JHEP11(2008)091 (charge) resented. 1 l July 30, 2008 November 4, 2008 Duality. November 28, 2008 ived in detail from . Furthermore the appearing with the 11 Received: ′ E α Accepted: weights of the Published: e pp-wave, M2, M5 and the ensions of the content of the the exotic, KK-brane charges in three, four, five, six, seven d in the relevant decomposition 11 11 E E representation of 1 l Published by IOP Publishing for SISSA (charge) representation of 1 l to the fundamental objects of M-theory and string theory is p 11 E Supergravity Models, Global Symmetries, M-Theory, String The particle, string and membrane charge multiplets are der [email protected] [email protected] representation, with the string coupling constant and 1 l Department of Mathematics, King’sLondon College, WC2R 2LS, U.K. E-mail: Dipartimento di Fisica, Universit`adi Romavia “Tor della Vergata”, Ricerca Scientifica, 1,Scuola Roma Normale 00133, Superiore, Italy, and Piazza dei Cavalieri, 7,E-mail: Pisa, 56126, Italy SISSA 2008 c Peter West Abstract: Paul P. Cook the decomposition of the and eight spacetime dimensions. A tension formula relating Charge multiplets and masses for representation of tensions of all the Dp-branes ofof IIA and the IIB theories are foun The reliability of thecharge formula multiplets. is tested The by formulaKK-monopole reproduces reproducing from the the the low masses t level for content th of the expected powers. The formulaof leads M-theory. to a classification of all Keywords:

JHEP11(2008)091 5 9 3 5 11 14 16 20 24 27 30 38 40 44 48 12 56 52 heory. Gen- . These arguments 7 E Serre relations to the a into representations of rams of finite dimensional the deletion of one node of D relevant to 11D, IIA − to make this conjecture were avity, leading to its formula- 11 11 erators of E E belongs to a class of algebras known as representation 40 11 has been conjectured to be the algebra de- 1 E l 11 E , are not well understood and analysis of their – 1 – 11 E representation of 11 1 l E The Kac-Moody algebra charges algebra in eleven dimensions encoding the symmetries of M-t p 11 representation of E 1 l 3.3.1 Compactifications3.3.2 of M-theory IIA3.3.3 supergravity IIB supergravity 1.1 The eleven-dimensional1.2 theory The ten-dimensional1.3 IIA theory The ten-dimensional IIB theory 2.1 General decomposition2.2 Rank 2.3 The particle2.4 multiplet The string2.5 multiplet The membrane2.6 charge multiplet p-brane charges associated to general weights of 3.1 The truncated3.2 group element as A a tension3.3 vielbein formula Brane tensions from weights of the 35 and IIB SuGra Lorentzian Kac-Moody algebras whichthe have the defining property Dynkin that diagram leavesgroups behind [2]. a set Progress of has Dynkin diag been made by decomposing the algebr content is hampered byputative the generators of lack the of algebra. a simple However way of applying the Introduction. Contents 1. The scribing the symmetries of M-theorybased [1]. upon The previously arguments unnoticed used tion properties as of a D=11 nonlinear supergr realisation which included the Borel gen eralised Kac-Moody algebras, such as lead to an 2. Exotic charges 3. Mass and tension in toroidally compactified backgrounds 3 4. Conclusion A. Low level weights in the JHEP11(2008)091 r 1 l as [3] the 11 11 E E time by ets from represen- 1 . l representation 1 l representation into 1 l in this way one can re- o carry a Kac-Moody ltiplet the dimensional e from the adjoint rep- er unit volume. Indeed one can associate a half- representation of 11 1 ing connection with areas the l 11 symmetry the translation erive this tension formula nd analysing the content. E brane charge multiplets in e a clear interpretation in g the proposition that the ve been found inside the ity. In addition one finds all E ward is by the introduction ough the arguments used to theory. st levels of the decomposition 11 in a physical theory are those g theory by application of U- sentation, or the ex structure to be interpreted E weight of the tail in [5]. It has been proposed entified in all dimensions from een constructed as an empirical ory a non-perturbative concept n the ce and time are already present ion of ion of n symmetry algebra, but a natural [3]. The success of this approach is 11 E – 2 – representation. Furthermore one finds that the 1 l representation one can find a corresponding generator representation and the adjoint representation of 1 1 l l sub-algebra. In this paper we further the arguments in favou 8 E representation by deriving all the possible charge multipl ⊗ 1 2 l and to check our formula by duplicating the tensions found in A 11 symmetry considered only the bosonic fields of supergravity . It was proposed that one could more naturally include space E 11 11 E E However in the original non-linear realisation of the The associations between the If the conjecture is true that string theory, and M-theory, d representation contains the brane charges of M-theory, the representation contains generators having the correct ind 1 1 finite dimensional sub-algebras which are graded by a level a Analysing the low-level content of the adjoint representat produce the dynamics of thethe bosonic bosonic sector fields of of D=11 the supergrav D=11as supergravity well theory as at the the lowe dual gravity field leading to a dualised gravity generator had to beresentation added of in by hand. Space-time did not emerg enlarging the algebra to include its first fundamental repre that the translation generator is associated to the highest tation, as well as the adjoint representation of and appears at the lowest level in the as the central chargesconjecture of an the supersymmetry algebra, even th l and, in fact, for every generator of the adjoint representat three dimensions which are predicted by U-duality [6 – 10] ha well as other very-extended algebras havethat been the studied full in set de of brane charges of M-theory are contained i of the relevance of the its algebra in three to eight spacetime dimensions. algebra one may hope toof better string understand theory the that algebrasof by are group mak well theoretical understood.terms constructions of that One the string direction can theory. for that be The give most argued rise fundamental to properties to theinside hav the simplest algebraic dimensionful construction, quantities. viainterpretation Spa the for local mass Lorentzia is missing.amenable In to the study case is ofpreviously the string [6 – tension the 10] a of group BPStool theoretical p-branes, tension to formula discover or has the the b duality charge mass symmetries. multiplets p of M-theory Itin and is the strin our context principle of aim in this paper to d BPS brane solution [4] and in the representations of an representation [11], and thethree particle to multiplet eight hasreduction in been on id [12]. an 8-torus To was find carried the out three as dimensional a decomposition charge of mu l associated with the conserved charge on the brane. Followin U-duality charge multiplets. JHEP11(2008)091 ity ne charges algebra, and s the deleted representation 11 1 algebra and find l E 1 by attaching three vations later in this − root. Decomposing representation. The and at the first few D 1 l representations of the 11 A 11 lest class of KK-brane brane charges are listed t in sections 1 and 2 we representation, based on E ion 3, we will present a E ose the e of U-duality in [6 – 10]. e observation that most of 1 f l ing theories. t been presented explicitly K-branes, which are higher- plets in various dimensions e es. We present this original n the on associated to any root in sition of the , for example, [21, 22]). Some ormula that will be presented to the fundamental objects of ector space that has not been licitly the charge multiplets of multiplets confirm the findings e a higher dimensional origin to thin the algebra. In the remainder of section 3 D diagram, each extra simple root having the − relevant to the eleven dimensional theory as 11 8 algebra. In the first section we will give the 11 – 3 – E E 1 E l root to make it into an for D spacetime dimensions is given in section 2.1. 10 D representation and identify how the charge multiplets A 11 − 1 l E 11 E . ⊗ 8 includes all the exotic KK-brane charges expected by U-dual 1 E − 11 D representation of E A 1 l representation also offers a classification of all the KK-bra 1 literature previously that will greatly simplify our obser l sub-algebra, graded by the level, which is the number of time 11 10 representation of E A 1 l representation of representation are given in section 3.3 and the simplest KK- representation is associated to exotic charges carried by K 1 is described completely by its Dynkin diagram which is found 1 l 1 By deleting the exceptional node, labelled eleven, one finds A by-product of the tension formula that will be derived is th The discussion in the paper will be split into two parts. Firs l l 11 remaining simple root must be added to the same length as any root of the computations in thethe section 1. Expressions for the tensi E in their entirety. 1. The additional roots to the longest leg of the paper. In section 2, we study the the algebra in this way one finds the adjoint representation o we reproduce the particle, string and membrane charge multi and give the tensions oflater the in exotic this charges paper. accordingin The to tensions the the for f literature the but particlebefore. and the string membrane In multiplet thetension charges second formula have and part no use of itM-theory the to and string derive paper, theory the commencing in tensions section with associated 3.3 sect from the roots of th are organised inside the representation, we then derive exp M-theory in three to eight dimensions. The abstract decompo into representations of the corresponding representations in the decomposition of the In section 2.2 we give the criteria for finding rank p charges i conversely the the will give explicit decompositions of the expected in M-theory.charges expected We in will M-theory, as present well the as in full the set IIA and of IIB the str simp dimensional generalisations of the KK-monopole andcertain provid brane charges and KK-modes inof lower these dimensions (see KK-brane chargesThe have been observed as a consequenc well as the decompositiondecomposition to using the an two explicitused ten-dimensional basis in theori the for the root lattice v interpretation of KK-brane charges offers a new way to decomp transformations, and these are organised into finite sets wi JHEP11(2008)091 , s 11 . E (1.1) 11 E Dynkin 11 E is believed to representation representation , set to one. In 1 1 ∗ 11 l l α E shown below, and one ng node 9, which yields se fields are well-known 12 and the ions connected to the IIA e basis denoted herein by one of the most beautiful e Dynkin diagram leaving E ∗ . nd the two ten-dimensional six-form field and the dual = 1 has the form: es nodes of the α xtra root, ments about the root lattices respond to the choice of IIA an elegant way. he highest weight of a repre- 11 sionally reducing the algebra orresponds to the deletion of ∗ t to physicists are the decom- E . i m groups, which give generators in give the decomposition relevant tion will differ cosmetically from α weight lattice [3]. The decomposi- i 11 α i E , the first fundamental weight of m 1 l takes the first fundamental weight of =1 11 i X 11 E + – 4 – ∗ root lattice with , or charge, representation of α 1 l 12 = The Dynkin diagram of E Dynkin diagram (a line of nine connected nodes). One β 9 by the deletion of the node A other than the adjoint are also interesting and of direct 12 11 E Dynkin diagram with a node attached by a single line to the E Figure 1: 11 representation of sub-algebras. For example in the reduction to ten dimension E 1 1 l diagram, giving the Dynkin diagram of − weight lattice [3]. Alternatively one can obtain the D 11 A 11 E E by extending the is found at level one with highest weight vectors instead of the more usual simple root basis, 11 11 The representations of We recall that the A general root appearing in the } E E i , associated to the translation generator, and treats it as t e 1 relevance to theoretical physics. The contain all the brane charges of M-theory in the there is choice ofaspects which of nodes the to construction,or delete namely IIB and that theories the this in two gives choices ten rise cor dimensions. to More explicitly one delet levels one finds theto gravitational gravity field, field. abosonic With three-form fields the from field, exception eleven a to dimensional of D supergravity. the dimensions Dimen dual correspondsrepresentations to to of gravity deleting the different nodes of th can do this inthe two bosonic ways, fields by of deleting the nodes IIA 11 and and IIB 10 theories or respectively by in deleti diagram so as to leave behind an tion of this algebradifferent to nodes different of spacetime its dimensionspositions associated giving also Dynkin representations diagram. c of SL(11) Ofthe and interes algebra SL(10) conjectured sub- tosupergravity be theories the brane respectively. charges In ofto section M-theory M-theory and 1.1 a in we sectionsand will 1.2 IIB and supergravity theories. 1.3 we Themuch make presentation of in the this decomposit the sec literature{ in that it will make use of a vector spac l sentation in the of longest leg of the then restricts to just those roots with the coefficient of the e other words one decomposes Before beginning the decompositions of our algebra a few com of JHEP11(2008)091 } is 11 ij (1.3) (1.4) A appears . It will ,...e 1 3 2 β , e = ∗ e 2 { y position has been made , so that hen we give explicit roots 2 1 ebra. One consequence of ] = 0 (1.2) . the Dynkin diagram. These relations which, in terms of e of generators appearingch in of the the Serre relations, ot the generic root = n of the Serre relations in a . . . 12 ] lattice and a vector orthogonal 2 1 E ctor space basis L(11) tensors graded by a level j l have been applied. 11 , F i E n the paper but has not been much F and 11 are deleted and is called the [ are the simple roots indicated by the i ∗ algebra is given by the Dynkin diagram ij . . . α lattice. Another consequence is that any δ i = 2 and 1 l 10 , we recall that fundamental weights are 1 F l 2 ∗ = A 11 − α > E y j generators in the second relation) where – 5 – = , α i i ∗ F representation into representations of a preferred α ]=0 [ 1 < λ The Dynkin diagram of l . . . ] j 2 for roots in the , E i ≤ E generators ( [ 2 Figure 2: i , by writing β y E . . . i ) E ij [ A − is the first fundamental weight of denotes the i’th fundamental weight and i 1 l λ shown above when the nodes indicated by a sub-algebra, giving representations of SL(11). This decom 12 We first decompose the roots into components in the E 10 of Where gravity line in this decomposition.and The are generators believed will to be be S the brane charges of M-theory. for the root lattice thatused will in simplify the later literature calculations previously. i The preferred previously [5] but here we will present the results using a ve to it, which we shall call dual to the corresponding simple roots of an algebra: Where in the root latticethe depends Chevalley upon generators, the are, application of the Serre A of generalised Kac-Moody algebras are in order. Whether or n Here there are (1 and another readily definedthe condition, algebra on to be the discussedand index later, structur are are far sufficient easier tosimple capture manner to is mu apply an computationally. openin problem the The in mathematics lattice applicatio and at later high on levels w these are1.1 the The criteria eleven-dimensional that will theory Our aim here is to decompose the root appearing in the lattice must have connected support on the Cartan matrix associatedthe to Serre the relations is Dynkin that diagram of the alg Dynkin diagram of the algebra. In this case, JHEP11(2008)091 : 1 (1.6) (1.9) (1.8) (1.5) l (1.10) in this 12 gth is the E lattice and a 10 ised by more than A n the Cartan matrix are nto infinite copies of a finite o two. In this notation the j ess than or equal to two. It is 10 b define the Kac-Moody algebra hat leads to an infinite algebra. ∗ r our root system. We introduce = 11 . . . ) (1.7) j X , is often referred to as the level of the 2 i ) (1.11) 11 i 11 , a e α α 1 11 i i ∗ , e + α ∗ = 11 i m i X 8 + λ m = 1 9 · · · =1 11 , defined in relation to the simple roots of i i X − · · · + − =1 11 11 10 i X z i e + 1 + A b e 1 i − 1 ( l = + – 6 – lattice descended from the highest weight a . We represent the simple roots of e . 1 2 1 ( b l ∗ +1 − 10 11 11 i 11 = 11 e 3 − e y α i 11 X m E ∗ , the coefficient of + − e = Λ= = = 11 i 9 e e β z = m y = = i the root into components with roots in the 11 α α 1 literature . We write, endowed with the Lorentzian inner product: , but here, since we will consider decompositions character representation therefore takes the form: z l and similarly for 11 } . Explicitly, 1 l E i ij 11 δ e i a = ∗ . The fact that the orthogonal component has an imaginary len > ,...e = 2 11 i 1 11 j , e − P , α ∗ i e is the i’th fundamental weight of = = { is, i 2 < λ a λ z y by We note that in the 1 10 A Where vector orthogonal to it, So that the innerreproduced, products and between all thevector roots simple have roots length-squared encoded normalised i t We may further decompose be useful later to consider an explicit vector space basis fo the basis And defines the weight vectors in the A generic root in the Where sign of an indefinite algebrawe - deduce from that the all Serre relations rootsthe which in existence the of algebra a have negativeIt length-squared contribution l to also the allows root us lengthalgebra to t graded sensibly by decompose the orthogonal the imaginary infinite component. algebra i basis with, one level we will continue using the notation decomposition and denoted Where JHEP11(2008)091 k + 11 i m rep- ip 1 (1.17) (1.14) (1.13) (1.16) (1.19) l or as =1 j i . At level 11 3 P 1, implying a m and the new 2 as ≡ a ≥ i 1 j p 1 k a B R m eneral solution is s expression for t 8 8 and ≤ i enerators of the adjoint ip , SL(11). By taking the in terms of the blocks of aring at level , j> i 10 =1 10 i λ by rewriting A  i C rs. We recall that the p + 1 k P highest weight, the translation C 11 ≡ =1 10 i ˆ X Λ (1.12) ) (1.15) m C k A ≡ − , j i z + 8 α + 1 + 11 + 11 1 (1.18) j  i + 1 C i i B j 11 p p − m + ) ) we find the coefficients of the simple roots B i i i − m + 1 + 11 11 p j =1 10 i − X − − + λ 1) A m =1 10 3 2 – 7 – − 1) − i X − ( 8 (11 (11  − 1 3 times on the translation generator. Consequently A ≡ λ =1 =1 10 10 + # = 3 A i i k 11 X X = + 11 , we have, y in this decomposition. So that, m +  m 11 11 11 1 1 = 3 + 11 λ m m m #= β 1 8 = m − λ − (3 ( z 11 2 3 j j 11 11 m ˆ Λ= + ( y is three times the level [20]. Substituting the expression f = = j 11 ∗ we will find representations of SL(11) described by the m α E 11 . Let us explicitly realise the lower bound on m , by its commutator action with the three-form generator, i . We can now give a direct interpretation of a p C P is a constant greater than or equal to zero. Substituting thi P is an integer which is bounded from below by the condition tha .The simple root coefficients must be positive integers. So a g i C k p ≥ k the three form generator acts We note that , 10 i>j β 11 P ˆ into the expression for the level, At each level Where we have defined the useful, integer expressions given by parameter j Where, that Where Λ is a highest weight representation in the weight space of antisymmetrised indices that appear on the SL(11) generato Where in inner product with the fundamental weight resentation as a representation ofgenerator, SL(11) is formed from its m we can express the number of indices, #, of any generator appe representation of By the same analysis one can see that number of indices on the g we have, JHEP11(2008)091 1 2 l be : (1.23) (1.21) (1.24) (1.20) 11 (1.22) aining lattice E 12 E  control the i 2) , a two-form, a p − P is bounded from i )( k j i B , controls the number − i p C , which corresponds to which is bounded above occur in the i hat is the representation of (11 p =1 10 8 8 j t on the Dynkin diagram k i i X 1 p l i ≤ P p 11 ces, i.e. not including volume )+ . An interesting class of roots , while is satisfied by the weight labels t an interesting choice is to set ke = k = 1, therefore any root in the , j> j>i en indices are proportional to the X Ak ∗ + . . . k n m 2 + 1 e + 4 j − ) + 2 representation at each level:  k , , in eleven dimensions, and correspond i i B ǫ 0 p , i.e. , 10 i 22 − − k n p A 10 = − i X jk = 2 are all connected by the links of the Dynkin diagram 2 =1 10 (11 i i X , j 2 k − p 12 β – 8 – )+ k ≥ 22 E j corresponding to the into the simple root coefficient expressions above + 1 =1  10 k i X − j − β 11 2 =1 B 10 A X n (8 1 since we are considering irreducible representations m − = 1) in terms of an integer − 11 + + 2 ∗ ≥ 2) ∗ 1. In our notation above, the variable m jk 2 e 1 gives an upper bound on m − A ( ≥ m i = − 1 . . . )( = β i . These are tensors which have the correct index structure to 2 j m (8 (with − − 9 1 m , . At low levels one finds the translation generator, 11 abcde 0 , if we do this we exclude from our algebra any generators cont i = , 11 (11 Z m p 2 i 2 m p . β = 2 ǫ P =1 2 10 i = X β k 8+  1 9 . In this case, ǫ = 2 Substituting our expression for , and a five-form, That is, the non-zero coefficients of a root in β for a given level 2 = 0and so ab i (and do not form a sub-algebra) and, by construction meaning that roots in the algebra must have connected suppor generators having no blocksforms, of eleven antisymmetrised indi occurs when we consider the lower bound for The fact that representation must have below in terms of the weight coefficients of the The first term gives the index structure of an SL(11) tensor, t and below. We observe that Consequently we find the generic root at a particular level when laid out on top of the diagram. blocks of indices in our generator of length less than eleven We recall that a putative root exists when of blocks of eleven antisymmetrised indices.completely Blocks antisymmetric of tensor, elev or volume form To reiterate we have solved the problem of finding which roots C to trivial representations of SL(11). We therefore note tha the volume form, Z we have: Such that, p JHEP11(2008)091 . 9 (1.25) (1.31) (1.30) (1.27) (1.28) (1.26) ,...α 2 , α 1 α representation given in section 1 l 11 Λ (1.29) α − 8 i w ν = 9 ≤ and i  p ∗ 10 revious section. We denote, α 9 , j heory to the ten dimensional =1 m i y be obtained from the eleven X he well-known supersymmetry d earlier we will use a different iven in [5] and in the notation lely with bosonic fields but also on a circle [23]. In terms of the + ebra in eleven dimensions. This on in table 25 in the appendix. + 1 ≡ lattice. For the j i 1 9 11 11 ν α , j B e i A m − . To find integer coefficients we find − m − i 8 11 ) w p + 1) w 9 1) 9 =1 10 , and related to the two deleted nodes of ν 1 i − X lattice containing the roots 11 10 8 e k − 11 9 i>j 9 − − 1 m 11 − 10 A P − · · · 9 w z ν j 8 and m − + 9 2 3 + ν – 9 – = 10 q + 1  + 11 + − e i m 10 ( + algebra, specifically we have, 11 z y m α ip + 1 z 9 10 m 8 = =  =1 A + 8 ν j i lattice that gives an ∗ = we find, 11 + 2 11 11 A α P j 10 α w m m − A ν in addition to the deletions of A ( ( ≡ + + j 1 10 10 10 j 2 3 1 α ν B (  = + j and Λ= y i m = ip β and =1 9 i 10 11 P = = are the fundamental weights of the 2 i w A ν Where 1.1. The procedure is an extension of that carried out in the p this means the deletion of Where interpreted as the central charges of the supersymmetry alg is quite surprising not least becausebecause we the have been charge working algebra so ischarges. infinite and We continues list beyond t theThe low-level results content of derived this in computationalof this problem this secti section, were originally and g to much higher levels, in1.2 [16]. The ten-dimensional IIA theory We now carry out the reductionone, from which the has previously eleven-dimensional t beennotation carried useful out to in our [5], later but computations.dimensional as note supergravity The theory IIA theory by ma dimensionalalgebra reduction we delete a node of the Taking the inner product with So that solutions parameterised by two variables, Therefore, Where Λ is a highest weight in the In this decomposition we have, JHEP11(2008)091 and is an , the b (1.33) (1.34) (1.32) (1.37) (1.38) (1.39) (1.40) a k and let R Z 2 i p by writing we have, e appendix. k ≥ − =1 2) (1.41) 9 i k corresponding a − P 8 i ( actions of = i j ...a branes of type IIA 1 p B k 10 a i 11 i e p p Z m ) , , 10 9 6 2 =1 j>i X i X a m 1 n the variable ...a arge algebra corresponding a + 1 − a R k nsidering Z controls the blocks of ten anti- 11 wever for general 8 (1.35) 2) + 2 − ated to trivial representations of , , m C 4 q ≤ − C ...a − + ( 2) 1 + 1) a 1 (1.36) 10 qk − k + 10 Z 6 i − i , ke ( , j p 2 − a ) + 9 10 a ( i + ) 1 i k to simplify the notation and q a + 1 m k n p we neglect these trivial representations whose − q e j q, Z − i + − in the algebra. Denoting the number of indices adjoint actions of p  B =1 10 , the charge of the fundamental string i – 10 – i + 10 + A X q c k, (10 11 p ( Z 11 − =1 P − A 11 10 i . The solution corresponds to SL(10) tensors with n ( 9 + m A =1 m i k 10 X 2 − = 1 2 m q kj i X P 2) + + = ≡ = 2 − = = = = − = k ≥ − i k j 9 k # ( 10 11 q 2 i  m m + 2 p m m 2 9 =1 0 is a constant. We see that q 9 X n and even forms =1 and i X ≥ i 5 + p ∗ C ...a e 1 =1 1) + = 0and so where we have shifted a 9 i =1+2 = Z − a C P 2 β a β where + ≥ (2 k C 1 indices, associated to a k = + − theory and the weights at low levels are shown in table 26 in th i q p 10 part of the Dynkin diagram: =1+ m =1 IIA 10 i 2 11 + β P E 11 = m the We note that 2 one action of the translation generator, Where in the last equality we have expressed the lower bound o on a generator appearing in the algebra by #, we have, k integer. In this case we have, generators carry blocks of ten antisymmetrised indices. Ho symmetrised indices appearing in the algebra,SL(10). which By are setting rel The results of thisto section the are readily used to compute the ch NS5 brane charge, us rewrite Amongst the charges we find a scalar, And, to the Ramond-Ramond branestring charges theory. of the D0, D2, D4, D6 and D8 As mentioned, the interesting set of solutions is given by co JHEP11(2008)091 8 12 E ) of 9 (1.46) (1.44) (1.42) (1.45) (1.43) (1.49) (1.48) (1.50) (1.52) (1.51) (1.53) ,...α m 2 , α 1 α . Let us use # . The solution i a p P 11 , 8 i>j Λ (1.47) 2 1 P we find, 5 2 − − j 9 z 8 = = + A i earing in the algebra, then,  s not been presented in the 2 i 2 = 11 ≤ µ Dynkin diagram dual to the i 10 ip p 9 m C , j A 11 2 5 =1 from the Dynkin diagram of , 8 =1 j i i 8 X 1 , v ) and find, − 10 ≤ = 11 P + 1 µ + 10 k ≡ 11 9 α j e i − 11 = m B 5 4 α p , j v i , j j + − 2 1 and + B m ) 1 2 + 1 )+ i 9 9 − v p 11 8 + 1) j + 1 + 10 , x α , =1 10 ) 5 2 i k, j − 8 m X µ x 9 i e B , 11 k and A ( ∗ e  − m − − − + – 11 – − 1 5 1 − − α 9 ( 11 8 x v y + 1 indices, counting the number of actions ( − 2 1 p 1 + 2 − µ + 10 + 8 )+ m jk x (10 9 9 ) algebra whose positive simple roots are = = = − ) − 10 9 = A A ( m 8 m 9 + 9 9 9 ∗ e e =1 10 9 i 10 X − 2 − A i α α A ( ( + = m α m m · · · · · · ip 1 j 1 = = ≡ j 10 10 µ + + contracted with the translation generator m =1 # 8 i ( 1 1 2 e e a ( ( P = 1 Λ= +(1+ a 1 1 2 5 j the fundamental weights of the y = Z i m = = = µ . We have introduced the vectors, A which have 2 v } x β 11 10 α A , 8 ,...α 1 . In this decomposition we delete, α 11 { α literature before, is obtained by deleting Now, 1.3 The ten-dimensional IIB theory The charge algebra corresponding to the IIB theory, which ha Where we denote by roots We parameterise the roots using gives tensors of and finding representations of the By taking inner products with the fundamental weights of the Where, In this case we have the two form generator and to denote the number of indices on an arbitrary generator app JHEP11(2008)091 , 11 ν p i IIB p then, which (1.56) (1.57) (1.58) (1.59) gives a i 11 l p l [6, 7, 10, X − i< n 11 9 = , =1 8 i E m 10 )+2 P j = m − = 10 11 k e m ) (10 l , we disregard these j i p p n weight lattice. The − i i p y weights of ory. The Greek indices k 11 rane and the NS5 brane; , B =1 etry corresponding to the 8 i i gravity line of roots. This charges we find odd forms =8 1 (1.54) − p j>i j X 9 P 9 − corresponding to trivial rep- 11 e point that if direction: , =1 A m ebra corresponding to the i 8 X = = -theory are the Weyl reflections + 2 ν then so does k + 11 + ( p 10 kA − e ) (1.55) ) 2 ) . One can find the components of the corresponding to the brane charges of q i k ) − 10 . For the special case − − we find, taking the inner product with 2 − 2 α 1 l v 10 9 l 2 1 k αβγ α qν ( (10 m = 9 i ( ≡ + ( p 2 1 10 ...a n ν 1 = 0, and thus – 12 – 1 2 e 8 which are in the ) + 10 =1 m a = i 9 X  − C Z i m 10 − p = x, v n m − 2 11 11 , = p and l i 8 X ( l ) − )+ diagram is , we have, αβ i k ( l 1 lattice, 7 2  − highest weight as A 9 ...a 2 − 1 9 1 1 =1 A =1+2 x 9 (10 a A X n 2 i 2 Z m p + β , ∗ = α 8 =1 e 5 i X q = ...a 1 β a 1) + Z − , l 3 (2 l ...a root lattice with different brane states being represented b 1 transform under the SL(2) symmetry. a n Z E root that was deleted and not directly attached to the , =1+ aα 10 2 vectors orthogonal to the resentations of SL(10), and by setting The last term counts blocks of ten antisymmetrised indices, If we parameterise Consequently, And, representations. Note that inα this case we have an SL(2) symm Therefore by writing the SL(2), given by afundamental weight Dynkin in diagram this with a single node, has its ow We write the root length squared in this form to illustrate th The results of this section aretheory used to to compute low-levels the in chargeZ table alg 27 of the appendix. Amongst the the fundamental string and the D1 brane; the D3 brane; the D5 b 2. Exotic charges The U-duality transformations of toroidally compactified M D7, D9 branesα, as β . well . . as their S-dual charges of IIB string the of an is due to the Weyl reflection perpendicular to solution (i.e. root length squared less than or equal to two) β JHEP11(2008)091 er- = 4. (2.1) D ) and the string 248 ( mensional reduction g implicitly included the particle multiplet 1 λ imensions derived from MNPQR . This work is discussed n olutions found under the Z multiplets was given in [8, monopole. It is these extra E seven charges found in the n erived from the dimensional ions completed the particle s whose higher dimensional let in five dimensions found to five dimensions one recov- lution as a weight vector and ication of the Weyl reflections E rvations [5]. The Weyl group ociated to the dual to gravity l origin was unclear. ng upon reduction to y algebra reproduces only the as introduced as an empirical al superalgebra and are exotic nal supergravity, many exotic ng charges were recognised as tion of the antiicommutator of ctor corresponding to the brane imensional reduction of charges ally in the review [9]. y t weight e to the KK6 monopole, from the MNPQR αβ of SU(6), which is the representa- = 3 many more exotic states were Γ 2 1 contains the full set of brane charges D MNPQR 56 + 11 , Z E MN MN Z , Z MN αβ – 13 – M Γ which does not appear in the supersymmetry al- P algebra. For example upon dimensional reduction 1 2 n + E M = 4 were associated to the compactification of the KK6 P representation of D are the fundamental weights of αβ 1 contains an infinite set of charges in addition to the central i l λ MNPQRST,R 11 =Γ Z of SU(6) by assigning all the central charge indices to be int E } in the compactification. In β 7 ¯ 27 Q T , = α ), where of the degrees of freedom of the Q 6 { 49 ⊕ 3875 ( = 15 7 sub-algebras, appearing upon dimensional reduction, bein λ ⊕ 21 n 6 representation of ⊕ E fills out the U-duality brane charge multiplets. Foe example 1 l n In addition to the expected brane charges, coming from the di The conjecture that the 21 E ⊕ of charges associated tocharges brane were solutions also inorigin present eleven was dimensio in unknown. the Exotic charges,reduction U-duality in of charge other the words, central multiplet are chargesthe not appearing eleven d in dimensional the supercharges decomposi [25], in detail in the original papers [6, 7, 10, 17, 18], but especi multiplet to three dimensions (n=8) the particle multiplet has highes 9] and we will recover aspectssolution of encodes this analysis. the tension The of weight the ve of BPS brane states and appl was discovered by encoding thethen well-known applying particle brane the so U-dualityaction transformations of to all it.multiplet. the combinations In The this of new constructiontool. the s the U-duality The tension brane transformat weightfundamental multiplets vector representations containing w of the the particle and stri 17, 18]. A group-theoretic approach to uncovering U-dualit For example by dimensionally reducing The of M-theory gave anof eleven-dimensional origin the to these obse from the outset. Furthermore thefield in inclusion the of eleven-dimensional the algebra, chargeoutset the ass field explains giving the ris appearance of the exotic states appeari nal indices. Thisin agrees [6, with 7, thethe 10, U-duality central 17, charges particle of 18]. multip the7 eleven dimensional However supersymmetr the particle multiplet in four d charges of the supergravity algebra and the charge of the KK6 tion of the particleparticle multiplet multiplet in in four dimensions fourother are dimensions. than derived from the the usual The d central extra charges. charges of These the extra eleven states dimension in monopole, having charge uncovered in the charge multiplets whose eleven dimensiona ers the gebra, wrapping the JHEP11(2008)091 i 1 l λ and (2.2) . We 1 . Our D − algebra 11 they , in the iplet has D 1 D l A . . . α and , and now ∗ is a vector in D 8) and identify + 1) x α with components , D ≤ 11 11 . 8 in [12] in all cases E D = ( E D ≤ i − d in [11]. ≤ 11 ree dimensions of the D iD ) E mposition of the in terms of its mula that played a crucial ) ill derive the brane charge ≤ c charges appearing in the ⊗ l be derived in section 3, to a, but to do so we will first 11 11 tiplet of charges. 1 space of eletion of nodes E ( e all associated to KK-branes, − E expression given here will have the deletion of A D ( we delete the fifth node. The two i A , and when 6 λ 1 to E − D 6= D ⊗ 12 ,i 4 A =11 E X we delete the node labelled D, in figure 3, i by expressing the D’th root, A =1 i D D − − − 11 = 11 E – 14 – E representation of to various dimensions (3 ⊗ 1 1 ⊗ l − 11 1 D E − A D x, ν . Denoting the fundamental weights of the decomposed A D + − ν 11, which is the algebraic equivalent of compactification on 11 The decomposition of ’th component in the Cartan matrix of − E conjecture. Compared to the original work uncovering the U- = D < i, D 11 D E α has previously been carried out in [11] and the particle mult representation in dimensions, D, where 3 and then second we will apply a tension formula to the results 1 11 Figure 3: l is the representation of 11 representation of E 1 are the fundamental weights of 1 E 1 l l iD ) − ) 11 E ( ...D A sub-algebras. The details of this decomposition can be foun = 1 D To find the In this section we extend the work of [11] and outline the deco In this paper we will derive essentially the same tension for i − 11 for the root lattice orthogonal to all the remaining roots after are fundamental weights of Where ( form, carry out the decomposition to orthogonal to the vectors in the remaining root space after d E deleted roots may be expressed in terms of vectors in the root to obtain, For example, to find representations of to arbitrary dimensions, a (11-D)-torus and thenthe apply charges the that tension are formula, found. which wil 2.1 General decomposition Let us give the decomposition of the the results were in perfect agreement with the U-duality mul been derived from charges which give anU-duality brane eleven charge dimensional multiplets. originwhich These to we exotic will the charges consider exoti ar further in section 3. role in uncovering the U-dualitya multiplets in simple [8, origin 9] in but the the aim will be todecompose demonstrate the the validity of the tension formul duality multiplets we willmultiplets work from in a reverse sense. First we w the brane charge multiplets. We note that the reduction to th representation of JHEP11(2008)091 D − sub- (2.7) (2.3) (2.6) (2.8) (2.5) (2.9) (2.4) 11 (2.10) E D − 11 > E j ,µ 1 ) r <µ i ( , we have, µ tal weights of ∗ Λ i i 1) q λ D r, m 2) ( , 1 δ D − i = − − ) =1 r X X D r ( 1 − > ( i 10 (2) i λ y 8 j ≡ ∗ i α λ D − i > + λ 1 D, ,µ m i x j m − µ 1 x p ∗ − x  + − ,..., D and , λ D > ) D 1 m 6= > > i i r − 1 ,i ( 11 ) µ X 1 = 9 = 1 = 11 + − X 11 r ν ( i − =1 2 < λ <µ 1 (1) D = i D 2 (1) D ≡ x − 2 D D x , λ m + 1 D x z, i (1) i , λ λ m m µ ) λ + D (1) 1 i < ν,λ D (1) 1 D − − ) , α − sub-algebra, or 2, referring to the i r ( i , a general root of which is given by, respectively we obtain expressions for the root m D − D m n > > – 15 – < λ 1 α ) 2 − x j 12 < λ ) m − + j j r + 2 − = r ( i ∗ λ − E 1 ( i D x λ D + , λ )(11 ,µ z, i m (2) i i i (1) i A m (2) i ∗ D are positive integers. Substituting our expressions for 2 (1) 1 α α = = i i − α − D i − i λ z, i and m 3 ∗ i ) n 6= < λ (8 D l <µ , D − j D m m 2) i ,i l D 11 i m i D 1 µ X − m − q p + + = − m − =1 =1 =1 i D i = i i X  i ∗ X ( D 11 ˆ i i (11 ˆ and µ µ = β α − X X + − − D      y (1) i = = = ∗ m = = ) = j j m , r m i , ( n ∗ i m (1) (2) λ Λ − = m − m Λ Λ β and i n : : , we obtain, 1 we have, D is either 1, referring to the D − − i α l r D 11 A E by and 11 ∗ algebra. In this notation, Adopting the notation Where E Where, coefficients, Taking the inner product with For a positive root, We make use of formulae for the inner products of the fundamen α We now return to our consideration of derived in appendix A of reference [11], JHEP11(2008)091 . i D 1 µ − m D and, is the (2.12) (2.13) (2.11) (2.17) (2.18) (2.15) (2.19) (2.16) A 2 D z 0. The − − with the D ≥ 11 a sub-algebra k R = tions for 1 ) ) 2 − and as such we p p 1 contravariant z D diagram, − − − 11 A n E D D D E > ( ( m j ≤ ≥ 2 1 ) (2.14) 2 1 ,µ 1 11 − − rges with all indices in the − − e D D D D <µ ect to an 1, implying that dence of the n’th node and ˆ · · · + . We note that − = , j ≥ + ) 1 D ) > 1 D +1 − D j − D m m D kD representation of D e − ,µ p are the fundamental weights of ( 1 + kD l − 2) 1) i 2 1+ + . p = 11 D − µ 9 2 − 2 − 1 D µ − D n x ( 2)(11 − p D m adjoint actions of a generator D = 1= ( , j = +1+ − ) i − <µ 2 1 D p D − µ D – 16 – D i λ )+ D ( q D m D = m ( = D + + j m e m 2 D i D − 1 x X − > − ≡ m − · · · 2 D 1 > + 1 # ˆ µ µ 1) + equation in (2.9) we obtain, j , p 1 ( 1 1 + − e ,µ ˆ j µ − ( ) D 2 sub-algebra of our decomposition leads to constraints upon p p D x subalgebra, where 1 1 D < − − is given explicitly in our basis by: A 1 − . This corresponds to a representation of the = D , where we recall that = 1+ ( − p = p D x D n 2 1 . The resulting generator in the algebra has − ( 2 D − c A x λ ...a A α D 1 P <µ ( a µ Z = = j = 2 gives, m 2 D charges − α must be integer valued and negative we find a simple set of solu p j is a constant bounded from below because weight in the = 1 and from the m k ∗ p − − m We are interested in the decomposition of the D µ are the weights of the Normalising We note that the vector vector in the root space corresponding to the linear indepen These weights are derived by deleting the n’th node with resp Consequently, We commence by looking for p-brane charges, that is rank p cha 2.2 Rank spacetime algebra, Since having the form, with highest weight Where solution corresponds in the algebra to translation generator A The condition we must satisfy is that indices. Let us denote the indices by #, then, the values to be taken by the root coefficient take JHEP11(2008)091 , . i j λ D n − sub- sub- , for i j j 11 (2.20) (2.21) 1 λ D E − ≥ ≤ , − D D ) algebra, 11 A − D E = 0 we may D, i D, i 11 k E − − D 10 10 D D j j , D − ) − D, D, D − ≥ ≤ sentations of , D − , ) 10 − − − 10 ) 10 m D D 8 D 2 D D, i D, i D, D, D m D D, m = 9 = 9 ≤ = 11 − − e of fundamental weight − 3 ). By setting − − − − − − ) , j j j j − − c weights of the ) 8 8 D ) 8 − 8 D i ntal weights of D ) ≤ ≤ = 9 i = 9 = 11 ≤ − ≤ = 11 = 9 − m SL( j=11-D 2 − (6 D i − D − ) − i − 2 , j , j , j ) , j ) − , i ) ) ) , j D D D ) D 2) (2(11 D D )(11 2 D m m − m , j m − 2 − D − j ) m antisymmetrised indices in the SL( − D − − 3 D D − − i ((8 – 17 – i ( corresponding to a rank p charge. We commence ( j l ( m D + − (3 2 D , 2 − 2 2 D ) (4(11 ((8 − j D − − j D D − 2 2 − corresponding to a p-brane charge in the j − D )+ D i > m , i D − , i m D − − i 1 1 ( D − j m D ) (8 D D 11 2) − + 2) into the expression for the simple root coefficients, j + − j + m , λ − − − − D 2 2 − D                            1 (11 D i D D D D − − 2 ( m m − D m D − D 2 D D m − 1 − ( ( l < λ − 3 − l D 11 − − − in (2.9) and making use of equation (2.10) we find, D 2(11 i j D + i      10 3+ i 2(11          m λ D −      D D, = − i > = − − λ j i 8 D p , λ i corresponds to blocks of ≤ m = 9 = 11 i i i i P k                                                            < λ is a positive integer or zero, which indicates trivial repre = = l j n Having found a criterion for − algebra, we now turn our attention to restrictions on specifi algebra consistent with the values of by finding conditions for representations of single fundame neglect these representations and simplify the algebra. which we set and indicates the occurrence of trivial representations of The variable and is Where The simplest case with a solution is dependent upon the choic Substituting the solution for JHEP11(2008)091 i ), D j D m − (2.22) (2.23) (2.24) − j j D − ≤ ≥ 11 n ) algebra and to indicate an D, i D, i . We note the 0. Notice that D i D antisymmetrised a − − ≥ D − sub-algebra. This l 11 D D 10 10 − D D E , sub-algebra and then − D − − 10 D D, D, )) 1 − 11 i − D − D 10 E − − − lj, D, D j j D 10 + D − 8 − l − D D, A − − D ≤ ≥ ) by deleting node (11 − the D, D ≤ = 9 = 9 = 11 10 8 − − 2) − D j j j j D − − 8 = 11 = 9 ≤ − D, i D, i D, − 8 the spacetime SL( − ≤ = 9 = 11 being considered, and the index − − − D 10) + (2 i i = 11 = 9 ≤ ( , 1 leads to tensors with 3 8 8 l ) − l algebra, after the action of the local − 3 D − ≤ ≤ = 9 = 11 D , j ) D D , i ) A − i l 1 − i 2 , equations (2.18) and (2.21), ) )(( − 11 − j jl j D + D − E ) tensors. We use the index D m − 1 − , j − D ) antisymmetrised indices. Explicitly, deletion of D kD − − j – 18 – D 2) − lj, j )( − D (11 − j − − l , i l, 11 + − > − 2) ) 2(11 2 D , i − , we will discuss this possibility in section 2.6. jl, j is made manifest and we see that D − j 1 D ( 2) − − (11 − ( l, j D )+ l l − D − i 3 SL(11 , λ − − − − D                            1 − ) D − 11 ( − − ⊗ l i (11 D D E D ) l l, j 1 m kD, i ( kD (11 D l < λ 2 3 l − which give a rank p charge in the − (11 − − D controls the appearance of blocks of 11 − − )+ − − j − − − D i + l D      10 i 2) 2)          m − (11 m = 0 we find the charge content indicated in table 1, where the D − (11 l − − − D D, D k − = 0 we disregard the trivial representations of      > D − − D − ( l ( = j l 8 l l , λ + ≤ = 9 #= = 11 i indicates blocks of 11 2+ 2(11 i i i i l ) together with the deletion of node is decomposed into representations of SL(11                                                                 < λ D D = = = − − p j 11 n E − We choose values of given above we have, node (11 the parameter which we denote # so that, when The dependence on the parameter subtlety that a trivial representation in the indices. By setting This allows us to see that a non-trivial representation of sub-group, may give rise to non-trivial representations in amounts to equating our two conditions for In particular for read off the value of the corresponding fundamental weight in charges are indicated by SL( index in the spacetime associated to the weight of JHEP11(2008)091 f D D ⊗ − − has ) = 3 . In 11 11 D (with ) λ E D D 248 − , )(11 (63) D − ,k . The 7 that the charge i 1 ime indices will ) representation. − λ ...j )(10 D ranges from three D (1) (2.25) 1 D D j j − − 2 3 − − j j 11 D 11 1 2 (9 j j 1 E 3 ...j 1 j Z j 1 a 1 j 2 2 a 1 a a . 1 1 Z − i a a D e charge multiplet in − Z Z ...a 1 where whose highest weight is a sum just the highest weight in the 11 1 upon dimensional reduction indi- whose highest weight is ation of a ed to a fundamental weight of , Z , Z E D − n table 1. From table 1 we read D Z − D − D belong to a given representation of − 11 Highest Weight Charge (28) (56) 11 6 5 E 11 E D with highest weight, E − ...j ...j 1 8 1 1 − j j 11 E D E A ) that we have used to indicate the 2 1 with highest weight 3 D ⊗ − − − i D D D − D – 19 – − D − µ µ µ D − 11 µ ⊗ ⊗ Dynkin diagram, 11 ⊗ are the highest weights of the membrane, string and E ⊗ , Z D D SL D D The Dynkin diagram of i D − − , Z the internal indices in the decomposition to SL( − − λ − 9 11 10 11 , and so on for the other charge multiplets. It may be (56) λ 11 D λ λ 3 E D (28) − E 2 − ...j j 11 multiplet is shown in table 2. Many of the charge multiplets 1 1 11 j j E 1 ) tensors Weight of E l is associated to the (10-D)’th fundamental representation Figure 4: D = 3 [11], the charges associated to the fundamental weights o 1 i j − 1 D D λ and 1 of ) of D D a D D ) tensors: − − Z D − − − D 8 , Z , Z − 10 SL(11 λ − ≤ = 9 ⊗ (1) (8) = 11 = 10 10 i i j 8 ) i i Index of P D ...j D, ) tensors will vary throughout the multiplet, but the spacet 1 ), so we may label them by the highest weight of their SL(11 j Charges associated to fundamental weights of SL(11 D − D Z with their nodes on the ⊗ − − ) D For the case of For representations of D 11= nodes 11 particle multiplets [9]. The set of charges up to rank indicated therein are associated to representations of to eight derived from the highest weight It is convenient torepresentation, indicate it the is tensor this charge thatthat associated is such to indicated a by charge the charges i useful to associate the various charge multiplets associat E SL( cated by SL( Table 1: which is the first fundamental representation of remain unaltered. Let us consider the example of the particl to indicate internal coordinates coming from the represent SL(11 representation. transforms under. The fundamental weights of For example, if we consider the representation of SL(11 table 1 it is the highest weight of SL(11 it belongs to the representation of JHEP11(2008)091 in ges 7 ) 11 ) ) ) ) 1 1 1 3 1 = 4 it E , ...a ------, , , , 1 a 3 6 3 6 D 18 ( ( ( ( ( Z 6 , in ) ) epresentation 8 2 2 ...a ------E , , 5 1 70 50 45 a 6 3 ( ( of Z 5 ) ) ) representation of 248 1 1 3 ...a ------, , , 1 1 40 15 10 l a 8 1 1 320 126 120 ( ( ( Z we now discuss the identifi- .6, where we extract the full ntation whose highest weight 4 ) 2 ...a ------, 1 ects having no spacetime indices 1 24 16 27 a 3 144 351 1728 ( Z = 3 it is the sub-group. 3 D ) ), having a highest weight charge indi- D 1 1 ...a − - - - - - , 1 1 1 − 10 45 27 a 3 351 133 11 8645 1539 D ( Z E µ ⊗ ) 2 – 20 – 1 a 2 , i.e. it has only internal indices and it transforms 1 λ , so that in , 1 5 1 a D 16 78 56 1 248 912 D − 3875 ( Z 30380 − 147250 10 ). From table 1 we see that this charge corresponds to 11 i ...j ) j E 1 1 j a , 1 1 5 10 27 Z Z 3 133 248 representation associated to particle, or zero brane, char algebra. 3875 ( 1 1 l l ) 2 , 1 Z 56 10 16 27 3 248 ( ) tensor, D − 5) SU(2) and so on. In section 2.2 the condition on the highest weight r , 7 8 6 7 ⊗ G E E E E SU(5) SO(5 Charge multiplet representations of the group, G, from the of 8. SU(3) ≤ 56 To find the roots explicitly we recall that this is the represe D 3 5 4 7 6 8 D ), and a set of internal indices ( i ≤ a Table 2: cation of the weights of the 2.3 The particle multiplet We identified the highest weight of this multiplet above, and is the first fundamental weight of the representation with highest weight ( is the 3 of fundamental weights. We dealcontent with of these table cases 2 in from section the 2 cated by an SL(11 in detail. The particle( charge multiplet will consist of obj in the first fundamental representation of the JHEP11(2008)091 . ) 8 11 E = 1 ∗ ). We (2.27) of m 11 we must ,...m 11 11 11 248 11 11 9 p +1 11 l 4 R 3 6 9 p p p R ...R ...R 1 4 12 15 18 p p p l l V l D R l l l ...R 2 9 6 248 10 ,...m R 7 V R dimensions we V R R . ,m +1 V R V R 8 i D D E +1 α i + 1 root coefficients D of ,m = 3 which has been m D representation +1 Mass (highest weight) 1 +1 D l D 248 11 D X = = 0 for the particle charge i e first rs of the fundamental repre- a number of sequential roots xactly match those found by p ive a formula, equation (3.6), + l than in the present example. ents (1 D 1 (2.26) α = 3/The − + D 8 1 8 · · · 28 56 63 56 28 amongst the candidate roots. In more + ,...D 1 α 248 = 1 + j ∗ – 21 – α , where the charge multiplet’s highest weight is the Dimension of SL(8) tensor ≡ p correspond to the weights of the first fundamental ) 3 6 } = 1 11 ...k ...k representations have been shown to form the ,k ,k we can read off the charges in the particle multiplet and 8 5 j 1 1 7 8 2 1 j i>D = 1. Since we are considering the ,k ,k j l ...j ...j 1 m 11 8 8 ...j ...j j 1 1 P m 1 1 j j D E j j { Z ,...m ...j ...j Z Z 1 1 Charge m Z Z j j lattice with a mass. In table 3 we give the result of applying +1 Z Z which we will identify. For the reduction to indicated in table 2. To distinguish the roots of D The particle charge multiplet in 12 1 D E ,m 6) − 4) 5) 3) , ⊕ , , , 3) 3 11 +1 , 2 3 2 , , , , 2) 1 D E 7 , , 4 6 4 , 248 (1 , , , 1 3 ) 1) amongst these roots. By finding all roots of the form (1 , , Table 3: 6 9 6 7 , 11 , , , 2 5 8 2 0 , root , , 5 7 5 , 0 9 representation of E , , , 3 4 5 , , , , 4 5 4 1 12 9 7 l , , , 2 3 (1 , E , , 3 3 3 (1 ) has been written in terms of 5 7 , , , 2 , , 2 2 2 D 3 (1 5 , , , , 4 4 5 4 (1 (1 (1 (1 (1 of SL( charge associated to a p-brane. From equation (2.16), with multiplet we find that, studied already in [11], and the In addition to thissentation condition of we must identify the root vecto are 1’s. The most complicated case is that of the reduction to From equation 2.18 we have should find the particle multiplet being made up of roots whos we identify the representation of then the particle multiplet contains roots with root coeffici use the superscript notationshare as the a same shorthand coefficient, to e.g. indicate that The simple root coefficients complicated cases to followFrom this the process will be less trivia U-duality [8, 9]. this formula to the roots in the table, the resulting masses e these are listed in table 3. In section 3 of this paper we will g relating a root in the identify the root vectors of the fundamental JHEP11(2008)091 of , and 16 , and (2.29) (2.28) SU(2) 6 2 ). We ǫ M E D ⊗ − ∝ = 7 where of 11 ] mulae used 3 11 D 11 ...µ 11 5 R 1 3 6 9 p p p 27 ...k 1 l l l 11 R ...R µ 1 11 10 7 . . ǫ V R 6 Mass R ,k 3 6 p p R 7 6 1 l l R 8 R ...R 10 E E 7 ) of SU(3) Mass ...l R R 2 1 l of of , 3 , Z content listed in table 4. 56 27 5 dimension of the internal nerator and its dual may 1 = 6 we look for the l 6, and those at level 3 are ed to a generator, ...j , lume form( 1 h commute with each other indicating the volume of the me tensors. For example, in j D 5 te an invariant mass squared , Z n. By using the mass formula [ 11 = 4/The = 5/The 7 7 21 21 D D ...R = 5 we look for the , 6 6 15 shown in table 3 the invariant mass 2 2 ǫ ǫ +2 D D 2 ∝ ∝ 18 p R l V ] ] 248 6 +1 ,m , multiply to give an invariant squared mass, ∼ 7 ′ ...k D 1 Dimension of SL(7) tensor ...l R ,k M 1 – 22 – roots [16], we find the 2 248 l 8 Dimension of SL(6) tensor = 8, where we identify the ( ,k ...j 5 12 M , Z 7 2 1 j j E = 4 where the particle multiplet should belong to ,k j 5 D ...j 1 2 are dual to those at 4 7 2 ...j j 1 P j 1 j , j , Z ...j 1 j Z 1 D ...j j 1 2 Z P l Charge 1 , j Z j 1 Z l Z to indicate Charge Z Z [ [ V . In the reduction to 3) 9 p , V l 2) 2) 1 , , , The particle charge multiplet in The particle charge multiplet in 1) 1 1) , , 1 ) 3 ∼ ) , , 2 , 2 , , 6 7 0 2 ǫ ǫ we find the states of table 6. We repeat the process for 0 2 5 0 0 root , , . From tables of root , , , , 2 56 , 1 0 3 7 6 0 3 4 l ∝ ∝ 5 , , of SU(5), the appropriate roots are listed in table 7. The for 12 , , , 12 E 9 ] 2 ] M (1 9 2 3 (1 E 8 , E ,j , , (1 7 8 (1 Table 5: Table 4: 10 7 2 ...k , (1 1 (1 5 ...k , for the representation. For the k 1 2 (1 ,k 8 , Z M 8 ...l 1 ...j = l 5). From the 1 ′ j , In table 3 we have used , Z Z j [ P [ internal space. From tableto 3 form we generators can proportional identify generators to whic multiple copies of the vo squared is: MM self-dual, that is call such pairs ofbe generators combined dual. to form Thespace. a Young rectangular Such tableaux a tableau of Young whosetable a tableau height 3 ge is is the proportional the generators to at multiple levels volu 0 the mass associated to its dual generator, in this section are even robust up to Let us look at the reduction to By identifying the dual generatorsto in the table multiplet, 4 one can associa representations of SO(5 We are identifying a(to bilinear be Casimir derived for in the section representatio 3) one can see that the mass associat we look for the the corresponding roots are in table 5. For the reduction to JHEP11(2008)091 X 7 ...j ------(0) 1 11 11 5). j ), associated to 8 7 , R 3 6 3 p p representation in 1 Z l l 11 R ...R 0 10 8 7 1 R , 3 p Mass 1 l l R R R 9 10 6 Mass 8. R of SL(5) he ...j (5) ------of SO(5 (0) ≤ 1 j 7 10 28 D Z 16 . The columns indicate the ≤ l indicates a tensor carrying 5 1) and (1 , ) l 0 ...j indices. (2) (2) ( ------(2) (0) 1 , ) tensors together with the level j = 7/The 6 6 0 X 21 56 D , = 6/The Z D D 9 5 1 − 10 D − 4 6 4 ...j - - - - (0) 1 j 5 Z 3 – 23 – Dimension of SL(5) tensor ...j (4) - - (0) 1 Dimension of SL(4) tensor j 4 56 Z 2 j j 1 2 j Z j P j 2 1 j Z j P 1 (1) (1) (1) (1) Charge (1) (0) (1) j Z 6 3 3 Charge The particle charge multiplets in 3 10 28 21 15 Z 2) , j 1) 1) 1 ) (6) (3) The particle charge multiplet in ) , both having dimension three. , Z , , 3 8 7 5 j 0 The particle charge multiplet in 0 2 0 0 root , root P , , , , representation from the roots (1 0 0 8 3 Table 8: 7 , , , 12 12 1 9 (3) (1 9 2 l (1 (2) Z E E , and 1 (1 (1 64 7 2 Table 7: j (1 1 Table 6: j 6 7 8 3 4 5 D Z We summarise the appearance of the particle multiplet from t contained in the generators table 8. The table shows the dimension of SL(11 number of indices in the charge modulo blocks of 11 degrees of freedom appearing in the decomposition at level the generator appears at in the decomposition, so that JHEP11(2008)091 , ) ), ). 11 +1 11 11 . In D D (2.31) (2.32) (2.33) D − K = 1 and 1) in the ...m ...m 10 ,...m . We can , 6 5 4 λ p 0 aj , are listed in Z 0 ,m ,m ,m , 1 2 2 2 l 9 , , , 5 4 3 . D − in table 9 together the 10 = 2 for string charges. ing charge 5). This corresponds to λ , D m to the root (1 ssions in agreement with the and so on. From section 2.2. harges, by putting dex. From table 1 we see that with highest weight can be identified. It is a simple procedure of searching the low 7 roots of the string multiplet in o find an invariant mass squared 1). This root does not appear in E 1 (2.30) D of SO(5 , 12 0 − − , lattice corresponding to the weight of E 11 0 3875 10 , E 12 4 2 D take values which vary according to the E V 133 − V V 36 p 2 18 a 9 p = 0), that is the highest weight of the 9 p l ,...D 2 2 a a l l 2 R i , R R i>D p with highest weight roots with root coefficients (1 D = 1 . This charge has one compact index (j=11) ∼ m ∼ ∼ j D 11 12 – 24 – − 2 10 E D. 2 133 11 2 3875 Z M E M ). The , from which the and the corresponding roots from the M = 4 it is the by a series commutators with the generators 8 11 7 = 1 ), which we list in table 12. The associated invariant E D 11 j E . . This corresponds to roots of the form (1 = 4has 11 = 6 we look for the lattice corresponding to the string multiplet have simple 8 6 of m of Z to the roots of the particle multiplet and find string charges D E 1 D 12 , in = 1 for the string, we find that, ,...m j 8 E of p a ...m ⊕ E +1 7 133 = 1 and all other K D 27 of ,m 248 D 2 ,m − , 2 ⊕ 6 , 10 , giving the charge p D 3875 8 ( 7 3875 D representation for roots with root coefficients of the form (1 − 1 l 11 ... K E , +2 of representation however, as discussed in [11], it is related D = 4 we look for the = 5 we look for the D 1 Using equation (2.20) we find the root in the l +1 − D = 3 this is the D = 3. These roots were originally given in [11] and we list them representation, with charge = 0 in equation (2.18) we find the simple root coefficient D 10 1 D From equation (2.16), with we recall that the string multiplet corresponds to 1-brane c We find the string multiplet in In k root coefficients (1 which we list in table 11. In 2.4 The string multiplet The string multiplet contains charges with one spacetime in such charges are associated to the representation of The candidate roots in the weight vector of the representation of roots of the form (1 In matter to pair the dual generatorsfor in the the multiplet: string multiplet, t mass squared for the representation in table 12 is also apply the generators associated masses given bystring equation multiplet (3.6) listed which explicitly giveslevels in expre of appendix the B of [9]. The and one non-compact index (a=D), and so is a component of a str K l the D reproduces the representation we are seeking to identify, is (1 µ table 10. The invariant mass squared for the multiplet is, in a similar manner. To commence we rediscover the JHEP11(2008)091 2 2 11 ) 2 11 11 11 11 11 R 11 11 2 11 11 11 R R 11 11 24 p 11 R R R l 11 11 11 11 10 24 p 18 p SU(3). 2 11 3 4 2 3 R 10 6 10 11 l l ...R ...R ...R 11 R R R R ...R ...R 10 10 10 24 p 33 p 27 p 12 p 15 p 18 p 18 p 21 p 11 5 5 8 8 9 l R l V l V l l V l l l V ...R ...R R 3 3 2 2 3 p 6 p 9 p 9 p 6 7 10 R 27 p 30 p 12 p 12 p 15 p 15 p 18 p 21 p 21 p R R R l l l l × ...R 4 5 3 4 3 4 4 3 4 4 4 3 2 4 11 8 5 9 VR R R l R l l l l l l R R l l 2 3 V V V V ...R 3 2 2 R 11 9 p 8 3 ...R 4 4 3 2 2 15 Mass p R R R R R R R R R R R R R R R ( l 3 3 3 3 6 R l ...R V 3 6 9 VR VR p p V V V 4 3 3 R R R R 5 12 p V V V l l VR V R R R R 5 R 8 4 3 3 3 3 3 l V R 4 3 3 3 R VR R R 3 3 3 R 4 R R R R R R R R 3 Mass R R R R R R R V R 4 R R 4 R R . . 7 8 E E ) of SU(2) 3 of of , 1 133 3875 8 8 8 8 1 28 70 70 28 36 56 63 63 56 36 420 216 216 420 168 504 720 504 168 7 7 35 49 35 = 4/The ), and we find only the root with = 3/The 11 D D Dimension of SL(8) tensor =8 we find the ( 7 4 ...m 2 9 D 1 1 m 1 ...m ...m 1 7 6 4 3 1 1 ,m ,m 2 Dimension of SL(7) tensor l ,m 5 8 1 1 1 ,m ,m ...l ...l ...l ...l 1 3 2 ) 6 5 8 ,m ,m ,l ,l ,l 7 l 2 1 1 1 1 1 2 ,l 8 8 7 4 7 ...l ...l 1 , ,l ,l ,l ,l 6 k ,l 1 1 – 25 – ...k ,l ...l ,k 3 6 7 8 7 8 ...k ...k ...k 7 1 4 1 8 ...l ...l ,l ,l 1 8 1 6 1 1 1 ...k ...k ...k k 8 8 1 1 ,k 1 ...k ( ,l 1 1 1 ,k , ,k ,k ,k ...j ...j 8 ...k ...k ...k ...k ,l ,l 1 ...k ...k 8 ,k 7 4 ...j ...k 8 7 8 8 1 1 ,k ,k ,k 1 1 1 1 8 8 1 1 6 a,j ...k ...k 1 ,k 1 8 8 8 1 ...j ,k ,k ,k ,k 1 1 8 ,k ,k ...k ...j Z a,j a,j ...j ...j ,k ...j ...j ...j 1 8 8 8 8 ...k ...k Charge 1 7 7 1 a,j ...j ,k ,k 1 1 1 1 1 8 ...j ...j ...j aj 1 1 Z Z of SL(5). This corresponds to roots of the form 8 8 1 ...j aj 1 1 1 ,k Z aj ...j ...j ...j ...j aj aj aj 1 aj ,k ,k a,j 8 Z ...j ...j ...j Z 1 1 1 1 aj aj aj aj 8 8 1 1 Z ...j ...j Z Z Z 1 aj Z Charge 5 Z 1 1 Z Z Z aj aj aj aj ...j Z aj Z aj aj ...j ...j 1 aj aj Z Z Z Z 1 1 Z Z Z aj Z Z aj aj Z Z Z The string charge multiplet in 4) 5) The string charge multiplet in 3) , , , 9) 9) 2 3 10) 11) 8) 8) 2) , , 1 7) 8) 6) 7) , , , , , , 6) 6) 6) 7) 5 5 , , , , 4) 5) 5) 4) , , 1) 6 7 5 5 , , , , 3) 4) 5) 4 6 , , 3 4 4 4 , , , , , 1 , , , , 3 , , , 3 3 3 4 3) , , , , , , 2 3 2 1 , , , , , 2 2 3 , , 0 7 9 8 8 11 10 , , , , 6 9 , , , 2) 1 12 14 10 10 7 6 7 8 , , , , , , , 2 5 4 6 5 4 , , , , , , , , , , , , 1) 4 4 6 , , , , , , 0 , , , , 1 3 root ), which we list in table 13. In 5 7 12 14 12 12 7 9 8 7 17 16 , , , 3 0 6 6 9 4 18 21 15 15 11 10 11 12 , , , , , , , , , , , , Table 10: Table 9: , , , , 2 5 , , , , , , , , 5 , , 11 6 7 6 6 4 5 , , 0 5 5 7 9 8 9 9 12 4 2 , , , , 3 10 11 10 10 14 13 Root , , , , 3 4 , , , , , , 14 17 12 12 , , 5 5 5 , , 5 , , , 2 , , , 6 4 4 5 , , 7 6 7 7 , , E 3 3 , , , , 8 8 8 8 4 , , , , 5 2 3 , , , , 2 9 9 12 , , 4 4 4 , , 4 11 10 , , , 3 3 3 , 2 , , 5 4 5 5 4 11 13 2 , , , , 6 6 6 6 , , E , , , , 3 , , , , 2 2 3 6 6 , , (1 ...m , , , 3 3 3 , , 3 8 7 2 2 2 3 2 3 3 3 3 , , 8 9 , , , , , , (1 4 4 4 4 , , 4 (1 , , , , , 2 2 2 8 , , 3 4 4 4 2 2 2 2 , , , , 5 4 (1 , , , = 7 we look for the , , 3 2 2 2 2 5 5 , , , , 2 2 2 2 , , , , , , 3 3 3 2 2 (1 , , , , , 3 3 3 , 3 2 2 (1 (1 (1 , , 3 3 3 3 2 2 3 3 3 3 , , ,m (1 (1 (1 , , 3 3 (1 (1 (1 (1 3 3 (1 (1 (1 (1 2 3 3 (1 (1 (1 (1 D (1 (1 , (1 (1 (1 (1 7 This corresponds to roots of the form (1 In (1 JHEP11(2008)091 7 multiplet ...j (3) 1 ------11 (11) 11 aj 8 11 224 ...R 11 3 6 9 p p p R Z l l l 8 V R 5 11 5 11 R Mass 5). R V ...R 3 6 p p 5 . R 3 6 p p R R , l l 7 8 l l 6 6 6 7 R R R Mass R Mass E R 6 of dimension 3 and ...j (8) 1 ------R (5) of 1 7 8. aj representation is sum- 484 aj of SL(5). Z of SO(5 ≤ 27 1 Z 5 l D 10 o find multiplets associated 5 harge and other charges up ≤ ...j (5) 1 - - - - aj 728 = 5/The Z = 7/The 6 6 15 D 4 1 = 6/The D 5 5 4 D ...j (2) (2) (2) 1 - - - (10) (2) 5 aj 70 35 15 70 Z Dimension of SL(6) tensor 3 – 26 – Dimension of SL(4) tensor ...j (7) Dimension of SL(5) tensor (4) 1 - 1 , indeed one may find multiplets corresponding to a 4 4 ,k aj 35 6 728 D 1 4 Z ...j ...j − 1 1 aj ...j a,j ...j 1), corresponding to a charge 1 11 a,j aj 1 Z , Z a,j 2 λ aj 0 j Charge aj Z Z Charge 2 Z (4) 1 , Z The string charge multiplets in 3 Charge Z 0 aj ⊗ , 5 2 484 Z , 2) − The string charge multiplet in 1) 3) 8 , 2) The string charge multiplet in D , , , 1 1) 2) 0 1 The string charge multiplet in µ , , 1 , 1) , (9) , , 2 0 aj , 1 0 (1) (1) (3) (1) (1) (1) (1) 3 root , , 2 , 0 8 in table 14. , , 7 6 6 5 4 3 8 Z , 3 0 , 2 Table 14: root 2 5 224 , , 3 , 0 12 root 2 ≤ , 3 , 2 , 3 , 4 12 E 2 , 2 4 1-brane, as discussed in [8, 9]. One might also find a fivebrane , 7 , 12 7 , D 2 3 2 Table 11: E 3 6 , Table 13: , 2 − (1 (6) E a , (1 (3) 6 5 , . The appearance of the string multiplet in the ≤ (2) 2 Table 12: (1 5 Z 1 D , (1 (1 11 49 847 5 (1 3 p R l 8 (1 R 5 6 7 8 4 3 D In addition to the string and particle multiplets, we may als tension simple root coefficients (1 whose highest weight is marised for 3 with the membrane charge, ato threebrane that charge, of a a fourbrane c JHEP11(2008)091 11 E 2 ) 2 11 2 11 2 11 11 R 11 11 11 11 11 11 R R 11 11 R R 11 11 2 11 10 R 15 p 18 11 2 11 18 p p 2 2 ...R 10 10 l l l R R R ...R ...R ...R 10 10 12 18 18 18 p p p p 9 2 3 6 6 10 l l l l ...R ...R R V V 2 2 2 9 9 7 8 p ) 12 15 15 18 p p p p R R l ...R 5 R 5 5 5 9 6 9 3 3 l l l l R R V R V R R 9 9 3 6 9 5 p p p p p ...R ( V V V 2 2 4 8 15 15 18 p p p l l l l l 4 4 4 4 5 5 2 3 3 R R R R l l l R R R R 7 8 3 3 3 V R R R V R V R V R V V 4 4 2 4 4 3 3 2 2 R R R R R R R R R 3 4 Mass V R R R R R 3 3 3 3 3 4 2 2 4 R R R R V R R R R R 3 3 2 2 2 R R R R R V R R R 3 2 2 R R R R ( 2 R R R R R 3 2 2 2 2 2 R R R 2 2 R R R R R R R 2 R R 2 R representation of R iven in the literature 1 in three, four, five, six, l as the particle and string ultiplets previously found find the remaining charge = 2 in equation (2.18) and p but are all a natural consequence 8 8 70 378 216 1 × × 56 28 × 420 216 280 336 378 280 × × 1344 3584 4200 2352 1344 1800 2 2 2 2 4 roots showing the 12 E symmetry. In the next section we will consider – 27 – 11 Dimension of SL(8) tensor E 1 3 2 2 1 1 l l 1 1 = 0. Using equation (2.16) we look for roots of the form 4 ,l ,l ...l ,m 1 1 6 5 3 ,l ,l 1 6 3 1 1 2 ,l ,l 4 7 1 ,l k ,l ,l 1 5 6 k ...k 1 7 4 ...k ...k ...k 5 ,k 1 1 ...k ...k 3 6 ,k 1 1 1 ...k ...k 8 ,k Table 15 — Continued on next page 1 1 ) in the tables of 5 = ,k ...k ...k ,k 1 1 7 ...j ,k ,k ,k ...k ...k 7 1 1 8 ...j ...j ,k ,k 1 6 7 6 ...j l 11 ,k ,k 1 1 1 1 8 8 ab ...j ,k ,k 1 ...j 7 7 1 ...j ,k ,k ...j 8 7 1 Z abj ...j ...j ...j 1 8 8 1 ...j ...j abj abj 1 1 1 ...j ...j abj 1 1 abj ...j ...j Z Charge 1 1 abj Z Z ...j ...j 1 1 abj abj 2 Z 1 1 abj abj abj Z abj abj Z 2 ,...m abj abj Z Z 2 abj abj Z Z Z Z Z 2 abj abj Z Z +1 4 2 Z Z Z Z D ,m 3 , 2 , 6 5 2 3 4 1 1) − Level D ( of the conjectured eleven dimensional variety of exotic charges whose interpretation is obscure, the membrane multiplets inmultiplets detail, shown in and table in 2. section 2.6 we2.5 will The membrane charge multiplet Until this point wefrom have focussed U-duality. on reproducing Thepreviously. the Here membrane we charge charge treat m thecharges multiplet membrane and has charge we in apply not the ourseven same been method and way eight g to dimensions. find the For a membrane multiplet membrane charge we take look for solutions when (1 which are listed at length in [16]. JHEP11(2008)091 ) ) 11 11 = 3 m 11 2 11 D 11 11 11 11 2 11 3 11 R R s ( 11 11 11 11 11 2 11 11 2 11 11 R R R R R R 10 10 . 24 24 p p ,...m 27 p R R R R R R R R l l 11 11 2 11 11 l 8 10 10 10 10 10 10 27 24 24 p p p R R 3 4 3 3 7 7 27 p ...R l l l 2 l R R R R 10 10 10 10 10 10 10 10 9 9 24 27 27 27 21 21 24 p p p p p p p R R R R R R +1 ) 8 6 6 6 E l l l l l l l R R V V V V 3 3 2 3 6 9 9 9 9 9 9 27 21 24 27 21 21 p p p 7 p p p R R R R R R R R R R 5 5 5 6 6 5 5 5 5 3 3 3 3 R l l l R l l l R R D V V V V 3 3 3 3 3 2 2 3 8 8 27 21 21 24 24 p p p p p R R R R R R 2 4 4 4 5 5 5 2 4 2 4 3 4 R R R R R R R R R l l l l l R R R R 3 3 3 3 of 3 3 3 2 2 2 ...R R R V V V V V V V V 4 2 4 4 2 4 4 4 2 4 5 5 2 2 2 2 R R R R R R R R R V ...R Mass R R R R 2 2 4 3 3 3 3 3 3 3 3 V V V V V V 4 4 2 4 ,m 3 4 R R R R R R R R R R R R R 2 2 2 2 3 3 3 3 3 3 R R R R R R R R R V V 2 4 2 4 3 R R R R R R R R R 2 2 2 2 2 2 2 2 3 3 ( R R R R R R , 2 R R 2 2 2 2 2 2 R R R R R R R R R R 2 R 2 2 , R R R R R R and so on. In with highest weight R R 1) 7 147250 D − E − D ( 11 of E 912 = 3/The D from which we can identify the root 63 28 36 56 8 945 720 945 420 168 504 1512 1280 2800 1 E × × × × 120 × × × × × × 2520 1512 2352 7680 1764 7680 2100 8820 1008 × × × 10584 5 3 3 3 2 4 2 3 3 4 of 2 2 2 1 = 4 it is the ⊕ D – 28 – 248 Dimension of SL(8) tensor , in 8 ⊕ E 3 3 4 8 2 2 2 1 1 2 2 1 1 1 1 1 m m m of 1 ,n m m ,n 1 1 1 1 5 3 4 6 ...m ...m ...m ...m 1 1 1 1 7 4 ,m ,m ,m ,m ,m 2 1 1 1 1 3875 ,m l 7 5 4 6 2 ,m ,m ,m ...l ...l ...l ...l 3 ,m 1 ,m ,m ,m ,m ...l ...l 7 6 7 1 1 1 1 1 ,m ,m ,m ,m ,l 6 3 4 3 ⊕ 1 1 ...l ...l ...l ...l ...l ,l ,l ,l ,l ,l 5 6 4 8 8 ...l 1 1 1 1 1 ,l ,l ...l ...l ...l 8 7 6 7 8 1 ...l ...l ...l ...l 7 7 ,l ,l ,l ,l ,l 1 1 1 ,l ...l ...l ...l ...l 1 1 1 1 8 7 8 6 7 ...k ,l ,l ,l 6 1 1 1 1 ,l ,l ,l ,l ...k ...k ...k ...k ...k 1 7 8 7 147250 ...k ...k ,l ,l ,l ,l 8 8 7 8 1 1 1 1 1 1 1 ...k ...k ...k ...k ...k ,k 8 7 8 8 ...k ,k ,k ,k ,k ,k 1 1 1 1 1 Table 15 — Continued from previous page 8 ...k ...k ...k ,k ,k 1 8 8 8 8 8 ...k ...k ...k ...k 30380 and we list the charges in table 15, where the charges at level 1 1 1 7 7 ,k ,k ,k ,k ,k ...k ...k ...k ...k 1 1 1 1 ,k 8 8 8 8 8 ...j 1 1 1 1 ,k ,k ,k 8 Charge ...j ...j ...j ...j ...j 1 ,k ,k ,k ,k ⊕ 8 8 8 ...j ...j 1 1 1 1 1 ,k ,k ,k ,k 8 8 8 8 1 1 ...j ...j ...j ...j ...j 8 8 8 8 ...j 1 1 1 1 1 abj ...j ...j ...j : The membrane charge multiplet in representation having coefficients of the form (1 1 ...j ...j ...j ...j abj abj abj abj abj 1 1 1 abj abj ...j ...j ...j ...j 1 1 1 1 Z 1 abj abj abj abj abj 1 1 1 1 Z Z Z Z Z l abj 3 Z Z abj abj abj 3 3 2 3 3 Z Z Z Z Z abj abj abj abj 147250 Z abj abj abj abj 5 2 4 4 2 Z Z Z = 3 it is the Z Z Z Z 147250 2 4 2 Z Z Z Z D Table 15 , in 9 8 7 D The membrane charge multiplet is a representation of Level − 11 the roots in the λ identify the vectors of the JHEP11(2008)091 (2.35) (2.34) 11 . . 7 R 11 6 12 p l 11 11 11 E 10 15 p 21 p 11 2 11 2 11 2 2 3 3 8 l l E 9 p R R R 15 p V R l 4 7 7 ) R l V V V V ...R 2 2 2 9 p 4 9 15 p 18 p 18 p 6 l of 7 6 4 4 4 4 9 VR VR VR R R R l l l 3 p 6 p 11 V V V of 12 p 12 p 24 p 5 R R l l R 3 4 4 4 6 6 2 5 5 5 R R l l l 11 R R R R R 4 4 4 3 2 5 R 6 5 3 3 3 3 4 R R R R R R R R R 11 Mass 2 R R R R R V 3 3 3 5 5 R R ...R R R R R R 3 3 3 5 4 78 V ...R 9 p 2 5 9 3 R R R R R 912 l R R R 5 9 VR R R 3 p 6 p R R R 12 p 15 p 6 n. l l 4 5 3 R l l ( R R ge multiplet in table 15 is 4 5 R R Mass R V 4 R R 5 4 R R ) and find the set of charges R 4 R 11 = 5/The = 4/The D D 6 ,...m 3 V 5 2 V ) 2 b 54 p ,m ) 1 7 7 1 l 35 21 28 28 21 35 b 140 224 224 140 R 3 27 p l a , R 2 a R 1 1 , 20 36 20 ( 3 R ( ∼ – 29 – ∼ . The associated invariant mass squared for the Dimension of SL(7) tensor 7 2 912 E 4 7 2 147250 Dimension of SL(5) tensor 2 l M 1 1 of ...l ...l 1 3 6 5 ,l ,l 1 1 M 2 ,l 4 7 ,l ,l k 6 7 7 ...k ...k ...k ,k 1 ,k 3 3 6 3 1 1 1 1 ...k ...k 5 5 as shown in table 17. The associated invariant mass squared ,k ...k 1 1 ,l 912 ,k ,k ,k 7 ...k ...k ...j ...j 1 ...j ...j 1 6 6 6 7 ...j ,k ,k 1 1 ...j 1 1 1 1 ab ab ab ,k 1 7 7 1 ...j ,k ,k 7 E Z Z Z ...j ...j ...j 1 7 7 ab,k ab,j ab,j 1 1 1 ...j ...j ab,j ab,j Charge Charge ab,j ...j ab,j 1 1 Z of Z Z ...j ...j The membrane charge multiplet in Z Z 1 ab,j The membrane charge multiplet in Z Z 1 1 ab,j ab,j ab,j Z ab,j ab,j Z Z Z ab,j 78 ab,j ab,j Z Z Z Z Z 5 1 2 3 4 Level 7 8 3 4 5 6 1 2 Table 17: Table 16: Level = 4 we look for roots of the form (1 D = 5 we find the The associated invariant mass squared for the membrane char In D 10 to 17 which are the duals of those at levels 8 to 1 are not show In in table 16, which form the membrane charge multiplet in table 16 is JHEP11(2008)091 . D m (2.36) repre- 1 l = 6, the = 8. We 7 (6) D ...j (14) D 1 ------representation in the abj 1 1856 l 11696 we will complete Z D 5) in ) indices appearing , um of fundamental − 6 D 8. − D 11 (11) ...j (3) (3) ≤ 1 ------11 SL(2) in E we considered all had a − E D abj D 420 147 ⊗ ⊗ D − Z . Now we will continue to 16408 from the ≤ − 1 of SO(5 D 11 5 − 11 − E (8) D E ns of ...j (5) 11 16 1 - - - - - (16) A f E traint for the root coefficient abj 56 273 = sub-algebra. Previously we have 23472 Z 5 D 2 4 − ⊕ V (5) ...j (13) 11 2 (7) 1 - - - ) , with highest weight more general than a 10 E b 35 had a single set of antisymmetric indices 18 p abj D l 5152 5152 R ⊕ =(1.22) of SL(3) Z 1 − a − 1 1 3 11 R = 0 in equation (2.21), implying that we have D ( E (10) ⊕ ...j l A – 30 – (2) (2) (2) (4) (2) 1 - - (2) ∼ 1 4 56 35 20 20 10 abj Z 2 78 23472 sub-algebra but we will generalise our considerations 1 2 M − j (7) 1 (4) (15) D abj A 273 420 Z 16408 The membrane charge multiplets in 3 = 7 and the (12) (4) (6) abj (3) D 5 Z 147 1856 11696 Table 18: (9) . The representations of (3) ab (1) (1) (2) (1) (8) (1) (5) (1) (1) (17) 1 1 1 1 1 1 1 1 1 Z 11 1 36 29128 E of SL(5) in 7 8 4 5 6 3 D to include those representations whose highest weight is a s 5 D in table 18. − = 11 11 So far we have not considered the significance of blocks of (11 4 E E ⊕ single fundamental weight, provide a straightforward cons weights. As we shall see, representations of of highest weight which was a single fundamental weight of consider p-brane charges in the and corresponded to p-brane charges, the representations o By including the most general highest weight representatio sentation of restricted ourselves to the constraint that in the index structure of the generators in the In the previous sections we considered representations of 2.6 p-brane charges associated to general weights of of table 2. 1 summarise the appearance of the membrane multiplet charges for the membrane charge multiplet in table 17 is We can continue this process and find the JHEP11(2008)091 ]. i D i D − . Let D i (2.37) (2.39) (2.41) 11 m D D m − − 11 4 D m e D D D 11 D 3 ) − ...p me indices D D − D D − − E D 2 − D − − − − 11 i − − − − 8 p D ,p when, , as: D 11 − 10 1 8 , we find, after h p j i ≤ = 10 = 9 = 11 p p 11 m tensor. However ) +3 n λ = 10 = 9 = 11 ≤ p 2 j j j j i 2) (2.38) ) + D D D j j p − − e simple root coeffi- − D − D − 10 D 11 − P p − − 9 2) D ...j ( 11 p 1 p h − +2 j +2(8 ) + without an index indicates ǫ D ) (2.40) + +(11 D D D p h − labelled by [ ( − 9 ) D e parameter, D r values for − h e p 10 D − l representation of D p + ( − 10 + + ) − +4 p 10 i ) +(8 11 (11 D e hj p · · · D ) ip E D we now have − − D + i + D − 10 − + 3 − λ p i 8 D 9 C ) +1 p − +(10 p D +(8 P + +4 11 − h − + D i D p − D p D 2 − 9 h sub-algebra, and 9 > − (10 e − j 9 p D p p D ) j p )( + 3 D ) =1 − ) i + ( X – 31 – 8 k − D + , λ D D 9 D + 2 1 11 D − − = − ) − he E h j 10 m h D ) indices may be acted upon by the generators of the h j p < λ − + − root is: D i (9 n + 4 D which are identical up to the +2(8 p p e )+(1+ +(10 +2(10 + 3 + 2 i 12 − i i p D m j D  D E ip D D i − − ≥ − ip ip − − i 8 + 4 p − − D D 11 9 i D D 11 e − n 11 > p D P p 8 − − 8 8 E ip h p = − j j ≤ + i X 8 ≤ ≤ i i i D + 2 + + + 4 , λ =1 − i + 2 − i i P i so that a set of trivial internal indices may acquire spaceti · · · X 8 P P D D p 3 h ip D − − 2 4 ) + h 11 j = h h j −  < λ 10 10 1 2 ≤ E 2 2 i i 11 p p e D − − 1 − − 1 1 +1 ( p p D are: i D D D m k P + + D ( − X + 8 = D D D D + j                                  X n − D − − − ∗ ≥ 8 i 9 9 − e + 11 = = 9 p p are the Dynkin labels for the is some positive integer, or zero. For this solution we find th E p 2 2 P j = i n h p          β − sub-algebra of = j 10 n making use of equation (2.10), and the remaining internal indices form part of a non-trivia ignored representations of A Putting this into equation (2.9), where instead of us consider a generic highest weight representation of the appearance of a block of (11 We see that we have a general solution giving positive intege Where cients in With these coefficients the full Where the corresponding p-brane charge multiplet. We can write th JHEP11(2008)091 D C − 11 (2.43) (2.44) E with a looking = 1 with 12 s of E D 0 then from m = 3 is given in of representation. D D h> − D − 1 representation of gth squared may l 11 = 1, and we will 11 ) (2.42) 1 p l , this implies that D D hp D in the decomposition, , whose highest weight D 6 − − m elations. Consequently D D 10 m − − of roots having the form − p = 2. From equation (2.38) 2 = ). The highest weight root D 11 (11 11 − n D 1 et associated to this particle E 11 C ) tensor representations the − p r the trivial representation is: 10 he cases with m D D + commencing as before with the i − hp = 3 we can satisfy − p 4 ) indices. We note that the lower and on of 11 = 0, and ) i p ,...,m D − D D p D − appearing in the − D +1 − − = 1and 9 − = 1 is the only possible representation, 11 D D D D 9 p p . In n = 0. Earlier we found the particle charge 1 − 2 − 4 = 0 we find, − 18 p p D ,m hp 2 11 l h V 11 k 4 − − 8 p E 11 ∼ (11 D − − E − D i 2 1 – 32 – D − =1 10 ip i 0. The extra particle multiplet in X − p M 10 2 9 = 2 and so it is present in the D D p ≤ =1 − = i 2 . Setting − X 2 8 2 9 β D = 3 a new possible particle charge multiplet appears D h β p − 2 2 m 2 i m D − − ip − j 2 representation into SL(11 D ≥− p h D i =1 − i X D − 8 C 2) ip − 11 + j − 11 n ≤ p i equation (2.43) places constraints on which highest weight X E -independent case where D ( ) indices appearing on generators at level representation. Let us see how useful this equation can be by D +2 − 1 ) and subject to D ,... l # = 3 2 11 − = 1+ − 2 , β 0 , controls the appearance of blocks of (11 ,...,m = 1, and from equation (2.43) we see this corresponds to a root = 2 C +1 = 0 and under the action of the SL(11) sub-algebra the root len 2 = 0. Putative string charge multiplets have , h D β 2 . From equation (2.38) we see that indeed h for the particle, string and membrane charges. Consider the particle charge multiplet, having β 1 ,m = 0 λ 4 11 i Since # is given by, Let us repeat this processcase for the string charge multiplets, bound on the number of indices, # occurs when If we decompose the number of SL(11 is bounded from below by are present in the So that for all the highest weight representations of is E multiplet associated to the first fundamental representati and from equation (2.43) we find commence with the However if we now generalise our considerations to include t associated to a different highest weight of equation (2.38) we see that in p squared length of zero.charge We by now looking identify the for complete roots multipl having the form (1 has be lowered or heldthe constant, highest weight but representation not(1 may raised be - found due among to the the set Serre r table 19. From table 19 associated invariant mass squared fo JHEP11(2008)091 ) 1 l 11 6= 0 is the D (2.46) (2.48) (2.49) − ) string = 0. If 8 D 1 ≤ h − , D ,...,m i − p 10 tring charge +1 λ 11 11 248 D . p , 9 p 8 l = 0 for the first D ...R E 4 ,m − Mass 2 R 2 β of 10 , 3875 p 3 . Consequently the 2 in the second case. 1 2 D − − ≤− 11 D p 2 + 1) in equation (2.38) we − β ree (the p presentations of the internal , any weight with 11 = 1) do not appear in the es 20 and 21. From table 20 e form (1 or = 3/The p 2 representation when D p D D = 1) (2.45) 1 − − l 9 9 D 1 p p − 4 =2=( = 3. We find, = 2) (2.47) 10 − D p D 4 h respectively. The string of roots in the ( ( 4 m D V 36 p V 2 a 1 − l 36 p = 2) and ( 2 a , l , R 10 1 R p p ∼ D D representation and associate an invariant mass ∼ Dimension of SL(8) tensor = 1) = 1) − − – 33 – 9 2 1 2 1 2 11 p 2 248 p and the p 0 in the first case, and 2 ( ,p = 0) we notice that we face the restriction 248 M M − + 1 = 2 we observe that the remaining positive terms ≤ − h Z , p j 2 = 1 D 6= 0 satisfying 248 p β = 1) representation having highest weight − i Charge h h 2 9 ( p ip D = 2) 2 j − 1 ≤ 3) p 10 − i X ( , are zero. From equation (2.43) we find that p 2 i 1 i , p ip +2 3 A second particle charge multiplet in , D 5 =1 − root , i X 8 4 , 12 + 3 E , p = 3, they are the 2 for the second case. So both of these are highest weights of s 2 , − 5 D Table 19: 1+ = (1 2 ≥ β representation such that 2 1 l The invariant mass squared is: must at least be accompanied by non-zero values of Bearing in mind that for the string putative string charge representations with ( must be balanced or outweighed by the negative terms. That is we now consider solutions when We note that the invariant mass squared is the same over all th we find three possible highest weight representations: However from equation (2.43) (with unique representation carrying the string charge in the charge multiplets in threegroup dimensions is - a the property Casimir of of the the spacetime re algebra. Where all the remaining multiplets in may find the remaining string charge multiplets in representation. Thus the ( representation may be found by searching amongst roots of th in the case and The precise roots formingwe these can multiplets identify are dual shown generators in in tabl the squared to the multiplet: JHEP11(2008)091 0 D = 11 11 11 11 − 11 11 2 R 2 11 ...R ...R 9 p R R ...R β 18 p V l 9 6 10 h > p l V 2 3 7 3 4 12 15 18 21 24 27 p p p p p p 4 3 2 R R l l l l l l V V R R 2 2 e shown R R V 3 3 Mass V R or 18 p V R 3 V V l 3 R R 3 3 3 R R Mass R D R R − 9 . = 1) (2.50) 8 p . E and so on. The 8 D , and the string = 3) (2.51) E − 1 8 of h = 1). When 11 ( E of p gth in equation (2.46) D ( 1 6 and they correspond 248 of harge multiplets. With − , rge multiplets there are tiplets from the particle − 11 1 , , brane charges with p 8 1 8 4 28 56 63 56 28 1 = 1) − ed for in [9] see for example , 1 = 1) tation carrying the membrane 2 e dimensions. = 3/The = 3/The ,p D − , D − D and the 0 = 2 10 , having ( ≤ h ,p ( 2 248 Dimension of SL(8) tensor β Dimension of SL(8) tensor = 1 1 , 6 3 1 l ,m 8 p 2 1 8 147250 l ...l ( 5 8 ,l 1 = 0 in equation (2.38). The possible highest 1 2 7 = 1) ...l ...k ,l ,l k supplemented by a singlet 1 1 8 8 ...k ...k 1 h 7 ,l , 1 1 D ...k ,k – 34 – . This means that at least either 8 ,k 1 8 i − ,k ,k ...k ...k 8 ...j 8 8 1 p 1 ,k 1 ...k 10 ...j 8 248 ,k ...j ,k 1 1 but also the aj = 1) ...j ...j Charge 1 8 8 ,k ,p 1 1 ...j appearing in table 2. For example the particle charge aj Charge 3 Z 8 aj 1 ...j ...j aj aj p Z 1 1 ( aj Z D ...j Z Z = 1 = 3 with 1 aj aj − Z respectively, as indicated in table 2. 3875 h aj Z Z 11 , D ( 1 6) Z E , m 4 , , 9) representation, the = 1) 8) and 8 , 6) 7) A third string charge multiplet in , 6) 2 1 , Dynkin labels, , , 5 4) 5) l , 5 3) , A second string charge multiplet in 4 4 , , , , 3 = 1) ,p 12 , , 2 3 D giving , 2 , 8 8 , , 11 2 1-brane multiplet in D dimensions. The results of doing so ar 248 − , 10 7 i , , , 4 6 , , , 4 root p 10 = 1 , , 11 ,p − , , 12 12 7 9 17 1 6 15 11 E 8 , , , , , 12 D , , , p , 6 7 ( = 1 5 9 E root 6 Table 21: , , 3875 10 10 14 , , 12 , , , 5 5 , h , 4 , 7 Table 20: 4 , , 8 8 ( 12 , , , 9 , , , 4 4 11 3 : , 5 E 2 , , 6 6 , , , 6 , , , 3 3 8 2 , 3 3 , , 4 4 , 2 , = 3) 4 , , 2 2 ,... 5 , 30380 , 2 (1 , , 2 2 , 2 1 3 = 2 we must find a negative contribution to the squared root len , 2 3 3 , , 2 p 3 , We can carry out the same analysis for the possible membrane c − The reader will notice that in addition to the well-known cha This process can be carried out to discover all the charge mul 3 3 ( p , (1 3 , (1 (1 3 (1 0 (1 (1 = 2, we look for (1 , (1 weights have non-zero Dynkin labels: must be non-zero. Consequently thecharge highest is weight unique represen in the For p coming from the we find the following highest weights carrying spacetime mem 2 in table 2 for three, four, five, six, seven and eight spacetim additional representations of multiplet in three dimensions is the to the multiplet to the Their highest weights have root lengths such that multiplet is no longer just the existence of some of these extra multiplets were in fact argu JHEP11(2008)091 e (3.1) repre- . The 1 l . . . were argued  6 7 ension encoded e group element E ...d 1 d imensional formula R 6 IB language appeared. ith the prediction that ...d 1 d red bound states formed ring tensions in M-theory, appeared mysterious. The nd the action of the Weyl s and offer a classification A s gravitational constant. In 0 brane in the IIA picture and particles corresponding ting the spectrum of bound ce the tensions of the brane 1 les of [9] were in D=5 where 6! , completing the spectrum of M-theory language, and then in these examples agrees with ble to interpret the most part D ations shown in table 2. The ic formula for deriving tensions ncovered in the BPS spectrum  11 generators of both the ts in this paragraph. E exp and two singlets of  3 c 6 and a 7 brane, now in the IIB picture. It is 2 56 c D 1 D c R representation. 3 c 1 2 l c 1 – 35 – c A 1 3!  at low levels is, )exp b 11 a E K s dimensional (sub-)vector space associated with spacetime + 1 dimensional vector space modulo the addition of the uniqu b ⊗ a D D 3 h 1 l 4 ) exp( as well as its adjoint representation. The restriction of th µ P 11 µ E x was achieved in [8, 9] by making use of a formula that gave the t charge algebra and show that it is consistent with the lower d n 1 l E = exp( The tension formula was justified empirically but its origin g See, for example, equation (3.13) on page 30 of [9] We thank the reviewer for drawing our attention to the commen 4 3 for to account for missing bound states of the was predicted and in D=4 where an additional from the particle andlooked string to charge multiplets, see given ifBy in identifying all the the the omissions boundstates and extra states arguing charge of in multiplets favour D-branesa were of in conjectured. singlet comple the corresponding The IIA examp to or a I bound state of a section 7.2 or appendix C therein. The authors of [9] conside Where the ellipsis indicates the exponentiation of further straightforward to confirm that thethe mass of masses the associated exotic to states natural the appearance matching of additional additional represent charge multiplets from sentation of vector orthogonal to a bound states in D=5there and exist D=4, further exotic isbut states an revealed beyond interesting fully those result here. previously tied u up w 3. Mass and tension in toroidallyThe compactified identity backgrounds between thegroup U-duality of symmetries of M-theory a in a weight vector. tension formula gave theto known a tension weight vector of in p-branes, a strings unique orthogonal vector was shownthis to section correspond we give to the Newton’ mainfrom result the of this paper: an algebra used in [6 – 10]. Weand apply the the solution IIA to and findcharge known IIB multiplets brane and derived theories, st in and section retrospectivelyof 2. to the Furthermore reprodu we charge are algebrascheme a as for these being exotic associated charges to within the KK-brane charge 3.1 The truncated group element asThe group a element vielbein of JHEP11(2008)091 for as- a ∗ (3.6) (3.5) (3.4) (3.2) p p ct pa- to run β · H . In this group 11 . However in the are dimensionless. E β i generators non-zero, · p . b circle, the line element a a H ) (3.3) q β K ]. So that if the radius of e in order to highlight the E π ra, by here are two natural bases ) ment of 2 a , n a truncated version of the hat the scaling of the Planck N me with respect to the eleven aximally oxidised supergravity . H of the − iagonalised vielbein and metric, 12 a a E , or charge algebra, which is defined , and the other uses the generators K symmetry. In that paper the non- a and the effect of the Weyl reflection 11 2 µ a aa E exp ((1 ′ p p 11 half-BPS solutions of the eleven dimen- l l K η C dx a i  E to the same constant [14]. p µ l β = R i ) · p . When we work in the basis of the Cartan 2 h i = ln +++ a 2 a e dξ G C i = ln , where we allow the inner product – 36 – K NH µ i ) = ( − 12 i p ′ ln . In this context it is natural to extend the form of p E 2 2 µµ C 12 e 1 g , occurring in the Cartan sub-algebra was singled out β ( E to β representation of − 11 1  l E . We now make a change of notation and substitute a representation we must also find an interpretation for H 1 l = exp β g in the worldvolume, we find by integrating, i R was denoted by a primed index. It was shown that the non-compa , we will label the fields pre-multiplying the a 11 associated to the generators, were set equal to the same constant H α i a is the Planck length, and appears so that the parameters is a circular coordinate taking values in the range [0 p h i p l ξ In the setting of the It will be fruitful to employ a change of notation at this stag Let us consider a spacetime with a dimension compactified on a in the root rameters encoded the scaling ofdimensional Minkowski metric metric via, of spaceti compact element the part of a general root, sociated to the additionalconstant coordinate. under It T-duality was shown was in predicted [14] by t an using the inner product of the root with the Cartan sub-algeb present paper we work with the the half-BPS group element of to the Cartan subalgebra, leaving only the components in the twelve-dimensional lattice of This group element encoded half-BPS solutions as a group ele Where corresponds in the nonlinear realisation to considering a d over the twelve generators of the Cartan sub-algebra of the field physical role played byto the work variables in, in one the uses group the element. diagonalised generators, T For the non-compact directions we set all the of the Cartan sub-algebra, subalgebra, for such a direction becomes, compactifcation is Where We recall the success in reconstructing the sional, IIA and IIB supergravitytheories theories [4] [13] as from well the asgroup the coefficients element m of given by, the Cartan subalgebra i JHEP11(2008)091 i p (3.7) (3.8) (3.9) his is . We ∗ p theory will and , specifically , and we also p 1 27 l 27 p p = algebra related to introduces dimen- ∗ p i algebra [15]. 27 p pret the radius of the . K 11 ultiplets of section 2 to additional coordinate is E e compact parameters ength. Ultimately we will 2 ++++ of mass in Planck units. A 24 ociate with  mpact the D 6), it appears appears to be s p node, i.e. l l as shown in the latter sections . We propose to treat the extra ∗ ynkin diagram is shown below, ≡ i  p , and two more, 28 i p l R K = ln )= p 1 l  extension of the  3 2  = ln s which are well-known identifications but p 1 p g 1 1 i l l l 3 p s  l  = ln(  s g C = ln – 37 – , i.e. = = ln = ln i  ∗ 11 3 p ’s related to the spacetime coordinates, which we p l i p E 11 p p l R and  The Dynkin diargam of s l s g = ln 11 = , and see if it leads to sensible and consistent results, but t p 2 ) 11 s p l l . R Figure 5: ∗ p representation as such a non-compact parameter and we set: = ( 1 being a massive parameter means that the fully-compactified l 27 ∗ p p While this interpretation of the vierbein associated to the Upon reduction to the ten dimensional IIA theory we may inter One can make similar speculations for the In this case we have twenty-six would like to make a similar association with energy to the coordinate in the sionful parameters into theconsequence group of theory with the dimension also have a dimensionful parameter. motivated as a particularlyon simple the correct solution footing, of having equation a (3. comparable form to that of th compact direction in terms of the string length, coordinates, and also beingrest upon dependent the only successful upon reproductionof the of this Planck the paper brane l tensions andjustify the on choice the of correct tensions in the brane charge m And, indeed, beyond the scope of the present work. for all non-compact coordinates. This choice for the non-co One way to satisfy this scaling is by setting one might take Where we have used have the dimensionful constant of string length which we ass which may be independently derived solely from considering the twenty-six-dimensional bosonic string, for which the D can interpret in a similar way as for JHEP11(2008)091 , β (3.11) (3.12) (3.13) root, 12 E nt is the Cartan sub- , tors. We note that the ) (3.10) b 12 are projected out as we β α · E a > the original brane solution io to find the change in the formula for an ab on (3.3) generalised to the transformation. is: α ) A β > a ) β , of | i q β , whose physical interpretation i y laced algebra act on the root ( l ( l − a i a a a a ˆ n S S | )) α ) prior to applying the expression in K Z β ) (3.14) , whose algebraic interpretation we · = H X β a · β = exp ( b b a · ≡ H ) (3.15) α α α β > a a i b ( | )) ˆ > q > α n Cartan subalgebra. It is the expectation β > ab β a a | b a > l a ) q q A < β 12 i b S , α , α H ( q E exp ( adjoint algebra there is a corresponding brane · · a a | b a − – 38 – ) α α 11 β > ) β b | a ( < α < α β E representation is described equivalently by roots in q < β,α q = exp ( = exp ( lattice transforms the radii of the compact physical a 2 Z exp ( lattice. For each extremal half-BPS brane the brane 1 1 , ( β l < β ∗ 12 β a − Z 11 11 n 9 11 KK ...a R n 1 10 ,...a R 1 9 representation of 9 , respectively. In tables 22, 23 and 24 we list +6 1 i R n ...a l , appearing in the algebraic decomposition of = 0 three of the tensions, given by division by 8 9 p 1 1 k l a n R contained all the dualised versions of the tensors – 54 – 7 5 and pp-wave charges are the exceptions. , R R 11 3 -brane charges ( M n j E 2 , appearing in the generators of the j 0) the tensions all diverge in the non-compact setting V ǫ 2, 1 Dp , ,j n 9 M 11 n> R ...a n 1 +3 10 n 9 p R l ,...a n 1 9 , and squared radii, representation, being composed mostly of mixed symmetry fie V p ...a brane charges in the , 1 1 1 l i a . For the case where 11 n 11 R , R Dp V n c 9 p ,b l n 9 ...R 3 ...a n 1 R 0. At first sight these generators, being all representation = ,...a ≥ 1 9 n V , are well-defined and the fourth, which is the ...a . The KK-branes may be labelled by the powers of the radii appe 1 1 V a 11 We have been able to classify the KK-brane charges that appea 9 An interesting infinite class of roots in the adjoint of ) E K π is associated to KK-branetheir charges. vanishing One in may the say case that of KK-brane the for the other roots in this class ( which may indicate some difference in their nature. Using the they are directly related tosection 1 the of parameter, this paper.number We of recall blocks that of this antisymmetricsymmetric parameter indices indices, played as or well volume as forms control which is quadratic into the the spatial number radii of and linear, may be labelled by (2 this sense much of the the tension found by using equation (3.10) is divergent in th of the full set of such formula. The most studied class of KK-branes, labelled Where the IIA theory and theU-duality IIB transformations theory of including the theis charges D7 of unclear. brane KK- A charge simple inan interpretation [21]. eleven of dimensional Th the origin KK-brane to charges the upon dimensional reduction asplay well a as more other importantKK-modes KK-brane part that than charge appear simply from book-keepingbe duality seen. the arguments bran One in may lowermay hope dimens reveal that significant the details extra about KK-brane the tower kinematics of and states, dyna simply, since the prototype KK-braneshed is more the light dual on graviton the th dual gravity theories. of massless dualised supergravity. The fieldstaking in the this class form, o Where ators with no blocks offound ten [20] or and eleven it antisymmetrised was indice highlighted that in eleven dimensions, SO(9), appearalgebra, but to if indicate we all consider possibl the charges associated to these r find that they have different ”masses”, namely, JHEP11(2008)091 (4.7) (4.8) (4.9) (4.12) (4.10) (4.13) (4.11) (4.16) ng in the 2 7 Σ 0 d olume elements + 11 nβ α + 11 + 3 α Kdtdy U Marie Curie research 10 2 52. The research of P.W. α zeolla, Nicolas Boulanger, + 3 − 0 olution for each of the dual )) (4.14) (4.15) 2 7 he research contract MRTN- ices in the dual generators. 2 2 2 10 + 2 sa for their hospitality during Σ nβ ration during the preparation α 9 d dt dt dy ing to study this class of roots ) + α 3 + − − K R 9 )+ 11 2 2 2 5 2 2 11 + 4 ¯ ¯ α y y α R d d 8 G dt ( ( 1 α 2 1 3 3 + 3 − + 2 R ( − − 8 1 10 + 6 α = − )+(1+ N N α 7 2 N α 11 + 5 + dt )+ )+ 7 9 )( 2 6 2 3 9 p )+ + 5 ¯ ¯ α α – 55 – x x l 2 K ...R d d 6 ( ( 3 α dy − 3 3 1 2 + 4 + 2 ( R 6 8 N N (1 N + 4 α α = 5 − 0 α )+ )+ 2 2 β )+ + 3 + 3 0 0 2 2 2 2 ¯ 2 2 x Z 5 7 ¯ ¯ x x ¯ d x + 3 α α d d nβ nβ n d ( ( ) ( 4 ) ) + + . The set of roots are: 3 3 2 1 α + 2 + 2 K + + +1) 11 10 11 4 6 n n n ( ( ( + 2 E α α α α 3 N N N = = = = α , the number of blocks of nine antisymmetric indices appeari ∗ ∗ ∗ ∗ = = = (1+ = n ≡ 6 2 5 pp ∗ ∗ ∗ ∗ 0 6 5 2 M M β 2 pp β β β KK 2 M 2 M β ds 2 KK denotes a seven dimensional Euclidean line element. These v ds ds 2 7 ds Σ d can write down a line element associated to a half-BPS brane s roots in the adjoint of The line elements corresponding to the set of dual roots are, Where, generator of the associatedfurther. dual roots. It would be interest Acknowledgments P.P.C. would like to thank the Scuolavarious Normale stages Superiore of di Pi thisAugusto work Sagnotti and and to Chris Hull thankof this for Per paper. conversations Sundell, and P.P.C. is Carlo inspi fundedCT-2004-512194 by Ia and EU Superstring INFN, Network via by t MIUR-PRINis contract supported 2003-0238 by a PPARCtraining rolling network grant grant PP/C5071745/1 HPRN-CT-2000-00122. and the E is the rootFor controlling reference, the blocks of nine antisymmetric ind Where are dependent on JHEP11(2008)091 2 4 2 2 8 6 2 4 2 2 4 8 4 6 2 2 4 2 6 2 4 10 0 2 2 0 2 0 2 0 2 2 2 0 2 2 0 2 2 2 0 2 2 0 2 2 0 2 2 0 2 2 0 − − − − − − − − − − − − − − − − − − − − − − squared Root length 0) 4) 4) 3) 3) 3) 3) 3) 3) 3) 3) 3) 4) 4) 4) 2) 2) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 4) 4) 4) 4) 1) 1) 1) 2) 2) 2) 2) 3) 2) 2) 2) 2) 3) 3) 3) 3) 4) 2) 2) 2) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 2 3 3 3 2 2 3 2 3 3 2 3 2 2 2 2 3 3 3 2 3 3 3 3 3 3 3 2 3 4 3 3 1 1 1 1 2 2 1 1 2 2 2 2 2 2 3 2 1 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 2 2 3 2 2 2 2 2 3 2 2 2 2 2 2 2 3 3 2 2 3 3 2 3 3 3 2 2 3 2 3 2 0 1 1 1 2 1 1 1 2 2 2 2 1 2 1 2 1 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 2 2 1 2 2 2 2 2 2 2 2 1 2 2 2 2 3 2 2 2 3 2 2 3 3 2 2 2 3 2 2 2 0 1 1 1 1 1 1 1 1 2 2 2 1 2 1 1 1 2 2 2 , relevant to 11D, IIA , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 2 2 1 1 2 2 2 2 1 2 2 1 1 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 1 2 2 2 0 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 2 2 1 1 1 2 2 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 1 1 1 2 0 0 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 2 2 , 11 Root , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 basis) 2 2 1 1 1 2 1 2 1 1 1 1 1 1 2 2 1 1 2 2 2 2 2 1 1 2 2 2 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 , E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , i 0 12 e 1 2 1 1 1 2 1 1 1 1 1 1 1 1 2 2 1 1 1 2 1 2 2 1 1 1 2 2 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 , ( E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 1 1 1 0 0 0 1 1 1 1 1 1 1 2 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 0 1 1 1 1 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 0 1 1 1 0 0 0 0 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 − , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 7) 7) 7) 7) , , 7) 7) 7) 7) 7) 6) 6) 6) 7) 7) 7) 7) , , 6) 6) 6) 6) 7) 7) 7) 7) 7) 7) 7) 6) 6) 6) 6) 6) , , , , , , , , 5 5 , , , , 5) 5) 5) 5) 6) 6) 6) 0) 1) 2) 3) 3) 4) 4) 4) 4) 5) 5) 5) 5) 5) , , , , , , , , 4 , , , , , , , , 4 , , 4 4 3 3 3 4 4 4 4 4 4 4 , , , , , , , , , , , , , , , , , , , , , , , 3 3 2 2 4 4 4 4 3 3 3 3 3 3 3 3 , , , , , , , , , , , , 2 2 2 1 3 3 2 0 0 1 2 1 2 2 2 1 3 3 3 3 2 9 9 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 8 8 7 8 7 8 8 8 10 10 8 8 8 8 , , 6 7 6 6 8 8 8 8 7 6 7 6 7 7 6 7 , , , , , , , , , , , , , , 5 4 5 5 6 6 5 0 0 2 4 3 4 4 5 4 6 6 6 6 5 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , – 56 – 14 14 8 8 8 9 9 9 9 0 1 3 6 5 6 7 8 7 9 9 9 9 9 12 13 12 13 12 12 12 12 15 15 12 13 12 13 , , 10 11 10 10 12 12 12 12 11 11 11 10 11 11 10 11 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 6 7 7 8 8 7 8 0 1 2 5 4 5 6 7 6 8 7 7 7 8 8 9 8 8 9 9 9 9 12 8 9 9 8 9 9 8 9 12 , , , , , , , , , , , , , , , , , , , , , 10 11 10 11 10 10 10 10 13 13 10 11 10 11 , , , , , , , , , , , , , , , , , , representation of 5 6 6 7 7 6 7 0 1 1 4 3 4 5 6 5 7 5 5 6 7 , , , , , , , , , , , , , , 6 7 7 6 7 7 6 7 7 7 7 7 7 7 6 7 Root , , , , , , , , , , , , , , , , , , , , , 8 9 8 9 8 8 8 8 8 9 8 9 basis) , , , , , , , , , , , , , , , , 10 10 6 4 5 5 6 6 5 0 1 1 3 2 3 4 5 4 6 3 4 5 6 , , , , , , , , , , , , 11 11 1 5 6 6 6 5 5 6 5 6 5 5 6 6 5 4 5 , , i , , , , , , , , , , , , , , , , , , , , , 6 7 6 7 6 , , 6 7 6 7 6 6 6 12 , , , , , , , , , , , , , , , , l 8 8 5 3 4 4 5 5 4 0 1 1 2 1 2 3 4 3 5 2 3 4 5 , , , , , , , , , , , , 9 9 α 4 , 5 5 5 4 4 5 4 5 , 3 4 5 5 3 3 3 E , , , , , , , , , , , , , , , , , , , , , ( 4 5 5 5 4 , , 5 5 4 5 5 4 4 , , , , , , , , , , , , , , , , 6 6 4 2 3 3 4 4 3 0 1 1 1 1 1 2 3 2 4 1 2 3 4 , , , , , , , , , , , , 7 7 3 , 4 4 4 3 3 4 3 4 , 2 3 4 4 1 2 2 , , , , , , , , , , , , , , , , , , , , , 3 3 4 4 2 4 4 2 3 4 2 3 , , , , , , , , , , , , , , , , , , 4 4 3 1 2 2 3 3 2 0 1 1 1 1 1 1 2 1 3 1 1 2 3 , , , , , , , , , , , , 5 5 2 3 3 3 2 2 3 2 3 1 2 3 3 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , 2 2 3 3 1 , , 3 3 1 1 3 1 2 , , , , , , , , , , , , , , , , 2 3 2 1 1 1 2 2 1 0 1 1 1 1 1 1 1 1 2 1 1 1 2 , , , , , , , , , , , , 3 3 1 , 2 2 2 1 1 2 1 2 , 1 1 2 2 1 1 1 , , , , , , , , , , , , , , , , , , , , , 1 1 2 2 1 , , 2 2 1 1 2 1 1 Table 25 — Continued on next page , , , , , , , , , , , , , , , , 1 2 , , , , , , , , , , , , 2 1 , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 0] 0] 0] 0] 0] 0] 1] 1] 0] 1] 1] 2] 1] 0] 1] 0] 0] 1] 1] 2] 2] 0] 0] 0] 0] 0] 1] 0] 0] 0] 0] 1] 1] 0] 1] 3] 0] 0] 0] 0] 0] 1] 1] 0] 0] 0] 0] 1] 0] 0] 1] 2] 0] , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 0 0 1 0 0 2 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 2 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , weights 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 10 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , A 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 1 0 1 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 2 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , [0 [0 [0 [0 [0 [0 [0 [0 [1 [1 [1 [1 [1 [1 [0 [2 [0 [0 [0 [0 [0 [0 [0 [1 [1 [1 [0 [0 [0 [0 [0 [1 [0 [0 [0 [0 [0 [0 [1 [0 [0 [1 [1 [0 [0 [0 [0 [1 [0 [0 [1 [0 [0 ) 11 7 7 7 7 7 7 7 7 7 7 7 7 7 6 7 7 7 7 7 7 7 6 6 6 6 6 6 6 5 6 6 6 6 6 6 6 5 5 5 5 5 5 5 3 3 4 4 4 4 5 0 1 2 and IIB SuGra m Level ( A. Low level weights in the JHEP11(2008)091 2 2 2 2 2 4 6 8 2 4 6 2 4 6 2 2 4 2 2 8 6 8 2 2 4 10 14 12 10 2 2 0 2 2 0 2 2 0 0 2 2 0 0 0 2 2 2 0 squared − − − − − − − − − − − − − − − − − − − − − − − − Root length squared Root length 0) 1) representation 0) 1) 1) , − , , , 0 1 , 1 1 1 , l 0 , , , 0 , 1 1 0 , 0 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 5) 5) 5) 5) 4) 4) 4) 4) 5) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 4) 4) 4) 4) 4) 4) , , , 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 , 0 , , , 4 3 3 3 4 3 2 3 4 3 4 3 4 3 3 3 4 3 3 2 3 3 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 3 4 3 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 , 0 2 3 2 3 3 3 2 3 3 3 3 3 2 3 2 3 2 3 2 2 3 2 2 2 2 3 3 3 3 3 3 3 2 3 3 3 3 3 4 3 3 3 3 , , , 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 , Root 0 2 2 2 3 2 3 2 3 3 3 2 3 2 2 2 3 2 2 2 2 2 2 2 2 2 3 3 3 2 3 3 3 2 3 3 3 2 3 2 3 3 2 3 , , , 0 basis) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 , 0 , , , 2 2 2 3 2 2 2 3 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 2 2 3 3 3 2 3 3 2 2 3 2 2 3 2 2 i 0 12 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , e 0 0 0 , ( 0 E 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 3 3 2 2 3 2 2 2 1 2 2 2 2 2 , , , 0 Root , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 , basis) 0 , , , 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 1 2 2 1 2 1 2 2 2 3 2 2 2 2 2 2 2 1 1 1 1 2 2 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , i 0 0 0 , 12 0 e , , , 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 1 1 1 2 2 1 1 2 1 2 1 1 2 2 1 2 2 2 1 2 2 2 1 1 1 1 1 1 1 ( E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 − 0 1 1 2 1 2 2 2 1 1 1 1 1 2 2 2 1 1 1 1 2 1 1 1 1 2 1 1 1 2 1 1 2 2 1 1 2 2 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 (1 1 1 1 0 0 0 2 1 1 1 1 1 1 1 2 1 1 1 1 0 1 1 1 1 2 1 1 1 1 0 0 0 2 1 1 1 2 1 1 1 1 1 1 (1 (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 2 1 1 1 1 0 0 0 1 0 0 0 0 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , decomposition of the (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 0) 0) 1) 1) 2) , , , , , 8) 8) 8) 0 1 0 1 1 8) 8) 8) 8) , , , , , , , , 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) , , , , 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 8) 7) 7) 7) 7) 4 5 5 0 1 0 1 2 , , , , , , , , , , , , , , 7) 5 5 , , , , , , , , , , , , , , , , , , , , , 5 5 , , , , , , , , 4 4 4 4 4 4 3 , 5 5 5 5 5 5 5 , , , , 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 2 , , , , , , , 0 1 1 1 3 , , , , , , , 3 , , , , , , , , , , , , , , , , , , , , , , , , , , 8 9 9 8 9 9 8 10 11 10 , 10 10 8 8 8 9 8 9 9 8 9 9 8 9 8 9 8 7 8 8 7 8 7 10 10 , , , , , , , , , , 0 1 1 1 2 7 10 10 10 10 10 10 10 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , – 57 – 0 1 1 1 1 14 14 15 14 14 15 14 16 17 16 15 15 13 12 13 14 13 14 14 13 14 14 13 14 13 14 13 13 13 13 12 13 12 15 15 Root , , , , , , , , , , , , , , , 11 15 15 15 15 15 15 15 basis) , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 1 1 , , , , , , , , i , , , , , 12 12 13 12 12 13 12 9 14 15 14 12 12 13 10 10 10 11 11 11 11 11 11 11 10 11 11 11 10 11 11 11 10 11 10 12 13 0 1 1 1 1 , , , , , , , , , , , α 12 12 12 12 12 12 12 , , , , , , , , , , , , , , , , , , , , , , , , , E , , , , , ( 7 , , , , , , , Root 7 8 8 8 9 9 8 9 9 8 8 8 9 9 8 9 9 9 8 9 8 basis) 0 1 1 1 1 , 10 10 11 10 10 11 10 9 9 9 9 9 9 9 12 13 12 11D supergravity , , , , , , , , , , , , , , , , , , , , , 10 11 10 11 , , , , , 5 , , , , , , , , , , , , , , , , , i 5 6 6 7 7 6 7 7 , , 6 6 6 6 7 7 6 7 7 6 7 6 , , 7 12 0 1 1 1 1 , 8 8 9 8 8 8 9 6 7 6 6 7 6 7 , , , , , , , , , , , , , , , , , , , , , 8 9 8 9 α , , , , , , , , , , , , , , , , , , , 4 10 11 10 E ( 4 4 4 5 5 5 5 5 , , 5 5 5 5 5 5 4 5 5 4 5 5 , , 5 0 1 1 1 1 , 6 6 7 6 6 6 7 5 5 3 4 5 4 5 , , , , , , , , , , , , , , , , , , , , , , , , 6 7 6 7 , , , , , , , , , , , , , , , , , , , 3 8 9 8 3 3 3 3 3 4 4 4 4 4 4 4 4 4 2 3 3 3 3 4 3 , , , , , 4 4 5 4 4 4 5 4 4 2 2 3 3 3 , , , , , , , , , , , , , , , , , , , , , , , , 4 5 4 5 (1 (1 (1 (1 (1 2 , , , , , , , , , , , , , , 6 7 6 2 2 2 2 2 3 3 3 , , 3 3 3 3 3 3 1 1 1 2 2 3 , , 1 , 2 2 3 3 3 3 3 3 3 1 1 1 2 2 , , , , , , , , , , , , , , , , , , , , , , , , 2 3 3 3 , , , , , , , , , , , , , , 1 4 5 4 1 1 1 1 1 2 2 2 , , 2 2 2 2 2 2 1 1 1 1 1 2 , , 1 Table 26 — Continued on next page , 1 1 1 2 2 2 2 2 2 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , 1 1 2 2 , , , , , , , , , , , , , , 2 3 2 , , , , (1 0] 0] 1] 0] 0] , , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 , , , , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 Table 25 — Continued from previous page (1 (1 (1 (1 0 0 0 1 0 (1 (1 (1 , , , , , 0 0 0 0 0 2] 0] 1] 2] 1] 0] 1] 0] 1] 0] 1] 1] 2] 1] 2] 0] 0] 0] 0] 0] 0] 0] 0] 1] 0] 1] 0] 1] 1] 1] 2] 1] 1] 2] 3] 1] 0] 0] 0] 0] 0] 0] 0] , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 1 1 1 0 0 0 1 0 1 0 2 0 1 0 2 0 1 0 0 0 0 1 0 1 0 1 0 2 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 2 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 , , , , , weights , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 , , , , , 9 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 A , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 weights 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 1 0 2 0 0 0 0 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 0 0 [1 [0 [0 [0 [0 10 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , : Low level weights in the A 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 ) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 2 2 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 1 1 1 2 11 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0) 0) 1) 1) 2) [0 [1 [1 [1 [2 [0 [0 [0 [0 [0 [1 [1 [1 [1 [1 [0 [0 [0 [0 [0 [0 [0 [0 [1 [1 [1 [1 [2 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [1 [1 [1 [0 , m , , , , , ) 11 (0 (1 (0 (1 (1 Level 10 E 11 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 7 7 7 7 7 m m ( Level ( Table 25 of JHEP11(2008)091 4 2 2 4 2 2 2 4 2 4 2 2 2 2 0 0 2 2 0 2 0 2 0 2 2 0 2 2 0 2 0 2 0 2 2 2 2 0 2 0 2 2 2 0 2 0 2 2 0 2 2 2 2 2 2 − − − − − − − − − − − − − squared Root length 1) 1) 1) 3) 3) 3) 3) 2) 2) 2) 2) 2) 2) 2) 1) 1) 1) 1) 1) 1) 1) 1) 1) 0) 0) 0) 0) 0) 0) 4) 4) 4) 3) 0) 2) 1) 1) 0) 0) 3) 2) 2) 2) 1) 1) 1) 1) 1) 0) 0) 0) 0) 4) 3) 3) − − − , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 2 3 2 2 2 2 3 3 3 2 2 2 2 3 3 3 3 4 2 2 2 3 3 3 2 2 3 2 1 1 1 2 1 2 1 1 2 2 1 2 2 2 3 2 2 2 3 1 2 2 2 1 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 2 1 2 2 2 2 1 2 2 2 2 2 2 2 2 3 2 1 2 2 2 3 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 1 1 2 2 1 1 1 2 2 1 1 1 2 2 1 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 1 1 1 2 2 2 1 2 1 2 2 2 2 1 2 1 2 1 2 2 2 1 2 2 2 2 1 2 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 2 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 1 1 1 2 2 1 1 1 1 1 2 2 2 1 2 1 1 1 2 2 2 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 1 1 1 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Root 2 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , basis) , , , 2 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , i 12 , , , e 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( 1 1 1 E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 6) 6) 6) 6) 6) , 6) 4) 4) 4) 4) 4) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 2) 3) 3) 3) 3) 3) 3) 4) 4) 4) 4) 4) 4) 4) 4) 4) , , , , , , , , , , , , , , 6 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 4 4 4 4 5 5 5 5 5 2 2 3 3 3 , 2 4 4 4 4 5 1 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 5 2 1 2 2 3 3 4 1 2 2 2 3 3 3 3 3 , , , , , , , , , , , , , , 9 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 7 7 7 7 8 8 8 7 7 6 6 7 7 7 , 5 6 6 6 5 7 5 5 5 5 4 6 6 6 6 5 5 5 6 6 6 6 5 7 3 3 4 3 5 4 6 4 5 4 4 6 5 5 5 4 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 12 9 9 8 8 8 9 9 9 8 8 8 9 9 9 9 9 8 8 9 9 8 9 9 9 4 5 6 5 7 6 8 7 8 6 7 9 7 7 8 7 10 10 10 10 11 11 11 10 10 10 10 11 11 11 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 8 8 6 7 7 8 8 8 6 7 7 8 7 7 7 8 6 7 8 7 7 7 8 8 3 4 5 4 6 5 7 6 7 5 6 8 5 6 7 6 8 8 8 8 9 9 9 8 8 8 8 9 9 9 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 10 , , , , , , , , , , , , , , – 58 – 7 7 5 6 6 7 7 7 5 6 6 7 5 5 6 7 5 6 7 5 6 6 7 7 2 3 4 3 5 4 6 5 6 4 5 7 4 5 6 5 , 7 6 6 6 7 7 7 7 6 7 6 7 7 7 Root , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 8 basis) , , , , , , , , , , , , , , , 6 6 4 5 5 6 6 6 4 5 5 6 3 4 5 6 4 5 6 4 5 5 6 6 1 2 3 2 4 3 5 4 5 3 4 6 3 4 5 4 6 4 4 5 6 5 5 6 5 6 5 6 5 5 i , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 6 12 , , , , , , , , , , , , , , , 5 5 3 4 4 5 5 5 3 4 4 5 2 3 4 5 3 4 5 3 4 4 5 5 1 1 2 1 3 2 4 3 4 2 3 5 2 3 4 3 α 5 2 3 4 5 3 4 5 4 5 4 5 3 3 E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ( 5 , , , , , , , , , , , , , , 4 4 2 3 3 4 4 4 2 3 3 4 1 2 3 4 2 3 4 2 3 3 4 4 1 1 1 1 2 1 3 2 3 1 2 4 1 2 3 2 , 4 3 4 3 4 1 2 4 1 2 3 4 2 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 4 , , , , , , , , , , , , , , , 3 1 2 3 1 2 2 3 3 3 1 2 2 3 3 3 1 2 2 3 1 1 2 3 1 1 1 1 2 1 2 1 1 3 1 1 2 1 1 1 3 2 3 2 3 1 1 3 1 1 2 3 1 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 3 , , , , , , , , , , , , , , , 2 1 1 2 1 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 2 1 2 1 1 2 1 1 1 2 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 , , , , , , , , , , , , , , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 Table 26 — Continued on next page 1] 1] 0] 0] 0] 1] 0] 0] 0] 1] 1] 0] 0] 0] 0] 1] 1] 0] 1] 3] 0] 0] 0] 0] 1] 0] 0] 2] 0] 0] 1] 0] 0] 2] 0] 0] 0] 0] 2] 0] 1] 0] 1] 0] 0] 0] 0] 1] 0] 0] 0] 1] 2] 0] 0] , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Table 26 — Continued from previous page 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 2 0 0 0 0 0 2 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , weights 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 9 0 0 0 0 0 0 0 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 A , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , [0 [1 [0 [0 [1 [0 [0 [0 [0 [0 [1 [0 [0 [0 [1 [0 [0 [1 [1 [0 [0 [0 [1 [0 [0 [0 [1 [1 [0 [0 [0 [0 [1 [1 [0 [0 [0 [1 [0 [0 [0 [0 [0 [1 [0 [1 [0 [0 [0 [0 [0 [1 [0 [0 [0 ) 11 6) 6) 6) 6) 6) 6) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 4) 5) 5) 5) 5) 5) 5) 4) 4) 4) 4) 4) 4) 4) 3) 3) 4) 4) 4) 4) 4) 4) 2) 3) 3) 3) 3) , m , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (2 (2 (2 (3 (3 (3 (5 (5 (5 (5 (5 (5 (6 (4 (4 (4 (4 (4 (4 (4 (3 (3 (3 (3 (3 (3 (4 (4 (5 (1 (2 (2 (2 (2 (3 (3 (3 (3 (4 (4 (4 (4 (3 (4 (1 (2 (2 (2 (3 (3 (2 (1 (2 (2 (3 Level 10 m ( JHEP11(2008)091 2 4 2 4 2 2 4 2 2 2 2 6 2 4 2 4 2 6 2 4 0 2 0 2 0 2 0 2 0 2 2 0 2 2 0 2 2 0 2 0 2 2 0 2 2 0 2 2 2 0 2 0 2 0 2 − − − − − − − − − − − − − − − − − − − − squared Root length 4) 4) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 0) 0) 0) 0) 0) 0) 0) 0) 0) 5) 4) 4) 4) 4) 4) 4) 4) 4) 3) 3) 3) 3) 3) 3) 3) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 1) 1) 1) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 3 4 3 3 3 3 3 3 3 3 4 4 4 2 2 3 3 3 3 3 3 4 2 2 2 2 3 3 3 3 3 2 3 3 3 3 3 4 2 2 2 3 3 3 3 3 3 3 3 4 4 4 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 3 3 2 3 2 3 3 2 3 2 2 2 2 3 3 2 2 3 2 2 2 2 2 2 3 3 2 2 3 2 3 2 2 3 2 2 2 2 2 3 2 2 3 2 3 3 2 3 2 2 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 3 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 2 2 2 3 2 2 1 2 2 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 1 1 2 1 2 1 2 2 2 1 2 2 2 2 1 2 2 1 1 2 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 1 1 1 2 2 2 1 2 2 1 1 2 2 2 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 1 1 2 1 1 1 2 2 2 1 1 2 2 2 1 1 2 1 1 1 2 2 2 Root , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , basis) 2 1 1 1 1 2 2 1 1 2 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 2 1 1 1 2 1 1 1 1 1 1 1 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , i 12 e 1 1 1 1 1 1 2 1 1 1 1 1 1 2 2 1 1 1 2 1 2 1 1 1 2 2 1 1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 2 2 ( E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 2 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 6) 6) , , 7) 7) 7) 7) 7) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 7) 7) 6) 6) 6) 6) 6) 6) 6) 7) 7) 7) 7) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6 6 , , , , , , , , , , , , , , , , , , , , , , 6) 6) 6) 6) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 3 3 3 3 3 , 5 5 5 5 5 5 6 6 6 6 6 6 2 3 , , , 4 4 4 5 5 5 6 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 3 , , , , , , , , , , , , , , , , , , , , , , 4 3 3 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , 8 8 7 7 7 , 9 9 9 8 8 8 10 10 9 9 9 9 9 8 7 8 , , , 8 8 8 7 7 7 9 7 7 7 6 6 6 6 7 7 7 7 7 7 7 7 6 6 8 8 8 8 8 7 , , , , , , , , , , , , , , , , , , , , , , , , 6 6 6 5 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 13 13 12 12 12 9 13 13 13 12 12 12 14 14 12 13 13 12 13 12 12 13 9 9 9 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4 4 , , 5 5 5 4 5 5 5 5 5 4 4 , , , , , , , , , , , , , , , , , , , , , , , , , , , 4 4 3 4 , , , , , , , , , , , , , , , , , , , , , , 6 6 4 4 3 4 4 4 3 4 3 3 4 3 4 4 2 3 3 4 2 3 4 4 1 2 2 3 3 , , , , 3 3 4 2 3 3 4 4 3 3 3 4 4 4 4 3 3 4 2 , , 4 2 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 3 , , , , , 2 3 , , , 3 4 4 3 3 2 3 3 3 2 3 2 2 3 2 3 3 1 2 2 3 1 2 3 3 1 1 1 2 2 , , , , 1 2 3 1 2 2 3 3 1 2 2 3 3 3 3 1 2 3 1 3 1 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 , , , , , 1 2 , , , 2 3 2 2 2 1 2 2 2 1 2 1 1 2 1 2 2 1 1 1 2 1 1 2 2 1 1 1 1 1 , , , , 1 1 2 1 1 1 2 2 1 1 1 2 2 2 2 1 1 2 1 , , 2 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 1 (1 (1 (1 (1 , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 Table 26 — Continued on next page 0] 0] 0] 0] 1] 1] 0] 1] 2] 1] 1] 2] 2] 0] 0] 0] 0] 1] 1] 0] 1] 2] 0] 0] 0] 1] 1] 2] 2] 0] 0] 0] 0] 0] 1] 0] 1] 0] 0] 1] 1] 0] 1] 2] 0] 0] 0] 0] 1] 1] 0] 1] 0] 0] 0] , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Table 26 — Continued from previous page 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 2 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 2 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , weights 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 9 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 A , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , [0 [1 [0 [0 [0 [0 [1 [1 [0 [1 [0 [0 [1 [0 [1 [0 [0 [0 [0 [1 [1 [0 [0 [0 [1 [1 [0 [0 [1 [0 [0 [1 [0 [0 [0 [0 [0 [1 [1 [0 [0 [1 [1 [0 [0 [0 [1 [0 [0 [0 [0 [0 [1 [1 [0 ) 11 7) 7) 7) 7) 7) 6) 6) 7) 7) 7) 7) 7) 7) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) , m , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (3 (3 (3 (3 (3 (6 (6 (2 (3 (3 (3 (3 (3 (6 (6 (6 (6 (6 (6 (6 (5 (5 (5 (5 (5 (5 (5 (4 (5 (5 (5 (5 (5 (5 (5 (4 (4 (4 (4 (4 (4 (4 (3 (3 (4 (4 (4 (4 (4 (4 (3 (3 (3 (3 (3 Level 10 m ( JHEP11(2008)091 . 8 6 2 4 6 2 2 4 2 2 0 2 0 2 0 2 2 0 2 2 2 4 2 2 − − − − − − − − − 2 0 2 2 2 2 2 2 2 0 2 2 2 2 0 2 0 2 2 0 2 2 0 2 0 11 − − − − − − E squared squared Root length Root length 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) , , , , , , , , , , , , , , , , , , 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 0) 0) 1) 1) , , , , , , , , , , , , , , , , , , 0) 0) 0) 1) 0) 2) 1) 0) 0) 3) 2) 1) 1) 1) 0) 0) 0) 2) 2) 2) 1) 1) 1) 1) 0) 0) 0) , , , 2 2 3 3 2 3 3 2 3 3 3 2 3 3 2 2 3 2 − , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 , , , , , , , , , , , , , , , , , , , 2 2 0 0 1 0 1 0 2 0 1 0 0 2 1 1 3 0 0 0 1 1 1 1 0 0 2 , 2 2 3 2 2 3 2 2 3 3 2 2 3 2 2 2 2 2 − 0 − , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 , , , , , , , , , , , , , , , , , , , , , 2 2 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 1 1 , 2 2 2 2 2 2 2 2 3 2 2 2 1 2 2 2 2 2 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 , , , , , , , , , , , , , , , , , , , , , 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 1 1 1 1 , 2 2 1 2 2 2 2 2 1 2 2 2 1 1 2 2 2 2 0 0 1 Root , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 , , , , , , , , , , , , , , , , , , basis) , , , 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 , 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 2 1 2 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 , , , , , , , , , , , , , , , , , , i representation of 12 , , , 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , e 2 2 1 1 1 1 2 2 1 1 1 2 1 1 1 2 1 1 Root 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , ( E 0 1 , , , , , , , , , , , , , , , , , , basis) , , , l 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , i 0 , , , , , , , , , , , , , , , , , , 12 , , , e 1 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 1 ( 0 0 1 E , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 , , , , , , , , , , , , , , , , , , , , , 1 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 , 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 , , , , , , , , , , , , , , , , , , , , , 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 1 1 , 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , of the (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 1 , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 − 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) , , 7) 7) 7) 7) 7) 7) 7) , , , , , , , , , , , , , , , , 4 4 4 4 4 4 4 4 4 4 4 , , 4 4 4 4 4 4 4 , , , , , , , , , , , , , , , , 9 9 7 8 8 8 8 8 8 7 7 3) 4) 4) 4) 4) 3) 3) 4) 4) 4) 0) 0) 0) 1) 2) 2) 3) 3) 2) 3) 4) 4) 3) 3) 4) 3) 3) 4) 4) 4) 4) , , 8 8 8 8 7 7 7 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 3 3 3 3 3 3 4 4 4 0 0 1 1 1 2 1 2 2 3 1 2 2 2 3 3 3 4 2 2 2 14 14 12 12 13 12 13 12 13 12 12 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 12 12 12 12 11 11 11 , , , , , , , , , 6 6 6 6 6 6 6 6 6 6 0 1 1 2 3 3 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 9 9 9 9 8 9 9 12 12 10 10 11 10 11 10 11 10 10 8 9 8 8 8 7 8 9 8 8 0 1 1 2 4 4 6 6 5 6 8 8 6 7 8 6 7 8 9 8 8 , , , , , , , , , – 60 – , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 7 7 6 7 7 7 7 Root 8 8 9 8 9 8 9 8 8 7 8 6 6 7 6 7 8 6 7 0 1 1 1 3 3 5 5 4 5 7 7 5 6 7 5 6 7 8 6 7 basis) , , , , , , , 10 10 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , IIA supergravity 6 5 5 5 6 6 5 , , i 6 6 7 6 7 6 7 6 6 6 7 4 5 6 5 6 7 5 6 0 1 1 1 2 2 4 4 3 4 6 6 4 5 6 4 5 6 7 5 6 12 , , , , , , , 8 8 Root , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , α basis) 5 3 4 4 5 5 4 , , E ( 4 5 5 4 5 4 5 5 4 5 6 3 4 5 4 5 6 4 5 0 1 1 1 1 1 3 3 2 3 5 5 3 4 5 3 4 5 6 4 5 , , , , , , , 6 6 i , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 12 3 4 2 3 3 4 4 , , 3 4 4 2 3 3 3 4 2 4 5 2 3 4 3 4 5 3 4 2 1 2 4 4 2 3 4 2 3 4 5 3 4 0 1 1 1 1 1 2 α , , , , , , , 4 4 , , , , , , , , , E ( , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 3 1 2 2 3 3 , , 2 3 3 1 1 2 2 3 1 3 4 1 2 3 2 3 4 2 3 1 1 1 3 3 1 2 3 1 2 3 4 2 3 0 1 1 1 1 1 1 , , , , , , , 3 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 2 1 1 1 2 2 , , 1 2 2 1 1 1 1 2 1 2 3 1 1 2 1 2 3 1 2 1 1 1 2 2 1 1 2 1 1 2 3 1 2 0 1 1 1 1 1 1 , , , , , , , 2 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 0 1 1 1 1 1 1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 Table 27 — Continued on next page (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 1] 2] 0] 0] 1] 0] 0] 1] 0] 0] 0] 1] 1] 1] 2] 2] 0] 0] , , , , , , , , , , , , , , , , , , Table 26 — Continued from previous page 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 2] 1] 0] 0] 0] 1] 2] 1] 0] 0] 0] 1] 0] 0] 0] 0] 1] 0] 0] 1] 0] 1] 0] 0] 0] 1] 1] 0] 0] 0] 0] , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 0 0 1 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 0 0 0 0 0 2 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , weights , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 , , , , , , , , , , , , , , , , , , 9 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , weights 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 A 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , : Low level weights in the , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 9 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 , , , , , , , , , , , , , , , , , , A , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , [1 [1 [0 [0 [0 [0 [0 [0 [1 [1 [1 [1 [0 [0 [0 [0 [0 [1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ) [1 [0 [0 [0 [1 [0 [1 [0 [0 [1 [0 [0 [0 [1 [1 [0 [0 [1 [0 [0 [1 [0 [0 [1 [1 [0 [0 [0 [0 [0 [0 11 ) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 7) 10 , m , , , , , , , , , , , , , , , , , , Table 26 3) 3) 3) 4) 4) 4) 2) 2) 2) 2) 3) 3) 3) 1) 2) 2) 2) 3) 3) 3) 4) 1) 1) 2) 1) 2) 2) 3) 0) 0) 1) (4 (4 (4 (4 (4 (4 (4 (4 (4 (4 (4 (4 (4 (4 (4 (4 (4 (4 Level 10 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , m 9 m (6 (6 (6 (6 (6 (6 (6 (6 (6 (6 (6 (6 (6 (5 (5 (5 (5 (5 (5 (5 (5 (2 (3 (3 (4 (4 (4 (4 (0 (1 (1 Level ( m ( JHEP11(2008)091 2 4 2 2 6 2 4 2 4 2 4 2 4 2 2 4 2 2 2 2 2 0 0 2 2 0 2 2 2 0 2 0 2 0 2 0 0 2 0 2 2 2 0 2 2 0 2 2 0 2 2 0 2 0 2 − − − − − − − − − − − − − − − − − − squared Root length 1) 1) 1) 2) 0) 1) 0) 0) 4) 3) 3) 3) 2) 2) 2) 2) 2) 2) 2) 2) 1) 1) 1) 1) 1) 1) 1) 1) 0) 0) 0) 1) 1) 1) 0) 3) 3) 2) 2) 2) 2) 1) 1) 1) 1) 1) 1) 1) 1) 0) 0) 0) 0) 0) 0) , , , − − − , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 1 , , , 2 2 0 1 1 1 0 0 0 0 0 2 2 2 1 1 1 1 1 1 1 1 0 0 0 3 3 3 2 0 0 1 1 1 1 0 0 0 0 2 2 2 2 1 1 1 1 3 3 2 − 1 − − 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 2 2 2 2 3 2 2 3 2 2 2 2 3 2 2 3 2 3 3 2 3 2 2 2 2 2 3 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 1 2 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 2 2 2 2 1 2 1 2 2 2 1 2 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 2 2 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 1 2 2 1 1 2 2 2 2 1 1 2 1 2 2 2 1 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 2 2 2 2 2 1 1 2 1 2 2 2 2 1 1 2 2 2 1 1 2 1 1 1 2 2 2 1 1 1 1 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 Root 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , basis) , , , , , , 1 2 1 1 2 2 1 1 2 1 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 1 2 2 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , i 12 , , , , , , e 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( 1 1 1 1 1 1 E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 6) 6) 6) 6) 6) 6) 6) 6) 6) 6) 4) 4) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 4) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) , , , , , , , , , , 4) 4) 4) 4) 4) 3) 4) 3) 4) 4) 4) 4) 4) 3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 4 4 4 5 5 5 , , , , , , , 2 3 3 3 4 4 5 5 5 4 5 5 3 3 3 3 3 3 4 4 4 4 4 4 4 4 2 2 3 3 3 3 4 4 4 , , , , , , , , , , 4 4 4 4 4 4 5 4 2 3 3 3 3 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 8 8 8 8 8 8 , , , , , , , 8 8 8 8 8 8 8 8 8 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 , , , , , , , , , , 8 7 7 7 7 7 7 6 7 7 7 7 7 7 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 9 9 9 8 9 9 9 12 12 12 12 12 12 8 9 9 9 8 9 9 12 12 12 12 – 61 – 10 10 10 11 11 10 10 10 10 11 11 10 11 10 10 11 11 10 11 10 10 11 10 10 10 10 10 10 10 10 10 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 8 8 7 7 7 8 8 7 8 8 7 7 7 8 8 8 8 9 9 8 8 8 8 9 9 8 9 8 8 9 9 8 9 7 8 9 8 8 8 8 8 8 8 8 8 , , , , , , , , , , , , , , 10 10 10 10 10 10 10 10 10 10 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 7 7 5 6 6 7 7 , , , , , , 6 7 7 5 6 6 7 , , , , 7 6 7 7 7 6 7 6 7 7 7 6 7 7 7 7 7 6 7 6 6 7 7 6 7 6 6 6 7 6 6 Root , , , , , , , , , , , , , , 8 8 8 8 8 8 8 8 8 8 basis) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 6 6 4 5 5 6 6 5 6 6 4 5 5 6 , , , , , , , , , , 6 5 6 6 5 5 6 5 6 6 5 5 5 6 6 6 5 4 5 5 5 5 6 5 6 4 4 5 6 4 4 i , , , , , , , , , , , , , , 6 6 6 6 6 6 6 6 6 6 12 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 5 , , , , , , 5 3 4 4 5 5 , , , , 5 5 3 4 4 5 4 α 5 4 5 5 3 3 4 4 5 5 5 3 3 3 4 4 4 4 5 4 5 5 4 5 2 3 4 5 2 3 5 E ( , , , , , , , , , , , , , , 5 4 4 5 4 4 5 5 4 4 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 4 , , , , , , 4 2 3 3 4 4 , , , , 4 4 2 3 3 4 3 4 3 4 4 2 2 3 3 4 4 4 1 2 2 3 3 3 3 4 3 4 4 3 4 1 2 3 4 1 2 4 , , , , , , , , , , , , , , 4 2 3 4 2 3 4 4 2 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 3 3 1 2 2 3 3 3 3 1 2 2 3 2 , , , , , , , , , , 3 2 3 3 1 1 2 2 3 3 3 1 1 1 2 2 2 2 3 2 3 3 2 3 1 1 2 3 1 1 3 , , , , , , , , , , , , , , 3 1 2 3 1 2 3 3 1 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 , , , , , , 2 1 1 1 2 2 , , , , 2 2 1 1 1 2 1 2 1 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 1 2 1 2 2 1 2 1 1 1 2 1 1 2 , , , , , , , , , , , , , , 2 1 1 2 1 1 2 2 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Table 27 — Continued on next page (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 Table 27 — Continued from previous page 1] 2] 2] 0] 0] 0] 0] 1] 1] 1] 0] 1] 2] 0] 0] 0] 0] 1] 1] 0] 1] 0] 0] 1] 0] 1] 1] 0] 1] 3] 0] 0] 1] 0] 0] 0] 0] 0] 1] 0] 1] 0] 0] 0] 0] 1] 1] 0] 1] 3] 0] 0] 0] 2] 0] , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 2 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 2 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , weights 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 9 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 A , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , [0 [0 [1 [0 [0 [1 [0 [0 [0 [0 [1 [1 [0 [0 [0 [1 [0 [0 [0 [0 [0 [1 [1 [1 [1 [0 [0 [1 [1 [0 [0 [1 [0 [0 [0 [0 [1 [0 [0 [1 [0 [0 [0 [0 [1 [0 [0 [1 [1 [0 [0 [0 [0 [1 [0 ) 10 5) 5) 5) 5) 5) 4) 4) 4) 4) 4) 4) 4) 5) 4) 4) 4) 4) 4) 4) 4) 3) 3) 3) 3) 3) 3) 3) 3) 4) 4) 5) 5) 5) 2) 3) 4) 4) 4) 4) 4) 4) 4) 3) 3) 3) 3) 3) 3) 3) 3) 4) 2) 2) 2) 3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , m 9 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (8 (7 (7 (7 (7 (7 (8 (8 (7 (7 (7 (7 (7 (7 (7 (7 (7 (7 (7 (7 (7 (7 (7 (6 (7 (7 (7 (7 Level m ( JHEP11(2008)091 4 2 8 6 2 4 6 2 2 4 2 4 2 2 8 6 2 4 6 2 2 2 0 2 2 0 2 0 2 0 2 0 2 2 0 2 0 2 2 2 0 2 2 0 2 2 0 2 0 − − − − − − − − − − − − − − − − − − − − − squared Root length representation 1 l 1) 2) 2) 1) 1) 1) 1) 1) 1) 2) 2) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 0) 0) 0) 0) 0) 0) 0) 0) 0) 4) 4) 3) 3) 3) 3) 3) 3) 3) 3) 2) 2) 2) 2) 2) 2) 2) 2) 2) , − , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 , 1 0 0 0 0 0 0 3 3 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 2 2 4 1 1 0 0 0 0 0 0 2 2 1 1 1 1 1 1 1 1 1 1 − , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 3 3 2 2 3 2 2 2 3 3 2 3 3 2 3 3 3 2 3 3 2 2 3 2 3 2 2 2 2 3 3 2 2 3 2 2 2 3 3 2 3 3 2 3 3 3 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 3 2 2 2 2 2 2 2 3 2 2 3 2 2 3 3 2 2 3 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 3 2 2 3 3 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 1 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 1 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 1 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 2 1 2 2 Root 1 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , basis) , , 2 1 1 1 2 1 2 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 2 1 2 1 1 1 2 2 1 1 2 2 2 2 2 2 1 1 2 2 2 2 1 1 2 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , i 12 , , e 2 1 1 1 2 1 1 2 2 1 1 1 1 2 2 1 1 1 2 1 1 1 2 1 1 1 1 1 2 2 1 1 1 2 1 2 2 2 1 1 1 1 2 2 1 1 1 ( 1 1 E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , decomposition of the (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 7) 7) 7) 7) 7) 7) 6) 6) 6) 6) 6) 6) 5) 5) 5) 6) 6) 6) 6) 6) 6) 5) 6) 6) 6) 6) 6) 6) 5) 5) 5) , , , , , , 5) 6) 6) 6) 5) 5) 5) 6) 6) 6) 6) 5) 5) 5) 5) 4) 6) 6) , , , , , , , , , , , , , , , , , , , , , , , , , 5 5 3 3 4 4 , , , , , , , , , , , , , , , , , , 5 5 5 5 5 5 5 5 5 6 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 , , , , , , 5 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 3 4 , , , , , , , , , , , , , , , , , , , , , , , , , 9 9 , 9 9 9 9 , , , , , , , , , , , , , , , , , 9 9 9 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 , , , , , , 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 9 9 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 14 14 14 14 14 14 12 13 12 13 12 13 12 12 12 12 12 13 13 12 13 12 12 13 12 13 12 13 12 12 12 , , , , , , – 62 – 11 12 12 12 11 11 11 12 12 12 12 11 11 10 11 10 12 12 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 12 12 9 12 12 12 12 9 9 9 8 9 9 9 9 9 9 8 9 8 9 8 9 9 10 11 10 11 10 11 10 10 10 10 10 11 11 10 11 10 10 11 10 11 10 11 10 10 10 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 7 7 6 7 7 7 7 7 7 6 7 7 7 6 7 7 7 7 Root 8 9 8 9 8 9 8 8 8 8 8 9 9 8 9 8 8 9 8 9 8 9 8 8 8 basis) , , , , , , , , , , , , , , , , , , 10 10 10 10 10 10 , , , , , , , , , , , , , , , , , , , , , , , , , , , 5 , , , , 5 5 5 6 6 5 6 5 5 5 6 6 5 5 6 6 6 i IIB supergravity 6 7 6 7 6 7 6 6 6 6 6 7 7 6 7 6 6 7 6 7 6 7 6 6 6 12 , , , , , , , , , , , , , , , , , , 8 8 8 8 8 8 , , , , , , , , , , , , , , , , , , , , , , , , , α 4 , , 5 3 4 4 5 5 3 4 4 5 5 4 5 , , 5 , , 5 4 4 E ( 4 4 5 5 4 5 4 5 5 5 5 4 5 4 5 5 4 4 5 5 5 5 4 5 5 , , , , , , , , , , , , , , , , , , 6 6 6 6 6 6 , , , , , , , , , , , , , , , , , , , , , , , , , 3 4 2 3 3 4 4 2 3 3 4 4 3 4 4 4 3 3 , , , , , , 2 3 4 4 2 3 3 3 4 4 4 2 3 3 3 4 2 3 4 4 4 3 3 3 4 , , , , , , , , , , , , , , , , , , 4 4 4 4 4 4 , , , , , , , , , , , , , , , , , , , , , , , , , 2 , , 3 1 2 2 3 3 1 2 2 3 3 2 3 , , 3 , , 3 2 2 1 2 3 3 1 1 2 2 3 3 3 1 1 2 2 3 1 2 3 3 3 1 2 2 3 , , , , , , , , , , , , , , , , , , 3 2 3 2 3 2 , , , , , , , , , , , , , , , , , , , , , , , , , 1 , , 2 1 1 1 2 2 1 1 1 2 2 1 2 , , 2 , , 2 1 1 1 1 2 2 1 1 1 1 2 2 2 1 1 1 1 2 1 1 2 2 2 1 1 1 2 , , , , , , , , , , , , , , , , , , 2 1 2 1 2 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 Table 27 — Continued from previous page 2] 1] 2] 0] 0] 0] 0] 1] 0] 0] 1] 0] 0] 0] 1] 1] 1] 2] 0] 1] 0] 0] 1] 0] 0] 0] 1] 1] 1] 2] 2] 1] 2] 2] 0] 0] 0] 0] 0] 1] 1] 0] 1] 2] 0] 0] 0] 0] 1] , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 2 0 0 0 0 1 0 0 0 1 0 0 2 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , weights 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 9 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 A , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 0 0 1 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 2 0 0 0 0 1 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , : Low level weights in the [0 [1 [1 [0 [1 [0 [0 [0 [0 [0 [0 [1 [1 [1 [1 [0 [0 [0 [0 [0 [0 [0 [0 [1 [1 [1 [1 [0 [0 [0 [0 [1 [1 [0 [0 [0 [1 [0 [0 [0 [0 [1 [1 [0 [0 [1 [0 [1 [1 ) . 10 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 5) 4) 4) 4) 4) 5) 5) 5) 5) 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 4) 3) 3) 3) 3) 3) 3) 3) 3) 5) 5) 5) 6) 3) 11 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , m E 9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (9 (8 (8 (8 (8 (9 Level m Table 27 ( of JHEP11(2008)091 . ]. ]. 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