The massless irreducible representation in E theory and how bosons can appear as spinors
aa r X i v : . [ h e p - t h ] F e b The massless irreducible representation in E theory and how bosons can appear as spinors.
Keith Glennon and Peter WestDepartment of MathematicsKing’s College, London WC2R 2LS, UK
Abstract
We study in detail the irreducible representation of E theory that corresponds to mass-less particles. This has little algebra I c ( E ) and contains 128 physical states that belong tothe spinor representation of SO(16). These are the degrees of freedom of maximal super-gravity in eleven dimensions. This smaller number of the degrees of freedom, compared towhat might be expected, is due to an infinite number of duality relations which in turn canbe traced to the existence of a subaglebra of I c ( E ) which forms an ideal and annihilatesthe representation. We explain how these features are inherited into the covariant theory.We also comment on the remarkable similarity between how the bosons and fermions arisein E theory. 1 . Introduction The symmetries of the non-linear realisation of E ⊗ s l with respect to the Cartaninvolution invariant subalgebra of E , denoted by I c ( E ), lead to equations of motionat low levels that are precisely those of maximal supergravity provided one discards thedependence on the coordinates of the spacetime that are beyond those we usually consider[1,2,3,4]. In particular the degrees of freedom they contain are those of the familiar gravitonand the three form. While there is no complete understanding of the role of all thehigher level fields a large class of them are known to provide different field descriptionsof the graviton and the three form degrees and freedom to which they are related byinvariant duality relations. The higher level fields fields also account for the gauging of themaximal supergravity theories. Indeed the non-linear realisation contains all the maximalsupergravity theories in their different dimensions including those that are gauged. For areview see references [5,6].In a different approach the irreducible representation of the semi-direct product of I c ( E ) with its vector representation l , denoted I c ( E ) ⊗ s l were formulated [7]. Thislatter algebra is the analogue of the Poincare group which can be written in the form SO (1 , D − ⊗ s T D where T D are the translations in D dimensions. The irreducible rep-resentations of the Poincare group were found in 1939 by Wigner [8] and one can use asimilar method in E theory. One important difference is that while the construction of theirreducible representations of the Poincare algebra begin by considering the possible valuesof the momentum, those in E theory involve the vector, or first fundamental representa-tion of E and these include all brane charges [2,9,10,11,12]. Thus while the irreduciblerepresentations of the Poincare algebra lead to all possible particles those in E theory leadto all possible point particle and branes, that is, all extended objects in E theory.The irreducible representation that arises when all the members of the vector repre-sentation vanish except for the usual momentum, which was taken to be massless, werestudied briefly in reference [7]. It was found that the corresponding little group was I c ( E )and it was argued that the representation only contains a finite number of states which arethose of the graviton and three form in eleven dimensions and so the degrees of freedomof eleven dimensional supergravity. In this paper we will study this irreducible representa-tion in detail. We will show that one can impose an infinite number of duality equationson the representation that are invariant under I c ( E ) and reduce the number of indepen-dent fields to be 128. These are the fields h i i = h ( i i ) , h ii = 0 and A i i i = A [ i i i ] , i , i , . . . = 2 , . . . ,
10 and they belong to the 128 dimensional spinor representation of I c ( E ) = SO(16). Corresponding to the duality relations we find that there exist an infi-nite number of generators of I c ( E ) which annihilate the representation and these form asubalgebra I that is an ideal. Indeed the Lie algebra I c ( E ) I = SO(16).To understand how it is that the bosonic states, and in particular the graviton, canbelong to the spinor representation of an algebra that involves spacetime symmetries wewill decompose the physical states into representations of the subalgebra SO(8) ⊗ SO(8)of SO(16). We find that they belong to the representations 128 = (8 v , v ) ⊕ (8 c , s ). Thegraviton belongs to the first representation.Given an irreducible representation of the Poincare algebra one can embed it in a largerrepresentation to find a Lorentz covariant formulation of the representation. The price is2hat the Lorentz covariant fields obey conditions which are the physical sate conditions andare subject to gauge transformations. We carry out the same procedure for the masslessirreducible representation of I c ( E ) ⊗ s l and make contact with the non-linear realisationof E ⊗ s l with respect to the Cartan involution invariant subalgebra of E , denoted by I c ( E ). Indeed the degrees of freedom of the massless irreducible representation are theones of the latter theory. This supports the asscertion that these are the only degrees offreedom that arise in the non-linear realisation and it explains the origin and structure ofthe duality relations that arise in the non-linear realisation.
2. The irreducible representation for a massless particle in E theory
The irreducible representations in E theory were discussed in reference [7]. To bemore precise this paper studied the irreducible representations of the semi-direct productof I c ( E ) and it’s vector ( l ) representation, denoted I c ( E ) ⊗ s l . This is a naturalextension of the method used to find the irreducible representations of the Poincare group[8]. The similarity comes from the fact that the Poincare group is a semi-direct productof the Lorentz group SO (1 ,
3) and the group of translations T , denoted SO (1 , ⊗ s T Indeed at the lowest level I c ( E ) ⊗ s l is just the Poincare group in eleven dimensions. Forthe case of the Poincare group one selects a value for the momentum corresponding to ifit is massive or massless and then one computes the little group that preserves this choice.One then takes a representation of this little group and finds a representation of the fullPoincare group by boosting, or said more technically, by taking an induced representation.In this way one can find all irreducible representations of the Poincare group.In E theory the vector ( l ) representation contains all the brane charges and so thefirst step is to select a preferred value of the vector representation and compute the littlealgebra that preserves this value. One then takes an irreducible representation of thislittle algebra and boosts it to the full I c ( E ) ⊗ s l algebra. At lowest levels the vectorrepresentation contains P a , Z a a , Z a ...a , . . . with a , a , ... = 0 , , ...,
10 where P a arejust the usual momenta and Z a a and Z a ...a the well known two form and five formcharges that first appeared in the supersymmetry algebra. In reference [7] the irreduciblerepresentation that arises when one takes the momentum to take a value correspondingto it being massless with the other components of the vector representation being zerowas discussed. We will refer to this irreducible representation as the massless irreduciblerepresentation. Taking some of the higher charges to be non-zero one finds the irreduciblerepresentations corresponding to branes.The purpose of this section is to fully elucidate the properties of the massless irre-ducible representation. To begin the construction we will take the momentum to have thevalues p = − m, p = m with all other components being zero. It will be more conve-nient to use light-cone notation, for which the components of a vector V a are defined as V ± = √ ( V ± V ) , V i , , ...,
9, and the Minkowski metric becomes η + − = 1 , η ij = δ ij sothat V ± = √ ( V ± V ). In light-cone notation the massless irreducible representation be-gins by taking the brane charges to be p + = p − = √ m , with all other momenta and otherbrane charges being set equal to zero. We next seek the subalgebra H of I c ( E ) whichpreserves this choice of brane charges. The brane charges l A transform under I c ( E ) and3o the subalgebra H is determined by the requirement that δl A = [Λ α S α , l A ] = 0 (2 . l A take the above valuesIt is straightforward to show that for a massless particle the parameters Λ α mustsatisfy Λ + a = 0 = Λ + − = Λ + ab = . . . , that is, any parameter Λ α with a lowered + indexis zero. The resulting algebra which leaves this choice invariant is [7] H = { J + i , J ij , S + ij , S i i i , S + i ...i , . . . } , i, j, . . . = 2 , . . . . (2 . J + i , J + j ] = 0 , [ S + i i , S + j j ] = 0 , [ J + i , S + j j ] = 0 , . . . (2 . I c ( E ) commutation relations withindices restricted to i, j = 2 , . . . ,
10. The result is that the subalgebra which preservesthe above choice of charges is the Cartan involution invariant subalgebra of E , denoted H = I c ( E ), given by the generators [7] J ij , S i i i , S i ...i , S i ...i ,j , S i ...i ,j j j , . . . (2 . J + i commute and in order toobtain a finite dimensional irreducible representation one takes a representation in whichthese act to give zero leaving us to take an irreducible representation of the algebra SO(D-2). The Dynkin diagram of E is given by deleting the first two nodes, which correspondto the + and − directions, from the Dynkin diagram of E ; • |⊕ − ⊕ − • − • − • − • − • − • − • − • ⊕ indicates that the node has been removed from the Dynkin diagram of E .In general the theory in D dimensions can then be obtained from the E Dynkindiagram by deleting node D and analysing the theory with respect to the algebra corre-sponding to the remaining nodes. Thus to find the theory in eleven dimensions we deletenode eleven, that is, the top node and analysis the theory when decomposed into the al-gebra GL(11), while in D dimensions we decompose into representations of A D − ⊗ E − D algebra. To find the irreducible representations in D dimensions one carries out the samesteps and so for the massless irreducible representation we end up deleting nodes one andtwo, as explained above, as well as node D . For the case of eleven dimensions we should4herefore delete nodes one and two to arrive at E and decompose this algebra in terms ofrepresentations of the A subgroup of E as shown in the Dynkin diagram below ⊗ |⊕ − ⊕ − • − • − • − • − • − • − • − • I c ( E ), rather than E itself,we actually decompose the I c ( E ) subgroup of E into representations of I c ( A ) = SO(9).Indeed the generators of equation (2.4) are listed in this decomposition as from the begin-ning of our discussion we had in effect deleted node eleven as we are concentrating on theeleven dimensional case.The next step is to choose an irreducible representation of I c ( E ). Such an reduciblerepresentation is provided by the Cartan involution odd generators of E which are givenby [7] T ij = η ik K kk + η jk K ki , T i i i = R j j j η j i η j i η j i + R i i i , (2 . T i ..i = R j ..j η j i ..η j i − R i ..i , (2 . T i ..i ,k = R j ..j ,l η j i ..η j i η lk + R i ..i ,k , (2 . T i ..i ,j j j = R m ..m ,l l l η m i ..η m i η l j η l j η l j − R i ..i ,j j j , . . . (2 . I c ( E ) because the involution operator I c is defined to act on the generators A, B, ... of a Lie algebra as I c ( AB ) = I c ( A ) I c ( B )and so I c ([even , odd]) = − [even , odd] which guarantees that the commutator will alwaysbe a Cartan involution odd generator. The generators of equation (2.8) are the Cartaninvolution odd generators of I c ( E ) when restricted to their indices taking only the val-ues i, j = 2 , ...,
10. Clearly one could for the massless case take a different irreduciblerepresentation but in this paper we will only consider this case.As a result we take our representation to consist of fields corresponding to the Cartaninvolution odd generators { h ij (0) , A i i i (0) , A i ...i (0) , h i ...i ,j (0) , . . . } , with i, j, ... = 2 , ..., . (2 . I c ( E ) ⊗ s l .We recognise h ij (0) and A i i i (0) as the degree of freedom of the graviton and threeform respectively. There are however, an infinite number of higher level fields. It wasproposed in [7] that these fields are connected to the gravition and three form by certainduality relations. The purpose of this section is to investigate these relations in moredetail.The fields of equation (2.9) live in the representation associated to the Cartan invo-lution odd generators and so we consider the quantity V = h ij T ij + A i i i T i i i + A i ..i T i ..i + h i ..i ,j T i ..i ,j + A i ..i ,j j j T i ..i ,j j j + . . . (2 . I c ( E ). In particular under the level one generator S i i i of I c ( E ) we take δ V = [Λ i i i S i i i , V ] + [Λ i ...i S i ...i , V ] + [Λ i ...i S i ...i , V ] + [Λ i ...i ,j S i ...i ,j , V ] (2 . E algebra to evaluate equation (2.11) we find that δh ij = 18 Λ ( i | k k | A j ) k k − η ij Λ k k k A k k k − η ij Λ k ...k A k ...k − ·
5! Λ k ...k ( i A j ) k ...k +7 · · ·
16 ( h k ...k , ( i | Λ k ...k , | j ) + 8 h ( i | k ...k ,l Λ | j ) k ...k ,l − η ij Λ k ...k ,l h k ...k ,l ) (2 . δA i i i = − · h [ i k Λ i i ] k + 5!2 A i i i k k k Λ k k k + 5!2 Λ i i i k k k A k k k + 7! ·
23 Λ k ...k ( h i i i [ k ...k ,k ] − h k ...k [ i i ,i ] ) (2 . δA i ..i = 2 Λ [ i i i A i i i ] + 112 h i ..i k k ,k Λ k k k + 112 h [ i ..i | k k k | ,i ] Λ k k k +12 Λ k [ i ...i h i ] k (2 . δh i ..i ,j = 2 (Λ [ i i i A i ..i ] j − Λ j [ i i A i ..i ] ) − · · k k k ( A i ..i k ,k k j + A [ i ..i | jk ,k k | i ] )+3 Λ [ i ...i A i i ] j − [ i ...i A i i j ] . (2 . i i i variations.We will now going to postulate duality relations and show they are preserved underthe I c ( E ) symmetry and as a result the number of fields in the representation is radicallyreduced, indeed, one can take the representation to contain only the graviton and the threeform. We first propose a duality relation between the three-form and six-form which isgiven by E i i i = A i i i + c ε i i i j ..j A j ..j = 0 (2 . c is a constant which we will fix by requiring that this relation is part of an infiniteset of relations that are, as a collection, left invariant by I c ( E ). Varying equation (2.16)under I c ( E ) we find that δE i i i = 2 c Λ j j j ε i i i j ..j ( A j j j + 1 c ε j j j i i i j j j A i i i j j j ) + . . . (2 . . . . denotes the gravity terms. Clearly we will only recover our original relation if c = ± and we choose c = − . For this choice the variation takes the form δE i i i = 16 Λ j j j ε i i i j ..j E j j j − E k [ i Λ ki i ] (2 . E i i i = A i i i + 112 ε i i i j ..j A j ..j = 0 (2 . ij = h ij − ε ir ..r h r ..r ,j = 0 . (2 . h [ i ..i ,j ] = 0 implies that for this relation to be consistent, the field h ij must be traceless, h ii = 0. With this condition h ij has . − δE ij = −
12 13 · ε ir ..r (Λ r r r ε r ..r j k k k E k k k − Λ jr r ε r ..r k k k E k k k )+12 · · k k k ε ir ..r ( E r ..r k ,k k j − E jr ..r k ,k k r ) = 0 (2 . E i ..i ,j j j = A i ..i ,j j j + 19! ε i ..i A j j j = 0 . (2 . A i i i and nine-three form A i ...i ,j j j .These duality relations E ij = 0, E i i i = 0, E i ...i ,j j j = 0 are the first of an infinitetower of duality relations showing that the fields at levels, two, three, four, etc... can beexpressed in terms of the fields h ij , A i i i . (2 . h i ...i ,j is related to the graviton by equation (2.20). All the higher level fields carry theindices of these fields as well a multiple sets of blocks of nine indices and we can expectthat in all the higher level duality relations these indices are carried by ε i ..i ’s in a waythat is similar to how they appear in equation (2.22) . These indices correspond to theaction of the affine generator of I c ( E ). As a result it should be possible of show that thecomplete infinite set of duality relations are invariant under the I c ( E ) symmetry.Hence we have found that the massless irreducible representation contains the fields ofequation (2.23) and as a result it contains the 44 + 84 = 128 bosonic degrees of freedom ofeleven-dimensional supergravity. Rather than write them as in equation (2.23) we can writethe degrees of freedom in terms of an E multiplet, or in our case here, an I C ( E ) = SO (16)multiplet, namely h i ′ j ′ , A i ′ i ′ i ′ , A i ′ ...i ′ , h j ′ ≡ ǫ i ′ ...i ′ h i ′ ...i ′ ,j ′ , i ′ , j ′ , . . . = 3 , . . . ,
10 (2 . i, j, . . . = 2 , . . . ,
10 and primedindices to range over i ′ , j ′ , . . . = 3 , . . . ,
10. The fields of equation (2.24) have been obtainedfrom those of equation (2.23) by expressing A i ′ i ′ in terms of A i ′ ...i ′ using the dualityrelation of equation (2.16) and also by expressing h j ′ in terms of h i ′ ...i ′ ,j ′ using the dualityrelation of equation (2.20). The fields of equation (2.24) also give 36 + 56 + 28 + 8 = 128degress of freedom. We note that h i ′ i ′ + h = 0 and as we have not included h we donot take h i ′ i ′ equal to zero. 7 . Reduction of degrees of freedom from the algebra viewpoint In the previous section we have seen that the higher level fields can be expressed interms of the fields h ij and A i i i at levels zero and one through the existence of dualityrelations such as those of equations (2.19), (2.20) and (2.22). In this section we will showthat the massless irreducible representation is annihilated by an infinite set of generatorsand this corresponds to the existence of the infinite set of duality relations. We begin byconsidering the generator of the form N k k k = S k k k + c ε k k k r ...r S r ...r (3 . c . Under the action of this generator the fields transform under Λ k k k N k k k as δh ij = 18Λ ( i | k k | A j ) k k − η ij Λ k k k A k k k − c ( · · k k k ε k k k r ...r ( i A j ) r ..r + η ij Λ k k k ε k k k r ...r A r ..r ) (3 . δA i i i = − · h [ i j Λ i i ] j + 60 A i i i k k k Λ k k k − c Λ k k k A r r r ε k k k r r r i i i − · c Λ k k k ( ε k k k r ...r A r ..r [ i i ,i ] − ε k k k r ...r j A i i i r ..r ,j ) (3 . δA i ...i = 2Λ [ i i i A i i i ] + 112 h i ..i k k ,k Λ k k k + 112 h [ i ..i | k k k | ,i ] ˜Λ k k k +12 c Λ k k k ε k k k j [ i ...i h i ] j − · c Λ k k k ε k k k r ...r A [ i ..i | [ r ..r ,r ] | i i ] − · c Λ k k k ε k k k r ...r r r A [ r ..r | [ i ..i ,i ] | r r ] − · c Λ k k k ε k k k r ...r A r ..r [ i i i ,i i i ] (3 . δh i ...i ,j = 2(Λ [ i i i A i ..i ] j − ˜Λ j [ i i A i ..i ] ) − · · k k k ( A i ..i k ,k k j + A [ i ..i | jk ,k k | i ] )+3 c Λ k k k ε k k k [ i ...i A i i ] j − c Λ k k k ε k k k [ i ...i A i i j ] (3 . c = − · in the generator of equation (3.1) becomes N k k k = S k k k − · ε k k k r ...r S r ...r , (3 . δh ij = 18Λ ( ik k E j ) k k − η ij Λ k k k E k k k = 0 (3 . δA i i i = 13! ε i i i k k k r r r Λ k k k E r r r − [ i i k E i ] k = 0 (3 . δA i ...i = 130 Λ k k k ε k k k [ i ...i | j E | i ] j + 28Λ k k k ε k k k r ...r E r ..r [ i i i ,i i i ] (3 . k k k ε k k k r ...r r r E i ..i [ r r r ,r r r ] = 0 (3 . N ij = J ij + 28! ε [ ir ...r S | r ...r | ,j ] (3 . ij gives δh ij = 2Λ ik E kj + 2Λ j k E ki = 0 , (3 . N i i i ≡ S i i i + 19! ǫ j ...j S j ...j ,i i i (3 . ǫ symbol.The set of all generators that annihilate the representation must form a subalgebra,denoted I , of I c ( E ) and we will now show that this is true for such lowest level generators.Indeed we find that[ N i i i , N j j j ] = 2 N i i i j j j − · δ [ i i [ j j N i ] j ] − ε i i i j j j k k k ˜ N k k k (3 . N i ...i = S i ..i + 112 ε i ..i k k k S k k k = 112 ε i ..i k k k N k k k (3 . I of a Lie algebra G is a subalgebra consisting of elements X ∈ I suchthat such that [ X, Y ] ∈ I for all Y ∈ G . One finds that[ S i i i , N j j j ] = − ε i i i j j j k k k N k k k − δ [ i i [ j j N i ] j ] (3 . S i i i , N jk ] = 3 η [ i | j ˜ N k | i i ] + 3 η [ i | j N k | i i ] (3 . S i i i lead to the whole of I c ( E ) it follows that theequations (3.16) and (3.17) together with their higher level analogues, which we have notshown, imply that the generators in the subalgebra I are an ideal of I c ( E ). Starting fromthe level zero field h ij (0) of the massless irreducible representation we can find all fieldsin the representation by the action of S i i i . Hence if the generators which annihilatethe massless irreducible representation form an ideal in I c ( E ) it follows that if all the9lements of I annihilate the lowest field, the graviton, then they will annihilate all fieldsin the representation.Given a Lie algebra G (group) which contains an ideal subalgebra (subgroup) I thenthe coset GI is also a Lie algebra (group). If A , A belong to G the corresponding equiva-lence relation is A ∼ A means A = A + i for some i ∈ I . Indeed I c ( E ) I is a Lie algebra.It is clear from the form of the generators in the ideal that all the generators in I c ( E )are related to the generators J i i and S i i i by the equivalence relations. Hence I c ( E ) I contains just these two generators. These generators contain 36 + 84 = 120 generatorsand, as we will show in the next next section, they generate S0(16). As a result I c ( E ) I = SO (16) (3 . I c ( E ).It contains only the graviton and three form. Indeed due to equation (3.18) these statesbelong to an irreducible representation of the much smaller algebra SO(16). The situationis a bit similar to the existence of highest weight states in representations of the Virasoroalgebra and the corresponding reduction in the number of states in the representation.
4. The bosonic states viewed as a spinor
In the section two we found that although the massless irreducible representation ofthe little group I c ( E ) contained, at first sight, an infinite number of states it was subjectto an infinite set of duality relations that reduced the number of degrees of freedom itcontained to 128. These can be listed as in equation (2.24), or as an I c ( E ) = SO(16)multiplet, in equation (2.24). The SO(16) algebra has two 128 dimensional irreduciblerepresentations both of which are spinors. The number of components of a spinor insixteen dimensions is 2 = 256. However, in sixteen, effectively Euclidean, dimensionswe can have Majorana Weyl spinors which have just 128 component. Hence it must bethat our 128 bosonic states belong to a spinor representation of SO(16). As E containsthe gravity line consisting of the nodes three to ten it contains the SL(9) algebra and as aresult I c ( E ) = SO(16) contains the SO(9) which acts on the spacetime time coordinates.The purpose of this section is to understand how the bosonic states can be assembled intothis spinor representation.We found that for the massless representation the little algebra is I c ( E ) which ariseswhen we delete nodes one and two in the E Dynkin diagram. Further decomposingthe I c ( E ) representations into those of I c ( E ) corresponds to also deleting node three inaddition to nodes one and two in the E Dynkin diagram as shown below • |⊕ − ⊕ − ⊗ − • − • − • − • − • − • − • A algebra. In the context of the massless irreducible10epresentation, which has little group I c ( E ), this implies that we should decompose withrespect to I c ( A ) = SO(9). Indeed the fields of equation (2.9), and so equation (2.23),appear as representations of I c ( A ) = SO(9). However, we wish to study the masslessirreducible representation from the viewpoint of I c ( E ) = SO(16) which is the algebrathat appears in the above E Dynkin diagram. As such we have in effect to undo thedeletion of node eleven. To better understand the manner in which bosonic fields belongto the 128 dimensional spinor representation we consider the fields from the viewpoint ofthe SO(8) × SO(8) subalgebra of SO(16).The first step is to identify the SO(8) × SO(8) generators amongst those of I c ( E ).The latter generators arise in E as { J i ′ j ′ , S i ′ i ′ i ′ , S i ′ ...i ′ , S i ′ ...i ′ ,j ′ } , with i ′ .j ′ , . . . , . . . ,
10 (4 . J i ′ j ′ generate an SO(8) This subalgebra is the I c ( A ) that appears as the gravity line consisting of nodes four to ten of the E Dynkindiagram. As such this SO(8) is just part of the familiar gravity line symmetries that areforemost in E theory discussions and act on the spacetime coordinates. The SO(8) × SO(8)subalgebra consists of the generators J ± i ′ j ′ = J i ′ j ′ ± J i ′ j ′ , where ˆ J i ′ j ′ = 16! ε i ′ j ′ k ′ ...k ′ S k ′ ...k ′ . (4 . E algebra one finds that they obey the com-mutation relations [ J + i ′ j ′ , J − k ′ l ′ ] = 0 , (4 . J + i ′ j ′ , J + k ′ l ′ ] = − δ [ i ′ [ k ′ J + j ′ ] l ′ ] , [ J − i ′ j ′ , J − k ′ l ′ ] = − δ [ i ′ [ k ′ J − j ′ ] l ′ ] . (4 . × SO(8). We note that the gravity line SO(8) discussedabove arises as the algebra which is the diagonal subalgebra of SO(8) × SO(8). We notethat we have three different SO(8) algebras.We will now decompose the spinor representation of SO(16), which contains the 128degrees of freedom, into representations of SO(8) × SO(8). To begin with rather thanconsider the fields we will consider the corresponding Cartan involution odd generatorsof equation (2.5-2.8). While these transform into each other in the usual way under theSO(8) rotations generated by J i ′ j ′ , under the S i ′ ...i ′ generator we find that[ S i ′ ...i ′ , T j ′ k ′ ] = − T [ i ′ ...i ′ j ′ δ k ′ i ′ ] − T [ i ′ ...i ′ k ′ δ j ′ i ′ ] , (4 . S i ′ ...i ′ , T j ′ j ′ j ′ ] = 60 T [ i ′ i ′ i ′ δ j ′ j ′ j ′ i ′ i ′ i ′ ] + 3 T i ′ ...i ′ [ j ′ j ′ ,j ′ ] , (4 . S i ′ ...i ′ , T j ′ ...j ′ ] = − T k ′ k ′ δ j ′ ...j ′ i ′ ...i ′ + 1080 T [ i ′ [ j ′ δ j ′ ...j ′ ] i ′ ...i ′ ] , (4 . S i ′ ...i ′ , T j ′ ...j ′ ,k ′ ] = 3360( T [ j ′ j ′ j ′ δ j ′ ...j ′ ] k ′ i ′ ...i ′ i ′ − T [ j ′ j ′ | k ′ | δ j ′ ...j ′ ] i ′ ...i ′ ) , (4 . T j ′ k ′ and ˜ T i ′ j ′ ≡ ε i ′ j ′ k ′ ...k ′ T k ′ ...k ′ transforminto each other as do T j ′ j ′ j ′ and T i ′ ≡ ε k ′ ...k ′ T k ′ ...k ′ ,i ′ .Motivated by the result just above we define the combinationˆ T i ′ j ′ = T i ′ j ′ + 2 ˜ T i ′ j ′ − δ j ′ i ′ T k ′ k ′ (4 . . T i ′ j ′ and ˜ T i ′ j ′ are symmetric and antisym-metric respectively. Under SO(8) × SO(8) the objects ˆ T i ′ j ′ transforms as[ J + i ′ j ′ , ˆ T k ′ l ′ ] = − δ [ i ′ k ′ ˆ T j ′ ] l ′ , [ J − i ′ j ′ , ˆ T k ′ l ′ ] = − δ [ i ′ l ′ ˆ T k ′ j ′ ] (4 . T k ′ l ′ as a vector and thesecond SO(8) transforms the second index on , ˆ T k ′ l ′ as a vector. Thus we recognise thatˆ T k ′ l ′ transforms as the (8 v , v ) representation of SO(8) × SO(8) where 8 v is the vectorrepresentation of SO(8).We now turn our attention to the objects T j ′ j ′ j ′ and T i ′ which have 56 and 8 degreesof freedom respectively. Under SO(8) × SO(8) they transform as[ J ± i ′ j ′ , T k ′ ] = 2 T [ i ′ δ j ′ ] k ′ ∓ T i ′ j ′ k ′ (4 . J ± i ′ j ′ , T k ′ k ′ k ′ ] = 6 T [ k ′ k ′ | [ i ′ δ j ′ ] | k ′ ] ∓ ε i ′ j ′ k ′ k ′ k ′ l ′ l ′ l ′ T l ′ l ′ l ′ ± δ [ k ′ k ′ i ′ j ′ T k ′ ] (4 . T k ′ and T i ′ i ′ i ′ and form the 56+8 = 64 dimensional representation of SO(8) × SO(8).We will now identify what representation this is. We observe that the eight dimen-sional gamma matrices in Euclidean space obey the equations γ i ′ j ′ γ k ′ = γ i ′ j ′ k ′ + 2 δ k ′ [ j ′ γ i ′ ] (4 . γ i ′ j ′ γ k ′ k ′ k ′ = 6 δ [ j ′ [ k ′ γ i ′ ] k ′ k ′ ] + γ i ′ j ′ k ′ k ′ k ′ − δ [ k ′ k ′ i ′ j ′ γ k ′ ] = 6 δ [ j ′ [ k ′ γ i ′ ] k ′ k ′ ] + 13! ε i ′ j ′ k ′ k ′ k ′ l ′ l ′ l ′ γ l ′ l ′ l ′ γ − δ [ k ′ k ′ i ′ j ′ γ k ′ ] (4 . γ ≡ γ ... (4 . T k ′ = γ k ′ , T i ′ j ′ k ′ = − γ i ′ j ′ k ′ (4 . γ i ′ j ′ has the same result as J + i ′ j ′ in the commutators ofequations (4.11) and (4.12) provided γ takes the value −
1. Similarly one can verify that12ight multiplication by − γ i ′ j ′ has the same result as J − i ′ j ′ in the commutators of equations(4.11) and (4.12).As a result we can think of T j ′ j ′ j ′ and T i ′ as belonging to a bi-spinor which is Weylprojected. A bi-spinor takes the formΓ = c I + c i ′ γ i ′ + . . . + c i ′ ...i ′ γ i ′ ...i ′ (4 . γ Γ = Γ then we find that c = ǫ i ′ ...i ′ c i ′ ...i ′ etc and we can takethe bi-spinor to be of the formΓ = (1 + γ )( c I + c i ′ γ i ′ + . . . + c i ′ ...i ′ γ i ′ ...i ′ ) (4 . γ = − Γ then find that Γ takes the formΓ = (1 + γ )( c i ′ γ i ′ + c i ′ ...i ′ γ i ′ ...i ′ ) (4 . T j ′ j ′ j ′ and T i ′ is a bispinor with the abovevalues of Weyl projection. The coefficients c i ′ and c i ′ i ′ i ′ correspond to the fields h i ′ ≡ ǫ j ...j h j ...j ,i and A i ′ ...i ′ of equation (2.24). It we denote the eight dimensional Majo-rana Weyl spinor representation of SO(8) with Weyl projection +1, that is γ ǫ = ǫ , by 8 c and the one with Weyl projection − s then the generators T j ′ j ′ j ′ and T i ′ belong tothe (8 c , s ) representation of SO(8) × SO(8).Hence we find that the the spinor representation of SO (16) which contains the 128bosonic degrees of freedom of the massless irreducible representation decomposes into the128 = (8 v , v ) ⊕ (8 c , s ) (4 . × SO(8). The (8 v , v ) contains the fields h i ′ j ′ and A i ′ ...i ′ ofequation (2.24) while the (8 c , s ) contains the A i ′ i ′ ,i ′ and h i ′ ...i ′ ,k ′ fields. The dualityequations relate these two representations. The higher level fields are related by dualityequations to the fields of these two representations and so these duality relations can bethought of arising from the action of the affine operator that takes I c ( E ) to I c ( E ).Rather than discuss the transformations of the Cartan involution odd generators wecan consider the transformations of the corresponding fields which we will now derive. TheCartan form containing these fields can be written as˜ V = ˆ A i ′ i ′ ˆ T i ′ i ′ + A i ′ T i ′ + A i ′ i ′ i ′ T i ′ i ′ i ′ (4 . V = [( ˆ A i ′ i ′ −
16 ˆ A k ′ k ′ δ i ′ i ′ )] T i ′ i ′ + A i ′ i ′ i ′ T i ′ i ′ i ′ + 26! ε i ′ ...i ′ k ′ k ′ ˆ A k ′ k ′ T i ′ ...i ′ + 18! ε i ′ ...i ′ A k ′ T i ′ ...i ′ ,k ′ (4 . δ ˜ V = [Λ ± i ′ i ′ J ± i ′ i ′ , ˜ V ] (4 . δ Λ + ˆ A i ′ i ′ = 4Λ + i ′ k ′ ˆ A k ′ i ′ , δ Λ − ˆ A i ′ i ′ = − A i ′ k ′ Λ − k ′ i ′ ,δ Λ ± A i ′ = 2Λ ± i ′ k ′ ˆ A k ′ ± + k ′ k ′ A k ′ k ′ i ′ (4 . δ Λ ± A i ′ i ′ i ′ = ∓
12 Λ ± [ i ′ i ′ A i ′ ] + 6Λ ± [ i ′ | k ′ A k ′ | i ′ i ′ ] ± ε i ′ i ′ i ′ j ′ j ′ k ′ k ′ k ′ Λ ± j ′ j ′ A k ′ k ′ k ′ (4 . I c ( E ) = SO(16), whose generators are given in equation equation (4.1), into representa-tions of terms of SO(8) × SO(8). The generators not included in SO(8) × SO(8) are S k ′ ≡ ǫ j ′ ...j ′ S j ′ ...j ′ ,k ′ and S k ′ k ′ k ′ and their commutation relations with the SO(8) × SO(8)generators are given by [ J ± i ′ j ′ , S k ′ ] = 2 S [ i ′ δ j ′ ] k ′ ∓ S i ′ j ′ k ′ (4 . J ± i ′ j ′ , S k ′ k ′ k ′ ] = 6 S [ k ′ k ′ [ i ′ δ k ′ ] j ′ ] ± ε i ′ j ′ k ′ k ′ k ′ l ′ l ′ l ′ S l ′ l ′ l ′ ± δ i ′ j ′ [ k ′ k ′ S k ′ ] (4 . S k ′ and S k ′ k ′ k ′ have the same commutators with SO(8) × SO(8) as T k ′ and T k ′ k ′ k ′ except for an oppositesign in the second term on the right-hand side of equation (4.27). Examining the gammamatrix algebra of equations (4.13) and (4.14) we see that we can identify S k ′ and S k ′ k ′ k ′ with γ k ′ and − γ k ′ k ′ k ′ provided we take γ in equation (4.14) to take the value +1. Hencethese generators belong to the (8 s , c ) representation of SO(8) × SO(8). Hence the 120dimensional adjoint representation of SO(16) consists of the120 = (28 , ⊕ (1 , ⊕ (8 s , c ) (4 . × SO(8). The (28 , ⊕ (1 ,
28) contain the adjoint representationof SO(8) × SO(8). We note that the decompositions we have found are not quite the sameas those one finds in certain books.
5. Spinors of I c ( E ) decomposed into representations of SO(8) × SO(8)Having decomposed the 128 dimensional SO(16) spinor representation that containsthe bosonic degrees of freedom in terms of representation of SO(8) × SO(8) it will be edu-cational to also analyse the 128 dimensional spinor representation to which the fermionicdegrees of freedom belong. Long ago it was shown that the fermionic degrees of freedomappear in maximal supergravity in D dimensions as a linear representation of the Cartaninvolution invariant subgroup of the duality group E − D , for example in four dimensions14his group is I c ( E ) = SU (8) [16]. It was therefore natural to take the fermions in E theoryto be a linear representation of the Cartan involution invariant subalgebra of I c ( E ). Infact fermions were first introduced [17-20] in this way in the context of the E theory andsubsequently [21] in the E theory. The key to these constructions was the realisation that I c ( E ) and I c ( E ) admit highly unfaithful representations. In particular it was shownthat I c ( E ) has a representation based on a spinor of SO(1,10), ǫ α in which the low lowestlevel generators take the form [2] J a a ǫ α = −
12 ( γ a a ) αβ ǫ β , S a a a ǫ α = 12 ( γ a a a ) αβ ǫ β ,S a ...a ǫ α = −
14 ( γ a ...a ) αβ ǫ β , S a ...a ,b = δ b [ a ǫ a ...a ] c c γ c c , . . . (5 . I c ( E ) commutation relations. Clearly, some parts of the higher level generators aretrivially realised and so the representation is unfaithful.We will be interested in the fermionic degrees of freedom from the viewpoint of theirreducible representations of I c ( E ) ⊗ s l and in particular as a representation of I c ( E ).The I c ( E ) transformations of the spinor ǫ α are given by equation (5.1) with the indicesrestricted to take the values 2 , . . . ,
10. It is straight forward to show that the representation, ǫ is annihilated by precisely the same generators of equation (3.6) and (3.11), that is, N i i i ǫ = 0 = N i i ǫ (5 . ǫ .Thus the spinor ǫ is annihilated by the same generators as the representation that containsthe bosonic degrees of freedom and so they form the same ideal I . As such the spinor isreally a representation of K ( E ) I = SO (16).The generalisation of the unfaithful representation of equation (5.1) to the gravitinowhich was first done in the context of E [17-20] and the result for E [21] is J a a ψ b = − γ a a ψ b − η b [ a ψ a ] S a a a ψ b = λ { γ a a a ψ b − γ b [ a a ψ a ] + 4 η b [ a γ a ψ a ] } S a ...a ψ b = − γ a ...a ψ b − γ b [ a ...a ψ a ] + 5 η b [ a γ a ...a ψ a ] S a ...a ,c ψ b = λ { γ b [ a ...a ψ c ] − γ ba ...a ψ c + 8( η b [ c γ a ...a ψ a ] − η bc γ [ a ...a ψ a ] ) − η c [ a γ a ...a ] ψ b − γ b [ a ...a ψ a η a ] c } (5 . λ to take the values λ = − λ = 1 and still have a repre-sentation of I c ( E ). This reflects the way the generator S a a a appears in the algebraresulting from its level one character. In reference [21] we took λ = 1 but in this paper wewill find that the other sign is better. 15hen the indices in equation (5.2) take the range 3 to 10 we can interpret this equationas containing the representation that contains the fermionic degrees of freedom as theyoccur in the little algebra I c ( E ).In this section we will decompose the spinor representation of equation (5.2) into thoseof SO(8) × SO(8) which is generated by J ± i ′ j ′ defined in equation (4.2), but we will beginwith the representation of equation (5.1). We find that J ± i ′ j ′ ǫ = − γ i ′ j ′ (1 ∓ γ ) ǫ (5 . γ = γ . . . γ . Defining ǫ ± = ( I ± γ ) ǫ we can rewrite this equation as J + i ′ j ′ ǫ + = 0 , J − i ′ j ′ ǫ + = − γ i ′ j ′ ǫ + ; J + i ′ j ′ ǫ − = − γ i ′ j ′ , ǫ − J + i ′ j ′ ǫ − = 0 (5 . ǫ + and ǫ − to be the 8 c and 8 s representations of SO(8) respectively then thespinor ǫ = ǫ + ⊕ ǫ − is in the ( I ⊗ c ) ⊕ (8 s ⊗ I ) of SO(8) × SO(8).We now wish to decompose the gravitino into representations of SO(8) × SO(8). Theaction of the SO(16) generators can be read off from equation (5.3) by taking the indices totake the values i ′ .j ′ = 3 , . . .
10. The gravitino obeys the condition γ i ψ i = γ · ψ + γ ψ = 0and so we can eliminate the ψ component in terms of γ · ψ ≡ γ i ′ ψ i ′ . We find that theSO(8) × SO(8) generators J ± i ′ i ′ act on the gravitino ψ k ′ as J ± i ′ i ′ ψ j ′ = − γ i ′ i ′ (1 ∓ γ ) ψ j ′ − η j ′ [ i ′ ψ i ′ ] ± γ γ i ′ i ′ γ j ′ γ · ψ − γ γ i ′ i ′ ψ j ′ ∓ γ η j ′ [ i ′ γ i ′ ] γ · ψ ∓ γ γ j ′ [ i ′ ψ i ′ ] ± γ η j ′ [ i ′ ψ i ′ ] (5 . γ k ...k n = ( − n ( n +1)2 m ! ǫ k ...k n j ...j m γ γ j ...j m (5 . n + m = 8. While the action of J ± i ′ j ′ on γ · ψ follows from equation (5.6) and it isgiven by J ± i ′ i ′ γ · ψ = − γ i ′ i ′ γ · ψ ∓ γ γ i ′ i ′ γ · ψ ± γ γ [ i ′ ψ i ′ ] . (5 . i = ψ i − γ i γ · ψ (5 . J ± i ′ i ′ Ψ k = − γ i ′ i ′ (1 ± γ )Ψ k ′ − η k ′ [ i ′ (1 ∓ γ )Ψ i ′ ] (5 . ± k = (1 ± γ )Ψ k then equation (5.10) can be rewritten as J + i ′ i ′ Ψ + k = − γ i ′ i ′ Ψ + k , J − i ′ i ′ Ψ + k = − η k [ i Ψ + i ] ,J + i ′ i ′ Ψ − k = − η k [ i Ψ − i ] , J − i ′ i ′ Ψ − k = − γ i ′ i ′ Ψ − k (5 . k = Ψ + k + Ψ − k belongs to the (8 c , v ) ⊕ (8 v , s ) representa-tion of SO(8) × SO(8). Thus both the fermionic and bosonic degrees of freedom belongto spinor representations of SO(16) and they have a rather similar decompositions intorepresentations of SO(8) × SO(8). Indeed the bosonic degrees of freedom belong to the(8 v , v ) ⊕ (8 c , s ) and one can interchange the bosons and fermions by interchanging 8 v with 8 c for the first SO(8) factor.It was observed in reference [18] that the fermionic degrees of freedom encoded in thegravitino carried a highly unfaithful representation of the Cartan involution algebra of therelevant algebra and as a result the representation should be annihilated certain generators.This paper also pointed out that this set of generators would form a subalgebra that wasan ideal. We will now compute the lowest level generators in the context of irreduciblerepresentation, that is I c ( E ). Using equation (5.3) with the indices taking the values3 , . . . , ,
10, we find, taking λ = − S i i i − ǫ i i i j ...j S j ...j ) ψ k = 0 (5 . J ij + 28! ε [ i | r ...r S r ...r , | j ] ) ψ k = 0 (5 . λ = 1and find the same ideal if one systematically changed our definition of γ by introducing aminus sign. Thus seen from the perspective of the irreducible representations of I c ( E ), thebosonic and fermionic states have a remarkably similar structure. They both belong to 128-dimensional spinor representations of SO(16), they have the same ideal which annihilatesthe two representations which carry a representation of the coset of I c ( E ) with the ideal I whose coset I c ( E ) I which is just SO(16).However, there is an important difference in the way the bosons and fermions appear.The original irreducible representation for the bosons contains an infinite number of fieldswhose number is reduced by duality relations, while for the fermions we just have thegravitino. This latter representation was not deduced from some general theory but byhand following the example of the maximal supergravity theories in lower dimensions. Itwould seem natural to introduce higher level spinors and have these linked to the gravitinoby duality relations. One possibility is to introduce the fields ψ α , ψ i ...i α , ψ i ...i α , ψ i ...i ,j ...j α , . . . (5 . ψ i ...i = γ i ...i j ψ j , . . . (5 . I c ( E ) and even E . Thislatter step could be possible if one exploited the fact that the algebras SL(D) do admitspinor representation. These have not been popular as they infinite dimensional, but thisis what we need in this context. [25].
6. The covariant formulation
In the above section we examined how the bosonic degrees of freedom of supergravityarose as the massless irreducible representation of I c ( E ) ⊗ s l . In particular we saw thatthey occurred as an irreducible representation of the little group I c ( E ) as carried by thefields of equation (2.9). To find the full representation of I c ( E ) ⊗ s l one has to carryout a boost. This procedure was discussed in reference [7]. In carrying out this last stepone does not introduce any additional fields and so even though all the symmetries arepresent they are not manifest. However, in field theory one usually requires representationsthat have some of the symmetries realised in a covariant manner. For example, for theirreducible representations of the Poincare algebra one usually requires the Lorentz algebrato be manifest. In our case we want the I c ( E ) symmetry to be essentially manifest.How to achieve this was very briefly outlined in reference [7] and in this section we willdiscuss this procedure in detail. We will also examine how the duality relations andannihilation generators which are present in the irreducible representation are inheritedinto the covariant formulation.Before we begin we will briefly recall the general theory for obtaining a covariantformulation of an irreducible representation of I c ( E ) ⊗ s l . We will use the same notationas in section four of reference [7]. Let the little algebra be denoted by H and then anygroup element g of I c ( E ) ⊗ s l can be written as g = e ϕ · S h for h ∈ H . Before the boostwe choose a linear representation u α (0) of H which we can take to transform as U ( g ) u α (0) = D ( g − ) αβ u β (0) , L A u α (0) = l (0) A u α (0) (6 . u α (0) which, as a result, must be a reducible representationof H .The next step is to boost the representation u α (0) to find a representation of I c ( E ) ⊗ s l in much the same way as we boosted the irreducible representation of H (see sectionfour of reference [7]). In particular, we take u α ( ϕ ) ≡ U ( e ϕ · S ) u α (0) where the symbol U denotes the action of the group element. However, u α ( ϕ ) does not transform covariantlyand so instead we consider the objects A α ( ϕ ) = D ( e ϕ · S ) αβ u β ( ϕ ) = D ( e ϕ · S ) αβ U ( e ϕ · S ) u β (0) (6 . U α ( ϕ ) transforms covariantly, and in particular U ( g ) A α ( ϕ ) = D ( g − ) αβ A β ( ϕ ′ ) (6 . g ∈ I c ( E ) and we have used the relation ge ϕ · S = e ϕ ′ · S h ( g, ϕ ) with h ( g, ϕ ) ∈ H .Although we have embedded the original irreducible representation into a bigger represen-tation of H we still have to ensure that we still have the irreducible representation and asa result we have to impose projection conditions and, for the massless case, equivalencerelations. How to do this in general has yet to be understood but we will do it for themassless irreducible representation.For comparison we very briefly give an account of the spin one massless irreduciblerepresentation of the Poincare algebra. We can choose all momenta to vanish except for p + = p − = √ m . The little algebra is SO(D-2) since we have to take the generator J + i tovanish in order to have a unitary representation. The representation of SO(D-2) is carriedby the fields A i (0), i = 1 , . . . , D −
2. We embed this irreducible representation into theSO(D-2) reducible representation A µ (0) , µ = 0 , . . . , D −
1. To get the same fields as in theoriginal irreducible representation we demand that A + (0) = 0 and require the equivalencerelation A − (0) ∼ A − (0) + p − Λ(0). The latter allows us to set A − (0) = 0. To find theeffect of these conditions in the covariant formulation in terms of A µ we can impose theseconditions in the rest frame fields, as given in equation (6.2), and we recover, in x -spacethe familiar gauge fixing condition ∂ a A a ( x ) = 0 and the well known gauge transformation A a ( x ) ∼ A a ( x ) + ∂ a Λ( x ).We will now carry out the above discussion in the context of the massless irreduciblerepresentation we studied in the earlier sections. The first step is to embed the fields of theirreducible of of I c ( E ) ⊗ s l contained in equation (2.9) into a larger representation. Thereis an obvious candidate, namely, we take the fields corresponding to the Cartan involutionodd generators of E rather than just those of E . As such we consider the fields A α (0) = { h a a (0) , A a a a (0) , A a ...a (0) , h a ...a ,b (0) , . . . } a, b, . . . = 0 , . . . , . (6 . I c ( E ). These correspond to the fields A α (0) of equation (6.2).We will now address the issue of the embedding condition that will ensure that we arereally still dealing with the same massless irreducible representation. We will first do thisbefore the fields are boosted as in equation (6.2). We take the fields A α (0) of equation(6.4) to be subject to two conditions. The first of these is K AB G A,BC = K AB ( D α ) B C ∂ A A α = 0 (6 . K AB is the metric on a tangent space which transforms under I c ( E ) as its indicessuggest [13], and the second is the equivalence relation A α ≃ A α + ( D α + D − α ) AB ∂ B Λ A . (6 . D α ) B C are those of the first fundamental representation and are definedin the equation [ R α , l B ] = − ( D α ) B C l C . As we will see equation (6.5) eliminates all fieldswith a lower + index, while the second equation (6.6) eliminating fields which take on alower − index, leaving us with the fields of equation (2.9), that is, those of the originalmassless irreducible representation. 19hen acting on the fields of equation (6.4), the only component of ∂ A which is non-zerois ∂ − = ∂∂x − . Thus the only non-zero component of K AB in equation (6.5) is K − + = η − + ,and so equation (6.5) reduces to ( D α ) + C ∂ − A α = 0. Since p − = √ m is a constant wehave the condition ( D α ) + C A α = 0. The above matrix occurs in the equation [ R α , P + ] = − ( D α ) + B l B and so our condition can be expressed as [ A α R α , P + ] = 0 which consists oftransformations A α R α that preserve the p + = 0 with the only non-zero momentum being p − . By definition this includes the little algebra transformations E and so the conditionof equation (6.5) places no constraints on the fields in equation (2.9). At levels zero, oneand two the condition of equation (6.5) lead to h ++ = 0 = h + i = h ii ; A + a a = 0 = A + a ...a (6 . A α R α , P + ] = 0 contains a factor δ a + times A α times the l generator at the corresponding level. Hence we find that the condition ofequation (6.5) implies that any field A α with a lower + index is set to zero.We can recover the same results by directly examining the component equations thatfollowing from equation (6.5). At levels zero, one and three, these are given by [13] ∂ e h ea − ∂ a h ee = 0 , ∂ e A ea a = 0 , ∂ e A ea ...a = 0 , . . . (6 . p − non-zero in equation (6.8) we recover the above result.We now consider the equivalence relation (6.6). In the rest frame, that is, for thefields of equation (6.4), this reduces to ( D α + D − α ) A − ∂ − Λ A which is proportional to( D α + D − α ) A − Λ A . This matrix occurs in the commutator [ R α + R α , l A ] = − ( D α + D α ) A − P − . At level one we have the commutator [ R a a a , Z b b ] = 6 δ b b [ a a P a ] and so werequire R a a − in order to get P − . At higher levels one also finds that this expression willonly be non-zero if the index α contains a lower − index. To see this we note that the levelzero P − on the right-hand side arises from the commutator of a level n l generator anda level − n level generator of E with a resulting structure constant that contains a δ a − factor. Hence all the fields A α with a lower − index are subject to an equivalence relationand for a suitable choice of Λ A they can be set to zero.The equivalence relation (6.6) have been worked out at low levels [14] and they leadto the expected gauge transformations; δh ( ab ) = ∂ a ξ b + ∂ b ξ a , δA a a a = ∂ [ a Λ a a ] , δA a ...a = 2 ∂ [ a Λ a ...a ] , . . . (6 . p − we find that at levels zero, one and two we can setto zero h − + , A − a a and A − a ...a , . . . , in agreement with the above discussion.So far we only considered the effect of the conditions of equation (6.5) and (6.6) in therest frame, that is, p − = √ m all other members of the vector representation being zero.To find the analogue of these conditions for the covariant theory we take the conditionsto act on u α (0) in equation (6.2) and then apply the boost and matrix multiplicationthat this equation contains. Since equation (6.5) contains the I c ( E ) covariant expression20 AB ( D α ) BC ∂ A it retains the same form under the boost and so we can take this equationto hold for the theory after the boost. Taking equation (6.6) to hold in the rest frame andcarrying out the boost and matrix multiplication,as in equation (6.2), we find that it takesthe form as in equation (6.6) but with a parameter Λ A ( ϕ ) = D ( e − ϕ · S ) C A U ( e ϕ · S )Λ C (0).We recognise this as a gauge transformation of reference [14].Hence we have shown that the conditions of equations (6.5) and (6.6) do not affectthe fields of equation (2.9). However for the fields of equation (6.4) we find that they setall fields with a lower + to zero and we can, using the equivalence relation remove allfields with a lower − index. As a result equations (6.5) and (6.6) ensure that A α , givenin equation (6.4), contains the same fields as occurred in the original massless irreduciblerepresentation, given in equation (2.9). Hence it is just the same massless irreduciblerepresentation of I c ( E ) ⊗ s l .In section two we found that the fields of the massless irreducible representationobeyed the duality relations of equation (2.19), (2.20) and (2.22) as well as similar higherlevel equations. While in section three we found that as a result this representation wasannihilated by the generators which belonged to an ideal, for example those in equations(3.6) and (3.11). As the covariant formulation of the irreducible representation is essentiallythe same as the original irreducible representation we can expect that the generators inequations (3.6) and (3.11), as well as those at higher level, imply the presence of analogousequations which we will not find. Indeed, starting from the equations that the generatorsin the ideal I annihilate the fields in the irreducible representation in the rest frame, wecan carry out a boost, as in equation (6.2), to find an infinite number of constraints in thecovariant theory. However, as the boost contains generators which levels greater than zeroit will transform the elements in the ideal with different levels into each other.Rather than carry out the boost, we will find some covariant operators that agreewith elements in the ideal in the rest frame. We begin by considering the operatorˆ N a ...a = P [ a S a a a ] + 12 . ε a ...a c ...c P c S c ...c + 14 Z e e S e e a ...a + 17 . ǫ a ...a c ...c Z e e S e c ...c ,e + 112 Z [ a a J a a ] (6 . N − i i i which is equal to p − N i i i ,where N i i i is defined in equation (3.6). However, as we have shown in section three N i i i vanishes on the massless irreducible representation of equation (2.9) and as a resultˆ N a ...a also vanishes on this representation on the fields the rest frame. The terms inˆ N a ...a that contain higher level elements of l vanish in the rest frame and the precisecoefficients that we have given will be derived below. Clearly, ˆ N a ...a is one of the covariantoperators resulting from the action of the boost on the annihilation operators in the restframe as discussed just above.We now consider if ˆ N a ...a really does annihilate the covariant states of equation(6.4). Since the covariant fields are only defined up to an equivalence, or equivalentlygauge symmetry, it simplifies the calculation considerably if we act on objects that are21auge invariant. At the linearised level such objects are ω a,b b = − ∂ b h ( b a ) + ∂ b h ( b a ) + ∂ a h [ b b ] , G a ...a = ∂ [ a A a a a ] , G a ...a = ∂ [ a A a ...a ] , . . . (6 . N b b b b G a a a a = − P a ε a a a b ..b c c c c E c c c c + 3 δ [ b b a a P a E a ] ,b b ] (6 . N b b b b ω a ,a a = − η a a P a E b ...b + 4 η a a P a E b ...b − P c η c a δ [ c c a a E b ...b ] − P c η c a δ [ c c a a E b ...b ] − P c η c a δ [ c c a a E b ...b ] + ∂ a ˜Λ a a b ...b (6 . E a ...a = G a ...a − · ε a ...a b ...b G b ...b = 0 , (6 . E a,b b = ω a,b b − ε b b c ...c G c ...c ,a = 0 , (6 . ∂ a ˜Λ a a b ...b = P a [12 δ a [ b G a b b b ] − δ a [ b G a b b b ] + 3 δ a [ b P a A b b b ] − δ a [ b P a A b b b ] + 38 ε b ...b c ...c P c η c a A a c ...c − ε b ...b c ...c P c η c a A a c ...c ](6 . E ⊗ s l with local subalgebra I c ( E ) [1,2,3,4]. Thus we find that N a ...a will indeed vanish on the covariant fields providedthe duality relations hold. The last term in equation (6.13) is defined in equation (6.16)and it just corresponds to the fact that ω a,b b is subject to local Lorentz transformations.We note that ˜Λ a a b ...b is indeed of the form of a local Lorentz transformation since the b . . . b indices can be contracted with a parameter with these indices.We will now consider the operatorˆ N b b b = P [ b J b b ] − ε [ b b | e ...e P e S e ...e , | b ] + . . . (6 . . . . means terms which contain higher order l generators. Acting on the fields ofequation (2.9) in the rest frame this operator we find that it obviously vanishes except forˆ N − i i = p − N i i where N i i is defined in equation (3.11). However, this generator alsovanishes on the rest frame states and so ˆ N b b b vanishes on the rest frame states. As suchwe would expect it to vanish on the covariant fields of equation (6.4) One finds thatˆ N b b b ω a ,a a = η a a P a E [ b ,b b ] − P a δ [ b ( a E a ) ,b b ] − η a a P a E [ b ,b b ] P a δ [ b ( a E a ) ,b b ] − P r ( η r a δ [ b b a a E | b | ,r r + η r a δ [ b b a a E | b | ,r r ] − η r a δ [ b b a a E | b | ,r r ] )+ ∂ a ˜Λ a a b b b = 0 , (6 . ∂ a ˜Λ a a b b b = P a { ε [ b b | r ...r P r ( η r a h a r ...r ,b ] − η r a h a r ...r ,b ] )+ P [ b ( δ b a h a b ] − δ b a h a b ] ) } (6 . N a ...a and ˆ N a a a , and presumably their higher level analogues,annihilate the covariant fields one would expect their commutator with the generators of I c ( E ) to give them back. At lowest order, and assuming we can find a missing factor oftwo, we find that[ S a a a , ˆ N b b b b ] = +9 . δ [ a a [ b b ˆ N a ] b b ] + 124 ǫ b b b b a a a c c c c ˆ N c c c c (6 . N a ...a and the higher level generators of I c ( E ). Itwould be interesting to compute the commutators of ˆ N a ...a and ˆ N a ...a , . . . with them-selves. Given the above result one would expect the result to be proportional to ˆ N ’s times l generators. A result which would again annihilate the covariant fields.The covariant fields are subject to the Casimir condition L Ψ = K AB L A L B Ψ = 0as well as higher level conditions [15]. We would expect that the generators ˆ N a ...a andˆ N a ...a , . . . will commute with these up to terms proportional to themselves. They shouldalso be consistent with the gauge transformation of equation (6.5).In this section we have found that the generators in the ideal I that annihilate themassless irreducible states of equation (2.9) in the rest frame lead to an infinite set ofoperators which annihilate the massless irreducible representation when expressed in termscovariant fields provided the covariant fields obey an infinite set of duality relations. Thesecovariant duality relations are precisely those that occur in the non-linear realisation of I c ( E ) ⊗ s l with local subalgebra I c ( E ). This is consistent with the observation that theduality conditions on the covariant fields are just covariant versions of the duality identitiesthat exist in the original massless irreducible representation [7]. Thus the existence of theduality relations which contain the dynamics arise from the irreducible representation andin particular many of them correspond to the action of the affine action of I c ( E ) on the I c ( E ) representation of equation (2.24).
7. An alternative formulation of fermions in E theory
The spinors were introduced in E theory by hand in the sense that they do not followin natural way from the E algebra but started from the familiar gravitino and insistedthat it carry a representation of I c ( E ), which was constructed by hand. In this section wewill take a different approach which generalises, in a natural way, the way spinors appear inthe context of the Poincare algebra. For the Poincare algebra, that is, SO (1 , D − ⊗ s T D T D leads to a D dimensional spacetime on which SO (1 , D − γ a matrix which act on a spinor.In E theory we start from the algebra I c ( E ) ⊗ s l which is also a semi-direct product.The vector representation l leads in the nonlinear realisation to the spacetime and wenow introduce matrices Γ A ’s which are in one to one correspondence with the vectorrepresentation, hence their label A . As a result they are also carry the indices of thespacetime coordinates in E theory. For example, in eleven dimensions we introduce thematrices Γ A = { Γ a , Γ a a , Γ a ...a , Γ a ...a ,b , Γ a ...a , . . . } (7 . A Γ B + Γ B Γ A = 2 K AB (7 . K AB is the I c ( E ) invariant tangent space metric. We now introduce a spinor Ψwhich carries a representation of the Γ matrices and transforms under I c ( E ) as δ Ψ = U ( S α )Ψ = −S α Ψ (7 . U ( S α ) is the action of the generator S α and the matrix S α its effect. We requirethat [ S α , Γ A ] = Γ B ( ˜ D α ) BA (7 . D α ≡ D α − D α . By its definition the matrix of the vector representation appearsin the commutator [ R α , l A ] = − ( D α ) AB l B . We recall that S α = R α − R α and that( ˜ D α ) AC K CB is an antisymmetric matrix. As such the commutator of S α with the Γ A ’sleads to a transform under I c ( E ) which is that of the vector representation.A generalised Dirac equation is given byΓ A ∂ A Ψ = 0 (7 . I c ( E ) transformations as the derivatives ∂ A ≡ ∂∂x A transforms as δ ( ∂ A ) = − ( ˜ D α ) AB ∂ B .At level zero I c ( E ) is just SO(1,10) and so at this level the above discussion justreduces to the standard discussion of the Dirac equation. The Gamma matrices correspond-ing to the higher level coordinates occur with derivatives with respect to these coordinates.Thus if we neglect the higher level derivatives this is just the familiar Dirac equation. Wenote that we can not take Γ a a , . . . to be proportional to the standard gamma matrix γ a a , . . . as this does not satisfy equation (7.2)The above applies if we replace E by any Kac-Moody algebra and the vector rep-resentation by anyone of its representations. To give a simple example we consider thenon-linear realisation of A +++1 ⊗ l which corresponds to gravity in four dimensions. Atlevels zero and one the generators of the vector representation are P a and Z a and so weintroduce the gamma matrices Γ A = Γ a , ˜Γ a , . . . with a = 0 , , . . . ,
3. The algebra I c ( A +++1 )contains the generators J ab and S ab at levels zero and one respectively. For an account24f this theory see reference [26] and the earlier references it contains. We can take thegamma matrices to be Γ a = γ a ⊗ I, ˜Γ a = 2 iγ ⊗ γ a (7 . γ a γ b + γ b γ a = 2 η ab are the usual gamma matrices in four dimensions, ( iγ ) = I and γ γ b + γ b γ = 0. Thus this spinor has eight components.The vector representation of the algebra I c ( E ) contains at level one the coordinates x a a and so the corresponding spinor will carry a representation of the matrices Γ a a which obey Γ a a Γ b b + Γ b b Γ a a = 2 δ a a b b (7 . γ a = ǫ ab b Γ b b and we can use our usual representation of gamma matrices in three dimensions. So werequire the spinor in this sector to have two components. If the index range is in four, forexample, Euclidean dimensions then we can defineΓ + a a = 12 (Γ + a a ± ǫ a a b b Γ b b ) (7 . ǫ a a b b Γ ± b b = ± Γ ± a a , and { Γ + a a , Γ − b b } = δ a a b b (7 . ± , Γ ± and Γ ± and a represen-tation is formed by defining the vacuum | > to satisfyΓ − | > = 0 = Γ − | > = Γ − | > (7 . +12 , Γ +13 and Γ +14 as creation operators acting on thisvacuum, that is, | >, Γ +12 | >, . . . , Γ +12 Γ +13 | >, . . . (7 . = 8 states, or spinor components, due to this sector of the spinor.Let us also consider the algebra I c ( E ) ⊗ s l . The vector representation of E hasdimension 248 and as explained in section four this decomposes into the 120 plus 128dimensional representations of I c ( E ) = SO(16). We can take Gamma matrices corre-sponding to both of these representations or just one of them. If we take just the spinor128 dimensional representation the we should take the Gamma matricesΓ i , Γ i i , Γ i i i , Γ ( ij ) , where i, j = 1 , , . . . , . ⊗ SO(8) decomposition of section four. Thegeneralised Dirac equation would have the form(Γ i ∂ i + Γ i i ∂ i i + . . . )Ψ = 0 (7 . γ matrices as above. The Γ i can be taken to be the usually gamma matrices in the firstblock and as their are eight of them this part of the spinor will have 2 = 16 components.More generally the spinor which carries a representation can be found by defining annihi-lation and creation operators. We might expect 64 creation operators and 64 destructionoperators and so the corresponding part of the spinor should have 2 components.The above considerations were at the linearised order but it is straightforward togeneralise it to the full symmetries of the E ⊗ s l by takingΓ A E AM ( ∂ M + Q M )Ψ = 0 (7 . E M A is the vielbein and Q M the connection found in the non-linear realisation.We end this section with some speculative remarks. To account for the gravitino wecan introduce the object Ψ A which is a spinor with the vector index A . We could take thisto obey the on-shell conditionsΓ B ∂ B Ψ A = 0 = Γ A Ψ A = ∂ A Ψ A (7 . a . We leave it to the future to examine how this fits into the full theory.We could also introduce a generalised supersymmetry generator Q A which is a gener-alised spinor and so with I c ( E ) generators it has the relation[ S α , Q ] = −S α Q (7 . { Q, Q } = Γ A l A (7 . In this paper we have analysed in detail the irreducible representation of I c ( E ) ⊗ s l corresponding to a massless point particle. The corresponding little algebra, I c ( E ) ⊗ s l , isan infinite dimensional Lie algebra and as a result one may expect it to contain an infinitenumber of degrees of freedom. However, the fields are subject to an infinite number ofduality relations which are preserved by I c ( E ) and these reduce the number of independentdegrees of freedom to be just 128 which belong to the spinor representation of I c ( E ) =SO(16). These can be taken to be h ij ( h ii = 0) and A i i i where i, j = 2 , . . . ,
10. Theyare indeed the bosonic degrees of freedom of eleven dimensional supergravity. The infinitenumber of duality equations relate all the other fields in the representation to these fields.A consequence of these duality relations is that the irreducible representation is annihilatedby an infinite number of generators of I c ( E ) which form a subalgebra I which is an ideal.The Lie algebra I c ( E ) I is SO(16). 26he 128 bosonic independent degrees of freedom belong to the (8 v , v ) ⊕ (8 c , s ) rep-resentations when decomposed into the subalgebra SO(8) × SO(8) subalgebra of SO(16).These two representations contain the fields ( h i ′ j ′ , A i ′ ...i ′ ) and ( A i ′ i ′ i ′ , h i ′ ...i ′ ,k ′ ) respec-tively where i ′ , j ′ = 3 , . . . ,
10. The SO(16) transformations not in the SO(8) × SO(8)subalgebra change these representations into each other. The remaining fields which ap-pear in the irreducible representation of I c ( E ) are affine copies of these 128 fields and arerelated by duality relations to the fields in the (8 v , v ) ⊕ (8 c , s ) representations. Like allsuch irreducible representations we can formulate the massless irreducible representationin a covariant manner. The infinite number of duality relations become covariant dualityequations which contain the dynamics and one finds a corresponding set of operators thatannihilate the representation.The dynamics that results from the non-linear realisation of E ⊗ s l with localsubalgebra I c ( E ) has been found at low levels and the resulting equations of motionagree precisely with those of maximal supergravity if one discards the dependence on thespacetime coordinates beyond those usually considered. The dynamics appears throughan infinite set of duality equations that relate an infinite number of the higher level fieldsto the graviton and three form and it is by taking space time derivatives of these thatone can find the standard equations of motion of maximal supergravity. The dualityrelations that arise in the covariant formulation of the massless irreducible of I c ( E ) ⊗ s l in the covariant formulation are essentially those that arise in the non-linear realisationof E ⊗ s l with local subalgebra I c ( E ) at the linearised level. This provides strongsupport for the strongly suspected fact that the only degrees of freedom contained in thenon-linear realisation are the 128 bosonic degrees of freedom of supergravity. We alsosee that the infinite number of duality relations that appear in the non-linear realisationare a consequence of the duality relations that occur in the irreducible representation andthis allows us to predict the structure of the duality relations in the former theory. Inthe non-linear realisation there are also higher level fields which have blocks of ten, oreleven, indices and so these fields do not appear in the irreducible representation. Thesefields obey equations predicted by the non-linear realisation and they lead to the gaugedsupergravities.In the irreducible representations in the E theory approach the bosonic and fermionicdegrees of freedom appear in a unified way. They belong to the two different spinorrepresentations of I c ( E ) = SO(16) which are associated with the two nodes at the end ofthe SO(16) Dynkin diagram. Thus swopping thee two nodes results in swopping bosonsand fermions. One puzzle with E theory is the way it leads to predictions that were usuallyseen as a result of supersymmetry. Examples are the appearance of the two and five branecharges and the BPS conditions [15]. Supersymmetry was discovered, at least from theRussian viewpoint, by demanding that internal and spacetime symmetries were containedin the same symmetry algebra. This required the introduction of the supersymmetrygenerators. However, E achieves the same objective as it does contain the symmetriesof spacetime, such as Lorentz symmetry, and also internal symmetries such as the internalexceptional symmetries of the maximal supergravities.In this paper we have concentrated on the massless irreducible representation of E ⊗ s l which corresponds to the maximal supergravity theories. The vector representation of27 contains the brane charges and by taking these to be non-zero we will find otherirreducible representations of E ⊗ s l corresponding to branes. In a future paper we hopeto examine some of these representations in detail and make the connection to the thebrane dynamics as also appears in the context of E theory, see for example references [27].The interesting paper [28] also remarks on the similar properties of bosons and fermionsalthough from a different viewpoint. It noticed that the SO(9) representations to whichbosonic (44 ⊕ K Ψ = 0. In this equation K is the Kostant operator which isof the form K = P a T a γ a where γ a are gamma matrices and T a are generators of F thatbelong to the coset F SO(9) . The algebra F , dimension 52, can be found from the algebraSO(9), dimension 36, by adding to the latter the 16 generators that belong to the spinorrepresentation of SO(16).One can speculate that this picture can be generalised to incorporate the features ofthe irreducible representations found in this paper. The bosonic degrees of freedom belongto one spinor representation of SO(16), while the fermionic degrees of freedom belongto the other spinor representation, both of which have dimension 128. The algebra E emerges from the SO(16) algebra if we add to the 128 generators belonging to the spinorrepresentation of SO(16) to the 120 generators of SO(16). We can then consider the cosetof E SO(16) . The corresponding Kostant operator would consist of 128 Gamma matricesmultiplied by the generators in the coset. Could it be that the solutions of the equation K Ψ = 0 contain the two 128 dimensional spinor representation of SO(16)? We can alsowonder what are the higher spin solutions? Speculating even further we can suppose thatsome of these 128 Gamma matrices can be taken to be the supercharges similar to whattook place in the discussion of the N = 2 hypermultiplet given in reference [28]. We hopeto study these matter further and how they might fit into E theory. Acknowledgements
Peter West wishes to thank the SFTC for support from Consolidated grants numberST/J002798/1 and ST/P000258/1, while Keith Glennon would like to thank Kings Collegefor support during his PhD studies.
References [1] P. West, E and M Theory , Class. Quant. Grav. , (2001) 4443, hep-th/ 0104081.[2] P. West, E , SL(32) and Central Charges , Phys. Lett. B 575 (2003) 333-342, hep-th/0307098.[3] A. Tumanov and P. West,
E11 must be a symmetry of strings and branes , arXiv:1512.01644.[4] A. Tumanov and P. West,
E11 in 11D , Phys.Lett. B758 (2016) 278, arXiv:1601.03974.[5] P. West,
Introduction to Strings and Branes , Cambridge University Press, 2012.[6] P. West,
A brief review of E theory , Proceedings of Abdus Salam’s 90th Birthday meet-ing, 25-28 January 2016, NTU, Singapore, Editors L. Brink, M. Duff and K. Phua,World Scientific Publishing and IJMPA,
Vol 31 , No 26 (2016) 1630043, arXiv:1609.06863.[7] P. West,
Irreducible representations of E theory , Int.J.Mod.Phys. A34 (2019) no.24,1950133, arXiv:1905.07324.[8] E. P. Wigner, On unitary representations of the inhomogeneous Lorentz group, AnnalsMath.40(1939) 149. 289] A. Kleinschmidt and P. West,
Representations of G+++ and the role of space-time ,JHEP 0402 (2004) 033, hep-th/0312247.[10] P. West, E origin of Brane charges and U-duality multiplets , JHEP 0408 (2004) 052,hep-th/0406150.[11] P. Cook and P. West, Charge multiplets and masses for E(11) , JHEP (2008) 091,arXiv:0805.4451.[12] P. West, The IIA, IIB and eleven dimensional theories and their common E origin ,Nucl. Phys. B693 (2004) 76-102, hep-th/0402140.[13] M. Pettit and Peter West, An E11 invariant gauge fixing , Int.J.Mod.Phys. A33 (2018)no.01, 1850009, arXiv:1710.11024.[14] P. West,
Generalised Space-time and Gauge Transformations , JHEP 1408 (2014) 050,arXiv:1403.6395.[15] P. West,
Generalised BPS conditions , Mod.Phys.Lett. A27 (2012) 1250202, arXiv:1208.3397.[16] E. Cremmer and B. Julia,
The SO(8) supergravity , Nucl. Phys.B 159(1979) 141[17] S. de Buyl, M. Henneaux and L. Paulot,
Extended E8 Invariance of 11-DimensionalSupergravity
JHEP (2006) 056 hep-th/05122992 [18] T. Damour, A. Kleinschmidt qand H. Nicolai
Hidden symmetries and the fermionicsector of eleven-dimensional supergravity
Phys. Lett. B (2006) 319 hep-th/0512163 .[19] S. de Buyl, M. Henneaux and L. Paulot
Hidden Symmetries and Dirac Fermions
Class.Quant. Grav. (2005) 3595 hep-th/0506009 .[20] M Henneaux, E Jamsin, A Kleinschmidt and D Persson; Phys. Rev. D (2009) 045008;arXiv:0811.4358[21] D. Steele and P. West, E11 and Supersymmetry , JHEP 1102 (2011) 101, arXiv:1011.5820.[22] A. Kleinschmidt, H. Nicolai, and J. Palmkvist, K ( E ) from K ( E ), JHEP 06 (2007)051, hep-th/0611314[23] A. Kleinschmidt, Unifying R-symmetry in M-theory , in V. Sidoraviˇcius (ed.) NewTrends in Mathematical Physics, Proceedings of the XVth International Congress onMathematical Physics, Springer (2009). hep-th/0703262.[24] A. Kleinschmidt, H. Nicolai, and A. Vigan`o.
On Spinorial Representations of Invo-lutory Subalgebras of KacMoody Algebras , in Partition Functions and AutomorphicForms, pp. 179-215. Springer, Cham, (2020). arXiv:1811.11659[25] Y. Ne’eman,
Gravitational interaction of hadrons Band-spinor representations of GL(n,R) ,Proc.Natl.Acad.Sci.USAVol.74,No.10,pp.4157-4159.[26] K. Glennon and P. West,
Gravity, Dual Gravity and A1+++ , Int.J.Mod.Phys.A 35(2020) 14, 2050068, arXiv:2004.03363.[27] P. West,
E11, Brane Dynamics and Duality Symmetries , Int.J.Mod.Phys. A33 (2018)no.13, 1850080, arXiv:1801.00669;
A sketch of brane dynamics in seven and eight di-mension using E theory , Int.J.Mod.Phys. A33 (2018) no.32, 1850187, arXiv:1807.04176.[28] P. Ramond,
Boson-Fermion Confusion: The String Path To Supersymmetry , Nucl.Phys.Proc.Suppl.101 (2001) 45-53, arXiv:hep-th/0102012; P. Ramond,
Algebraic Dreams , Contribu-tion to Francqui Foundation Meeting in the honor of Marc Henneaux, October 2001,Brussels, arXiv:hep-th/0112261; L. Brink, P. Ramond and X. Xiong,