math. 047 [ Fortschr. Phys. [ [ multiplet supergravity couplings supergravity theories in five dimensions Lett. potentials and geometric structure chiral supergravities systems B. de Wit, P.G. Lauwers and A. Van Proeyen, E. Cremmer et al., Grav. quaternionic spaces from symmetries B 658 Scherk-Schwarz reduction of [ [27] F. Riccioni and P. West, [26] P. West, [25] S. Araki, [17] S. de Buyl, M. Henneaux, B. Julia and L. Paulot, [23] A. Van Proeyen, [24] S. Helgason, [18] P. Fr´eet al., [20] M. and G¨unaydin M. Zagermann, [10] L.J. Romans, [22] B. de Wit and A. Van Proeyen, [21] E. Cremmer and A. Van Proeyen, [19] P.C. West, [16] M. Henneaux and B. Julia, [12] B. de Wit and A. Van Proeyen, [11] L. Castellani, R. D’Auria and S. Ferrara, [13] E. Cremmer, B. Julia, H. L¨uand C.N. Pope, [14] A. Keurentjes, [15] L. Andrianopoli, S. Ferrara and M.A. Lled´o, JHEP05(2008)079 , 09 , y extended JHEP ; , Class. and , Class. and Quant. , ]. ]. theories hysics and mathematical (2007) 286 +++ , invited talk given at G es Nuffield gravity workshop B 645 and the embedding tensor Cambridge Workshop 1980 Kac-Moody spectrum of 11 hep-th/9711048 ]. arXiv:0711.2035 chiral supergravity in six dimensions and E ]. 2) , and a ’small tension expansion’ of M-theory Phys. Lett. , ]. 10 = (4 11 E E – 35 – N (1998) 289 [ (2008) 069 [ ]. 02 Duality symmetries and hep-th/0207267 at low levels, gravity and supergravity 8 B 420 arXiv:0706.3659 A , invited paper presented at JHEP ]. , and arXiv:0706.0667 8 Dual fields and Non-propagating degrees of freedom in supergravity and ver E hep-th/0212291 Phys. Lett. (2002) 221601 [ , ]. (2008) 045012 [ 89 (2007) 038 [ 25 arXiv:0705.1304 11 (2003) 2393 [ Group disintegrations Very extended , Chicago U.S.A., July 6-July 16 (1982). 20 JHEP , 2 hep-th/0612001 [ Quant. Grav. (2007) 047 [ (half-)maximal supergravities solvable Lie algebras G AMS-SIAM summer seminar onphysics applications of group theory in p Grav. Cambridge U.K., June 22–July 12 (1980), published in Kac-Moody symmetry of gravitation and supergravity theori Phys. Rev. Lett. [28] F. Riccioni and P. West, [29] F. Riccioni, D. Steele and P. West, [30] E.A. Bergshoeff, I. De Baetselier and T.A. Nutma, [31] E.A. Bergshoeff, J. Gomis, T.A. Nutma and D. Roest, [32] J. Gomis and D. Roest, [35] P. West, [34] R. D’Auria, S. Ferrara and C. Kounnas, [36] T. Damour, M. Henneaux and H. Nicolai, [33] B. Julia,