aa r X i v : . [ h e p - t h ] M a r ULB-TH/11-22November 13, 2018
Tensor hierarchies, Borcherds algebras and E Jakob Palmkvist ∗ Physique Th´eorique et Math´ematiqueUniversit´e Libre de Bruxelles & International Solvay InstitutesBoulevard du Triomphe, Campus Plaine, ULB-CP 231,BE-1050 Bruxelles, Belgium [email protected]
Abstract
Gauge deformations of maximal supergravity in D = 11 − n dimensions genericallygive rise to a tensor hierarchy of p -form fields that transform in specific representationsof the global symmetry group E n . We derive the formulas defining the hierarchy froma Borcherds superalgebra corresponding to E n . This explains why the E n representa-tions in the tensor hierarchies also appear in the level decomposition of the Borcherdssuperalgebra. We show that the indefinite Kac-Moody algebra E can be used equiv-alently to determine these representations, up to p = D , and for arbitrarily large p if E is replaced by E r with sufficiently large rank r . ∗ Also affiliated to: Department of Fundamental Physics, Chalmers University of Technology,SE-412 96 G¨oteborg, Sweden
Introduction
Eleven-dimensional supergravity is the low energy limit of M-theory, and leads bytoroidal reductions to maximal supergravity in lower dimensions. General gauge de-formations of the lower-dimensional theories have been systematically studied in recentyears as a way of exploring the M-theory degrees of freedom beyond supergravity [1,2].These studies have exhibited features that also appear in other approaches to M-theory,developed during the last decade, where possible symmetries based on Kac-Moody orBorcherds algebras have been investigated [3–5]. The present work is an attempt torelate the different approaches to each other via the features that they share.We will in this paper consider maximal supergravity in D spacetime dimensions,where 3 ≤ D ≤
7. This theory can be obtained by reduction of eleven-dimensionalsupergravity on an n -torus, where n = 11 − D , and it has a global symmetry group Gwith a corresponding Lie algebra g = E n [6, 7]. It contains a spectrum of dynamical p -form fields, which are antisymmetric tensors of rank p = 1 , , . . . , D − r p of g .One way to algebraically derive the representations r p from only g and r is toconsider all possible gauge deformations of the theory encoded by a so called embeddingtensor. As shown in [1, 2, 8–11] this leads to a tensor hierarchy of p -forms whichcontains the dynamical p -form spectum. Another way is to embed g into an infinite-dimensional Lie (super)algebra, either a Borcherds algebra (which depends on g ) orthe indefinite Kac-Moody algebra E . In the level decomposition of the Borcherdsalgebra with respect to g , the representation content on level p coincides with r p , up tolevel D − E if the level decomposition is done withrespect to g ⊕ sl D , and restricted to tensors that are antisymmetric under sl D [14–19].It is remarkable that the same representations show up in both the tensor hier-archy and the level decomposition, although the approaches are seemingly unrelated.Moreover, both the tensor hierarchy and the level decomposition can be continuedto p ≥ D − D − D -formsthat are possible to add to the theory. Also these predictions are the same in the twoapproaches, apart from two exceptions in D = 3.In this paper we will explain why the tensor hierarchy and the level decompositiongive the same result up to D -forms for 4 ≤ D ≤
7. The paper is organized as follows.In section 2 we review the tensor hierarchy and in section 3 the level decompositions.Section 3 is more mathematical and divided into two subsections, devoted to theBorcherds algebras and E , respectively. Our main result is presented in section 3.1,where we show that the tensors defining the hierarchy can be interpreted as elementsin the Borcherds algebra. In section 3.2 we show that the Borcherds algebras and E lead to the same p -form representations in the level decompositions up to p = D (whichhas been explained differently in [13]), and for arbitrarily large p if E is replaced by E r with sufficiently large rank r . We conclude the paper in section 4.2 The tensor hierarchy
In this section we will briefly review how the tensor hierarchy arises in the embeddingtensor formalism of gauged supergravity. We will follow [1] and refer to this paper(and the references therein) for more information.We start with the vector field in maximal supergravity in D dimensions, whichtransforms in a representation r of the global symmetry group G, or of the corre-sponding Lie algebra g . We write the vector field as A µ M , where the indices M areassociated to r and µ = 1 , , . . . , D are the spacetime indices. In gauged supergravity,a subgroup G of the global symmetry group G is promoted to a local symmetry group,with the vector field as the gauge field. Accordingly, we can write the generators ofthe gauge group G as X M , with an r index downstairs. However, the generators X M do not have to be independent, so the dimension of the gauge group G can besmaller than the dimension of r .We let α be the adjoint indices of G and we let t α be its generators. Since thegauge group G is a subgroup of G, the generators X M must be linear combinationsof t α and can be written X M = Θ M α t α . (2.1)The coefficients of the linear combinations form a tensor Θ M α which is called the embedding tensor since it describes how G is embedded into G.It follows from the index structure of the embedding tensor that it transforms inthe tensor product of ¯ r and the adjoint of g . This tensor product decomposes into adirect sum of irreducible representations, and supersymmetry restricts the embeddingtensor to only one or two of them. This restriction is known as the supersymmetry con-straint or representation constraint . The requirement that G close within G leads toa second constraint on the embedding tensor, which is known as the closure constraint or quadratic constraint and can be written[ X M , X N ] = − ( X M ) N P X P . (2.2)Thus ( X M ) N P serve as structure constants for the gauge group, but because of thepossible linear dependence in the set of generators, ( X M ) N P is in general not anti-symmetric. Only when we contract ( X M ) N P with another X P the symmetric partvanishes.When we gauge the theory we replace the partial derivatives with covariant ones, ∂ µ → D µ = ∂ µ − gA µ M X M , (2.3)where g is a coupling constant. Then the field strength of A µ M becomes F µν P = 2 ∂ [ µ A ν ] P + g ( X M ) N P A [ µ M A ν ] N (2.4)but as shown in [1] this expression is not fully covariant. The recipe presented there(following [8, 9]) for regaining covariance is to3i) add a term to the gauge transformation of A µ M with a parameter Λ µ MN : δA µ M → δA µ M + 2 g ( X ( M ) N ) P Λ µ MN , (2.5)(ii) add a term to the field strength of A µ M involving an new field A µν MN : F µν P → F µν P − g ( X ( M ) N ) P A µν MN , (2.6)(iii) define the appropriate gauge transformation of the new field A µν MN .The new field A µν MN carries two r indices and thus transforms under g in the tensorproduct r × r . It always occurs contracted with ( X ( M ) N ) P and therefore only thesymmetric part ( r × r ) + of this tensor product enters. Furthermore, the supersymme-try constraint on the embedding tensor restricts A µν MN to only one of the irreduciblerepresentations within the symmetric tensor product ( r × r ) + . This irreducible rep-resentation of g is what we call r .By introducing a two-form field A µν MN we solve the problem with the field strengthof A µ M , but on the other hand it leads to the same problem for the field strength of A µν MN — it is not fully covariant. We can solve the problem in the same way asbefore, by introducing yet another field, which will now be a three-form A µνρ MN P with three r indices, transforming in a representation r ⊂ ( r ) of g .The procedure that we have described can be continued until we reach the space-time dimension in the number of antisymmetric indices. This gives a theory that isautomatically consistent and gauge invariant. In the end we can set the coupling con-stant g to zero but still keep the fields and parameters that we have added, and thusobtain an alternative formulation of the ungauged theory.Each time we introduce a new ( p + 1)-form field A µ ··· µ p +1 M ···M p +1 and a parameterΛ µ ··· µ p N N ···N p +1 we add a term − gY M M ···M p N N ···N p +1 Λ µ ··· µ p N N ···N p +1 (2.7)to the gauge transformation of the previous p -form, and a term gY M M ···M p N N ···N p +1 A µ ··· µ p +1 N N ···N p +1 (2.8)to the field strength. The intertwiners Y M M ···M p N N ···N p +1 are defined recursivelyby the formula Y M M ···M p N N ···N p +1 = − δ N hM Y M ···M p iN ···N p +1 − ( X N ) N ···N p +1 hM M ···M p i (2.9)where the angle brackets denote projection on r p . The lower indices of the tensor Y M M ···M p N N ···N p +1 then define r p +1 ⊂ r p × r so that, by definition, Y M M ···M p N N ···N p +1 = Y M M ···M p hN N ···N p +1 i . (2.10)4Obviously we also have Y M M ···M p N N ···N p +1 = Y hM M ···M p iN N ···N p +1 .) The recur-sion formula (2.9) is valid for p ≥
2. For p = 1 we have Y P MN = − ( X M ) N P − ( X N ) MP (2.11)as we have already seen in (2.5) and (2.6). (In [1] the symmetric part of ( X M ) N P isdenoted by Z P MN . Thus our Y P MN is the same as − Z P MN in [1], which meansthat the two- and three-form and the corresponding parameters are also normalizeddifferently compared to [1].) The second term in (2.9) is the component of X N in thetensor product ( r ) p projected on r p . By the definition of a tensor product we have( X M ) N N ···N p P P ···P p = ( X M ) N P δ N P · · · δ N p P p + δ N P ( X M ) N P δ N P · · · δ N p P p · · · + δ N P · · · δ N p − P p − ( X M ) N p P p . (2.12)The formula (2.9) defines a sequence of representations r p for all positive integers p — also for p > D since no spacetime indices enter. The only input is g itself, r andthe representation constraint (which is needed to determine r ). Below we list g and r p for 3 ≤ D ≤ ≤ p ≤ D g sl (5 , R ) so (5 , E E E r
10 16 c
27 56 248r s
78 912 3875 +
10 45 351 133 +
24 144 s + +
40 10 + s + + + Although no spacetime indices enter in the formula (2.9), the table shows that therepresentations know about spacetime. The duality between p -forms and ( D − − p )-forms is reflected by the relation ¯ r p = r D − − p — the corresponding representations areconjugate to each other. Furthermore, r D − is always the adjoint adj of g , and the lasttwo representations in each column are related to the constraints of the embedding5ensor: ¯ r D − is the subrepresentation of adj × ¯ r in which the embedding tensormust transform according to the supersymmetry constraint (except for an additionalsinglet in the last column), and ¯ r D is the representation in which the closure constrainttransforms. (In the tables in [1,2,10], the entry corresponding to r for D = 3 containsthe additional singlet of the embedding tensor representation, although only isincluded in r . There is also an issue with r for D = 3 that we will discuss in theconclusion, section 4.)Except for the last column, the representation r p coincides with the content on level(minus) p in the level decomposition of a certain Borcherds algebra or the Kac-Moodyalgebra E with respect to g or g ⊕ sl D , respectively (in the E case restricted totensors which are antisymmetric in the sl D indices). Therefore a natural question iswhether it is possible to derive the formula (2.9) from the Borcherds algebra or E .As we will see in the next section, the answer is affirmative. We will in this more mathematical section study the Lie algebra g of the global sym-metry group G, as a special case of a finite Kac-Moody algebra, and show how itcan be extended to a Borcherds algebra or to the indefinite Kac-Moody algebra E .Borcherds algebras are also called Borcherds-Kac-Moody (BKM) algebras or generalizedKac-Moody algebras. They were first defined in [20] and generalized to superalgebrasin [21]. For simplicity we will in this paper use the term Borcherds algebras also forthe superalgebras, and the extension of g that we will discuss is in fact a superalgebra.We will not introduce more concepts than necessary, but refer to [22, 23] for a compre-hensive account of Borcherds and Kac-Moody algebras. (As noted in [13], footnote 8,there is an error in [23], but this is not important for the cases that we consider here.)As can be read off from the table above, g is the exceptional Lie algebra E , E , E for D = 3 , ,
5. We will now extend this notation and write g = E n , where n = 11 − D ,also for D = 6 and D = 7. In fact g is the split real form of the complex Lie algebra E n , which is usually denoted by E n ( n ) , but here we keep the simpler notation E n alsofor the split real form.We recall that E n , as a special case of a Kac-Moody algebra, can be constructedfrom its Dynkin diagram n − n − n − n − n ✐ ✐ ✐ ✐✐✐ ✐
6y associating three
Chevalley generators e i , f i , h i to each node ( i = 1 , , . . . , n ), satis-fying the Chevalley relations [ h i , e j ] = A ij e j , [ h i , f j ] = − A ij f j , [ e i , f j ] = δ ij h j , [ h i , h j ] = 0 , (3.1)where A is the Cartan matrix corresponding to the Dynkin diagram. Any off-diagonalentry A ij is − i and j are connected, and 0 if they are not. The diagonalentries are all equal to 2.We let ˜ E n be the Lie algebra generated by e i , f i , h i , and h its Cartan subalgebra,spanned by the Cartan elements h i . Among the ideals of ˜ E n that intersect h trivially,there is a maximal ideal, generated by the Serre relations (ad e i ) − A ij ( e j ) = (ad f i ) − A ij ( f j ) = 0 . (3.2)Factoring out this ideal from ˜ E n we obtain the Kac-Moody algebra E n .By adding nodes to the Dynkin diagram E n can be extended to a bigger algebra.We will in the next two subsections study two such extensions, where the extendedalgebra is infinite-dimensional. In the first case we add only one node, but we alsomodify the construction of the algebra and let the added node play a special role. Thisleads to a Borcherds algebra. In the second case we just extend the Dynkin diagramwith D = 11 − n more nodes, and accordingly we obtain the Kac-Moody algebra E . Following [4, 13] we indicate the special role of the added node in the construction ofthe Borcherds algebra by painting it black, whereas the other nodes are white. Welabel it by 0, so we have the following Dynkin diagram. n − n − n − n − n ✐ ✐ ✐ ✐✐② ✐ The black node plays a different role than the white ones in two respects. First,the corresponding diagonal entry in the Cartan matrix is zero, A = 0 (insteadof A = 2). Second, the corresponding generators e and f are not even (bosonic)elements in an ordinary Lie algebra, but odd (fermionic) elements in a Lie superalgebra .Thus we consider the Lie superalgebra ˜ U generated by 3( n + 1) elements e I , f I , h I ( I = 0 , , . . . , n ) all of which are even, except for e and f , which are odd. The7hevalley relations (3.1) still hold if we generalize the ordinary antisymmetric bracket[ x, y ] of any two elements x and y to a superbracket [[ x, y ]], which is antisymmetricif at least one of the elements is even, and symmetric if both elements are odd. Inthe first case we write [[ x, y ]] = [ x, y ] as usual, and in the second [[ x, y ]] = { x, y } . Thecommutation relations among e , f and h are thus[ h , e ] = [ h , f ] = 0 , { e , f } = h . (3.3)Like the Lie algebra ˜ E n above, which gives rise to the Kac-Moody algebra E n , also theLie superalgebra ˜ U has a maximal ideal that intersects the Cartan subalgebra trivially.This ideal is generated by the Serre relations(ad e i ) − A iJ ( e J ) = (ad f i ) − A iJ ( f J ) = 0 , (3.4)where now i = 1 , , . . . , n and J = 0 , , . . . , n [23, 24]. Factoring out this ideal weobtain a Borcherds (super)algebra that we here denote by U n +1 . With the notationin [13], we have U n +1 = V D .The black node gives rise to a Z -grading of U n +1 which is consistent with the Z -grading that U n +1 naturally is equipped with as a superalgebra. This means that itcan be written as a direct sum of subspaces ( U n +1 ) p for all integers p , such that[[( U n +1 ) p , ( U n +1 ) q ]] ⊆ ( U n +1 ) p + q , (3.5)where ( U n +1 ) p consists of odd elements if p is odd, and of even elements if p is even. Inthe grading associated to the black node, e belongs to ( U n +1 ) − , whereas f belongs to( U n +1 ) , and all other Chevalley generators belong to ( U n +1 ) . It follows that ( U n +1 ) ,as a vector space, is the direct sum of E n (with the Dynkin diagram obtained byremoving the black node) and a one-dimensional algebra spanned by h . As a basiselement of ( U n +1 ) , the Cartan element h does not commute with E n , but can bereplaced with another Cartan element h that does. This element is in the case n = 8( D = 3) given by c = h + 2 h + 3 h + 4 h + 5 h + 6 h + 4 h + 2 h + 3 h . (3.6)In the cases 4 ≤ n ≤ h is obtained by removing the (8 − n )leftmost terms on the right hand side of (3.6), and relabelling the nodes according tothe Dynkin diagram above.According to (3.5) any subspace ( U n +1 ) p closes under the adjoint action of ( U n +1 ) ,and in particular of E n . Thus it constitutes a representation of E n , which we call s p .Such a decomposition of (the adjoint action of) a graded Lie (super)algebra is usuallycalled a level decomposition , where p is the level of the representation s p . Determiningthe representations s p explicitly in the different cases, one finds that they coincide withthe representations r p in the table above for 1 ≤ p ≤ D , except for D = 3 (where one8n addition finds a singlet on level 2, and an adjoint E representation on level 3). Forany level p , the representation s − p is the conjugate of s p .Starting with the fact that r = s , we write the basis elements of ( U n +1 ) − and( U n +1 ) as E M and F M , respectively. For p ≥ U n +1 ) − p is spanned bythe elements E M ···M p ≡ [[ E M , [[ E M , . . . , [[ E M p − , E M p ]] · · · ]]]] (3.7)and ( U n +1 ) p by the elements F M ···M p ≡ [[ F M , [[ F M , . . . , [[ F M p − , F M p ]] · · · ]]]] . (3.8)For any irreducible representation s within the tensor product ( s ) p we now want toknow whether s is contained in s p or not. In other words, we want to know whetherthe expressions (3.7)–(3.8) vanish if we project them on s . The lemma below is usefulfor determining whether the projected expression is zero or not, but first we need tointroduce one more concept.In the Borcherds algebra U n +1 we introduce a bilinear form, which we write as h x | y i for two elements x and y , and define by h h i | h j i = A ij , h e i | f j i = δ ij , h e i | e j i = h f i | f j i = 0 . (3.9)The definition can then be extended to the full algebra U n +1 in a way such that thebilinear form is invariant and supersymmetric, h [[ x, y ]] | z i = h x | [[ y, z ]] i , h x | y i = h x | y i + ( − pq h y | x i , (3.10)where in the second equation x ∈ ( U n +1 ) p and y ∈ ( U n +1 ) q , and furthermore satisfies h ( U n +1 ) p | ( U n +1 ) q i = 0 whenever p + q = 0. For level ± h E M | F N i = δ MN .With this bilinear form at hand we are now ready for the lemma. Lemma 1.
Let x be an element in ( U n +1 ) − p for any p . Then x = 0 if and only if [[ x, y ]] = 0 for all y ∈ ( U n +1 ) p − . Proof.
If [[ x, y ]] = 0 for all y ∈ ( U n +1 ) p − , then also h [[ x, y ]] | z i = 0 for all z in ( U n +1 ) ,and by invariance of the inner product x must belong to the ideal in U n +1 consistingof elements u such that h u | U n +1 i = 0. This ideal intersects the Cartan subalgebratrivially, and by the construction of U n +1 it follows that x = 0. The other part of thelemma is trivial. (cid:3) The lemma says that we can as well study the expression [[ E N ···N p +1 , F P ···P p ]] insteadof E N ···N p +1 directly, in order to know if this is zero or not. We can then use theJacobi identity subsequently to replace [[ E N ···N p +1 , F P ···P p ]] by expressions that onlyinvolve lower (positive and negative) levels, until we are left with a (zero or nonzero)9inear combination of the basis elements E M of ( U n +1 ) − . The Jacobi identity for theLie superalgebra U n +1 can be written[[[[ x, y ]] , z ]] = [[ x, [[ y, z ]]]] − ( − pq [[ y, [[ x, z ]]]] (3.11)where x ∈ ( U n +1 ) p and y ∈ ( U n +1 ) q . Applying it subsequently to expressions of theform [[ E N ···N p +1 , F P ···P p ]] is of course a tedious task, but the theorem below givesa number of identities which simplify it, although they only hold under a certaincondition.For any p ≥ s p as ( P p ) M ···M p N ···N p , withthe indicies such that( P p ) M ···M p N ···N p = ( P p ) M hM ···M p iN hN ···N p i , (3.12)where the angle brackets denote projection on s p − . Thus we have for example E M ···M p = ( P p ) M ···M p N ···N p E N ···N p = E hM ···M p i ,F N ···N p = ( P p ) M ···M p N ···N p F M ···M p = F hN ···N p i . (3.13) Theorem 2.
Let p ≥ be an integer. If there are real numbers a k such that h E M ···M k | F N ···N k i = a k ( P k ) M ···M k N ···N k (3.14) for k = 2 , , . . . , p , then the following identities hold: [[ E M , F N ···N p ]] = ( − p a p a p − δ hMhN F N ···N p i , (3.15)[[ E M ···M p , F N ···N p ]] = a p F hN δ M N · · · δ M p N p i , (3.16)[[ E M ···M p , F N ···N p ]] = a p (cid:16) { E hM , F N } δ M N · · · δ M p iN p + δ hM N { E M , F N } δ M N · · · δ M p iN p · · · + δ hM N · · · δ M p − N p − { E M p i , F N p } (cid:17) . (3.17)Note that if s k is irreducible, then there must be such a number a k , but not necessarilyif s k is a direct sum of irreducible representations, since the corresponding projectorscan come with different coefficients. Proof.
By the assumptions we have F N ···N p = 1 a p h E M ···M p | F N ···N p i F M ···M p , (3.18)10nd by linearity we get x = 1 a p h E N ···N p | x i F N ···N p (3.19)for any x ∈ ( U n +1 ) p . In particular[[ E M , F N ···N p ]] = 1 a p − h E M ···M p | [[ E M , F N ···N p ]] i F M ···M p = ( − p a p − h E MM ···M p | F N ···N p i F M ···M p = ( − p a p a p − ( P p ) MM ···M p N ···N p F M ···M p = ( − p a p a p − δ MhN F N ···N p i , (3.20)which gives (3.15), and[[ E M ···M p , F N ···N p ]] = h E M | [[ E M ···M p , F N ···N p ]] i F M = h E M ···M p | F N ···N p i F M = a p ( P p ) M ···M p N ···N p F M = a p F hN δ M N · · · δ M p N p i , (3.21)which gives (3.16). We can now prove the identity (3.17) by induction. It is triviallytrue already for p = 1. Suppose that it holds when p = q −
1, for some q ≥
2. Thenusing (3.15) and (3.16) we obtain[[ E M ···M q , F N ···N q ]] = ( − q [[ E M ···M q , [[ E M , F N ···N q ]]]]+ [[ E M , [[ E M ···M q , F N ···N q ]]]]= a p a p − δ M hN [[ E M ···M p , F N ···N q i ]]+ a q a { E M , F hN } δ M N · · · δ M q N q i = a q a (cid:16) { E M , F hN } δ M N · · · δ M q N q i + δ M hN { E M , F N } δ M N · · · δ M q N q i · · · + δ M hN · · · δ M q − N q − { E M q , F N q i } (cid:17) , (3.22)which gives (3.17) for all p ≥ (cid:3) We have now arrived at the main result of this paper.11 orollary 3.
For any integer k ≥ , set Y N ···N k M ···M k +1 = 1 a k [ F N ···N k , E M ···M k +1 ] , X M|N ···N k P P ···P k = 1 a k [ E M , [[ E N N ···N k , F P P ···P k ]]] . (3.23) Then, for p ≥ and with the conditions in the theorem, we have X M|N ···N p P P ···P p = X M|N hP δ N P · · · δ N p P p i + δ N hP X M|N P δ N P · · · δ N p P p i · · · + δ N hP · · · δ N p − P p − X M|N p P p i , (3.24) Y N ···N p M ···M p +1 = − δ M hN Y N ···N p iM ···M p +1 − X M |M ···M p +1 N N ···N p , (3.25) Y MN P = − X N |P M − X P|N M . (3.26) Proof.
The identity (3.24) follows directly from (3.17), whereas the Jacobi identitygives (3.26) and a p Y N ···N p M ···M p +1 = [ F N ···N p , E M ···M p +1 ]= − [ E M , [[ E M ···M p +1 , F N ···N p ]]] − ( − p +1 [ E M ···M p +1 , [[ E M , F N ···N p ]]]= − [ E M , [[ E M ···M p +1 , F N ···N p ]]] − a p a p − δ M hN [ F N ···N p i , E M ···M p +1 ]= − a p X M |M ···M p +1 N N ···N p − a p δ M hN Y N ···N p iM ···M p +1 , (3.27)using (3.15) and (3.24). (cid:3) The relations (3.24)–(3.26) are nothing but the definition (2.12), the recursion formula(2.9) and the initial condition (2.11), with X and Y replaced by X and Y . We havethus derived these formulas from the Borcherds algebra U n +1 .It follows that if r = s , then r k ⊆ s k for k = 1 , , . . . , p +1 as long as the conditionin the theorem is satisfied. The condition r = s must be inserted by hand, since the12eneral formulas (2.11) and (3.26) are not enough to determine r and s if we do notknow what ( X M ) N P and X M|N P are. We thus have to insert the definitions( X M ) N P = Θ M α ( t α ) N P , X M|N P = [ E M , { E N , F P } ] (3.28)in (2.11) and (3.26) to see which representations r and s are. A priori they can beany parts of the symmetric tensor products ( r × r ) + and ( s × s ) + . The correctrepresentation is then singled out ultimately by the Serre relation { e , [ e , e ] } = 0 (3.29)on the Borcherds side, and by the supersymmetry constraint on the tensor hierarchyside (the tensor product ¯ s × s must have a nonzero overlap with the representationto which the embedding tensor belongs). Remarkably, the Serre relation (3.29) andthe supersymmetry constraint give the same result, so that s = r , in all cases exceptfor n = 8 ( D = 3), where we have s = r + .For p ≥ s p from the formula (3.25) without knowing what X M|N P is — the Serre relation (3.29) is automatically taken into account. However,on the tensor hierarchy side we cannot a priori exclude the possibility that the super-symmetry constraint removes some part of s p (again by requiring that ¯ s p × s p − havea nonzero overlap with the representation to which the embedding tensor belongs).Thus we can a priori only conclude r k ⊆ s k for k = 1 , , . . . , p + 1, but in fact wehave r p = s p for p = 1 , , . . . , D , which shows that the Serre relation (3.29) is reallyequivalent to the supersymmetry constraint in this sense.For D = 7 the representation s is reducible, and therefore the condition in thetheorem is not automatically satisfied. It would still be satisfied if the projectors corre-sponding to the irreducible representations and came with the same prefactor,but a computation of the inner product on level ± U n +1 shows that the relative prefactors are 3 and 4. Nevertheless, we have r = s , whichmeans that the condition is sufficient but not necessary.We end this subsection by showing how the representations s p can be determinedin the general case, when the condition in the theorem is not satisfied. It is convenientto introduce the notation f N ···N p P ···P p = h E N ···N p | F P ···P p i = ( − p +1 h [[ E N ···N p , F P ···P p − ]] | F P i , (3.30)since, according to the lemma, the lower indices in (3.30) determine s p . We also write f MN P Q = h [ { E M , F N } , E P ] | F Q i (3.31)and note that ( U n +1 ) − can be considered as a triple system with (3.31) as structureconstants for the triple product,( E M , E N , E P ) [ { E M , σ ( E N ) } , E P ] = [ { E M , F N } , E P ] = f MN P Q E Q , (3.32)13here σ is the superinvolution given by σ ( E M ) = F M and σ ( F M ) = − E M .As an aside, we mention that ( U n +1 ) − with the triple product (3.32) is a generalizedJordan triple system , like the three-algebras considered in for example [25–27]. Howeverthe triple system ( U n +1 ) − is not an N = 6 three-algebra since the triple product isnot antisymmetric, f MN P Q = − f P N MQ (3.33)(and not an N = 5 three-algebra either since the triple product does not satisfythe generalized antisymmetry condition in [28, 29]). Nevertheless, the constructionof an associated Lie superalgebra from any N = 6 three-algebra [30, 31] can also beapplied to the triple system ( U n +1 ) − , and gives then back the full Borcherds algebra U n +1 . The non-antisymmetry of the triple product is reflected by the fact that thisLie superalgebra is not 3-graded but decomposed into infinitely many subspaces in the Z -grading.By applying the Jacobi identity repeatedly, we now obtain[[ F N , E M ···M p ]] = [[ { F N , E M } , E M ···M p ]] − [[ E M , [[ F N , E M ···M p ]]]]= [[ { F N , E M } , E M ···M p ]] − [[ E M , [[ { F N , E M } , E M ···M p ]]]]+ [[ E M , [[ E M , [[ F N , E M ···M p ]]]]]]= [[ { F N , E M } , E M ···M p ]] − [[ E M , [[ { F N , E M } , E M ···M p ]]]]+ [[ E M , [[ E M , [[ { F N , E M } , E M ···M p ]]]]]] · · · + ( − p +1 [[ E M , [[ E M , . . . , [[ E M p − , { F N , E M p } ]] · · · ]]]]= p − X i =1 p X j = i +1 ( − i +1 f M i N M j P E M ···M i − M i +1 ···M j − PM j +1 ···M p + ( − p f M p N M p − P E M ···M p − P (3.34)and then, using the invariance of the inner product, f M ···M p N ···N p = h E M ···M p | F N ···N p i = ( − p +1 h [[ F N , E M ···M p ]] | F N ···N p i = p − X i =1 p X j = i +1 ( − i + p f M i N M j P f M ···M i − M i +1 ···M j − PM j +1 ···M p N ···N p − f M p N M p − P f M ···M p − P N ···N p . (3.35)14 .2 The E approach We will in this subsection go from the Borcherds algebra U n +1 to the Kac-Moodyalgebra E in two steps: first replace the black node with an ordinary (white) one(thereby going from U n +1 to E n +1 ) and then add another 10 − n nodes, each oneconnected to the previous one with a single line (thereby going from E n +1 to E ).First we give the commutation relations for level 0 and ± U − n that we discussed in the previous subsection. They are { E M , F N } = ( t α ) MN t α + 19 − n δ MN h, [ t α , t β ] = f αβγ t γ , [ t α , h ] = 0 , [ t α , E M ] = ( t α ) MN E N , [ h, E M ] = − (10 − n ) E M , [ t α , F N ] = − ( t α ) MN F M , [ h, F N ] = (10 − n ) F N . (3.36)As before, t α are the basis elements of g = E n , and f αβ γ are the corresponding structureconstants. The adjoint E n indices are raised with the inverse of the Killing form in E n , which coincides with the restriction of the bilinear form in U n +1 . Thus we have h t α | t β i = δ αβ .Inserting (3.36) into (3.31) we get an expression for the structure constants of thetriple system, f MN P Q = ( t α ) MN ( t α ) P Q − − n − n δ MN δ P Q . (3.37)When we replace the black node with a white one we get E n +1 , which is not a(proper) Lie superalgebra but an ordinary Lie algebra. But still the added node givesrise to a Z -grading and a level decomposition with respect to E n , where we find r and ¯ r on level − E n +1 ) − and ( E n +1 ) as E M and F M , respectively. As for the Borcherdsalgebra we can consider the level − E n +1 ) − with the triple product( E M , E N , E P ) [[ E M , τ ( E N )] , E P ] = [[ E M , F N ] , E P ] = g MN P Q E Q , (3.38)where τ is minus the Chevalley involution, given by τ ( E M ) = F M and τ ( F M ) = E M .The commutation relations for level 0 and ± E M , F N ] = ( t α ) MN t α + 19 − n δ MN h, [ t α , t β ] = f αβγ t γ , [ t α , h ] = 0 , [ t α , E M ] = ( t α ) MN E N , [ h, E M ] = (8 − n ) E M , [ t α , F N ] = − ( t α ) MN F M , [ h, F N ] = − (8 − n ) F N . (3.39)15hus the only differences compared to the Borcherds case (3.36) are the eigenvaluesof h acting on E M and F N (and of course that we now have a commutator insteadof an anticommutator of E M and F N ). It follows that the structure constants of thetriple product (3.38) are g MN P Q = ( t α ) MN ( t α ) P Q + 8 − n − n δ MN δ P Q . (3.40)Finally we consider E with the following Dynkin diagram. D − D D +1 7 8 9 1011 ✐ ✐ ✐ ✐✐✐✐ ✐ ✐
The node that we added to the Dynkin diagram of E n in the construction of E n +1 isnow labelled D , and on its left hand side we have added D − A D − = sl ( D, R ) = sl D . This means that the node D gives rise to a grading of E where the subalgebra ( E ) is the direct sum of E n ,a one-dimensional subalgebra spanned by h , and sl D . It follows that any subspace( E ) p in the grading constitutes a representation of both E n and sl D . On level ± r and ¯ r as before, but now together with the fundamental and antifundamentalrepresentations of sl D . Thus we can write the basis elements of ( E ) − and ( E ) as E M a and F M a , respectively, where a = 1 , . . . , D . For p ≥
2, the subspace ( E ) − p isthen spanned by the elements E M ···M p a ··· a p = [[ E M a , [[ E M a , . . . , [[ E M p − a p − , E M p a p ]] · · · ]]]] , (3.41)and ( E ) p by the elements F M ···M p a ··· a p = [[ F M a , [[ F M a , . . . , [[ F M p − a p − , F M p a p ]] · · · ]]]] . (3.42)If we on any level p = 1 , , . . . , D antisymmetrize the sl D indices, we find that the E n representation is the same as s p in the corresponding level decomposition of U n +1 . Aswe saw in the preceding subsection, this representation s p in turn coincides with r p inthe tensor hierarchy for 4 ≤ D ≤
7. In this way the spectrum of p -forms that appearsin gauged supergravity can be derived from E . It can also be derived from E inthe same way if we neglect the D -forms, and from E if we neglect the ( D − E r for r >
11, and by choosing r sufficiently large for each p we get an infinite sequence ofrepresentations t p from E r , where p can be any positive integer. This sequence canthen be compared with the infinite sequences s p and r p coming from the Borcherds16lgebra and the tensor hierarchy, respectively. We end this section by showing that s p = t p for all p (and sufficiently large r ), and thus explaining why the Borcherds and E approaches give the same result. In [13] this has been explained in a different way,by showing that a parabolic subalgebra of E (via the tensor product with the exterioralgebra, and the restriction to invariant elements) gives back a parabolic subalgebraof the Borcherds algebra U n +1 . The idea to consider the infinite rank extension of E was also presented in [13].Again we consider the subspace at level − E ) − with the triple product( E M a , E N b , E P c ) [[ E M a , F N b ] , E P c ] = h M a N b P c Q d E Q . (3.43)As shown in [32] there is a simple formula that relates this triple product with the one(3.38) in ( E n +1 ) − above. In terms of the structure constants it reads h M a N b P c Q d = g MN P Q δ ab δ cd − δ MN δ P Q δ ab δ cd + δ MN δ P Q δ cb δ ad . (3.44)Note that the number 11 does not enter here — it is only the range of the indices a, b, . . . = 1 , , . . . , D + r −
11 that changes when we replace E by E r for some other r ≥ − D . Inserting (3.40) into (3.44) yields h M a N b P c Q d = ( t α ) MN ( t α ) P Q δ ab δ cd + δ MN δ P Q (cid:18)(cid:16) − n − n − (cid:17) δ ab δ cd + δ cb δ ad (cid:19) , (3.45)and if we antisymmetrize in a and c we obtain h M [ a |N | b P c ] Q d = ( t α ) MN ( t α ) P Q δ [ ab δ c ] d + δ MN δ P Q (cid:16) − n − n − (cid:17) δ [ ab δ c ] d = δ [ ab δ c ] d (cid:16) ( t α ) MN ( t α ) P Q − − n − n δ MN δ P Q (cid:17) = δ [ ab δ c ] d f MN P Q . (3.46)Thus we get back the structure constants (3.37) for the triple system based on theBorcherds algebra U n +1 , times δ [ ab δ c ] d . As we will see next, this relation between thetwo triple systems, based on the Borcherds algebra and E , respectively, can be viewedas the reason why the two algebras give rise to the same sequence of representations.As for the Borcherds algebra, we can define a bilinear form in E (or E r ) by therelations (3.9). This bilinear form is still invariant, h [ x, y ] | z i = h x | [ y, z ] i , but thistime completely symmetric, since E (or E r ) is an ordinary Lie algebra. By the samearguments as before, it is the lower r indices in h M a ······M p a p N b ······N p b p = h E M ···M p a ··· a p | F N ···N p b ··· b p i (3.47)17hat determine the representation t p of E n on level p . By applying the Jacobi identityrepeatedly, we obtain an expression for (3.47) analogous to (3.35). The differencecompared to (3.35) is that f is replaced by h , that each r index is accompanied byan sl D index and, most important, that the prefactors ( − i + p and − −
1, respectively. However, if we antisymmetrizethe sl D indices a , a , . . . , a p accompanying M , M , . . . , M p in (3.35) and use (3.46),then we can eliminate the sl D summation index accompanying P . If we furthermorerearrange the sl D indices in the order a , a , . . . , a p , then we pick up a factor of ( − i − on the first line and a factor of ( − p − on the second. After the antisymmetrizationand rearrangement, the only sign difference compared to (3.35) is thus an overall factorof ( − p +1 . It is then easy to show, by induction over p , that h M [ a ······M p a p ] N b ······N p b p = ( − p ( p − / δ [ a ··· a p ] b ··· b p f M ···M p N ···N p . (3.48)This implies that s p = t p for 1 ≤ p ≤ D + r −
11 and all r ≥ − D . We have in this paper considered maximal supergravity in D spacetime dimensions,where 3 ≤ D ≤
7. We have studied three sequences of representations of E n , theLie algebra of the global symmetry group, which we have denoted by r p , s p and t p ,where p can be any positive integer. The first one, r p , comes from the tensor hierarchythat arises when we gauge the supergravity theory, whereas s p and t p come fromlevel decompositions of the Borcherds algebra U n +1 and the Kac-Moody algebra E ,respectively. It was known already that r p = s p = t p for 1 ≤ p ≤ D −
2, and that thisgives the spectrum of dynamical p -forms in maximal supergravity. It was also knownthat r p = s p = t p for p = D − p = D , apart from two exceptions in the D = 3case. Thus the tensor hierarchy, U n +1 and E give the same predictions about non-dynamical p -forms that are possible to add to the theory. This agreement has beenconsidered somewhat mysterious, since neither U n +1 or E appears in the constructionof the tensor hierarchy. On the other hand, it is in line with the ideas of gauging as aprobe of M-theory degrees of freedom [1, 2], and of Borcherds or Kac-Moody algebrasas symmetries in M-theory [3–5].In this work we have removed much of the mystery by deriving the formulas definingthe tensor hierarchy from U n +1 . But it is still remarkable that the supersymmetryconstraint on the embedding tensor and the Serre relation (3.29) on the Borcherdsside restrict the symmetric tensor products ( r × r ) + and ( s × s ) + equally much,so that r = s , in all cases except for D = 3. It would be interesting to find anexplanation also for this fact.Once r = s , our results imply that r p ⊆ s p for all p ≥ U ± ( p − satisfy a certain condition. Namely, their inner product must be proportional18o the projector corresponding to s p − . This condition is automatically satisfied if s p − is irreducible, which is the case for 2 ≤ p ≤ D , except for D = 7, where s is a directsum of two irreducible representations. We have checked that the condition is notsatisfied in this case, but nevertheless, r = s . Thus the condition is sufficient butnot necessary for agreement. We do not know if r p = s p for 4 ≤ D ≤ p ≥ D = 3 is qualitatively different from 4 ≤ D ≤
7. In the end of section 2we mentioned that the embedding tensor, due to the supersymmetry constraint, trans-forms in the representation r D − in all cases except for D = 3, where the embeddingtensor representation contains an extra singlet missing in r D − = r . In section 3.1 wenoted that s = r in all cases except for D = 3, where we have s = r + . The con-clusion is that s D − is the representation in which the embedding tensor transforms,in all cases, even D = 3. Since s = r for D = 3 we cannot use the theorem and thecorollary to draw any conclusions about the relation between s and r . An explicitcomputation shows that s = r + , again a difference compared to 4 ≤ D ≤ s = r . It would be interesting to further investigate the role of theextra representations in s and s for D = 3, but we leave this for future work.The correspondence that we have derived between the components of the gaugegroup generators X M and the intertwiners Y on the one hand side, and the X and Y elements in the subspace ( U n +1 ) − of the Borcherds algebra on the other, is notfully satisfactory. For example, the intertwiners in the tensor hierarchy satisfy anorthogonality relation [1] Y P ···P p N ···N p +1 Y N ···N p +1 M ···M p +2 = 0 (4.1)but since the corresponding Y elements are no numbers we do not know how to multiplythem with each other. Only expressions that are linear in X and Y can be translatedinto corresponding expressions with X and Y . It is also not clear how to interpret the p -form fields themselves, the field strengths and their gauge transformations in termsof the Borcherds algebra.Concerning the representations t p coming from E we have established that s p = t p not only for 1 ≤ p ≤ D , but also for arbitrarily large p if we replace E by E r , where r is sufficiently large. An advantage of using E is that the antisymmetric sl D indicescan be naturally interpreted as the spacetime indices of the p -forms (if one restricts tospatial indices E can be used in the same way). Another advantage is that the samealgebra can be used for any D , we just decompose it differently, whereas the Borcherdsalgebra U n +1 depends on n = 11 − D . On the other hand E grows much faster withthe levels than U n +1 , and only a tiny subset of all the tensors are antisymmetric in the sl D indices. As long as only these tensors are of interest it is more economical to usethe Borcherds algebra. 19t would be interesting to in some sense extend our results to higher (and lower)dimensions. In D = 8 we do not expect anything special to happen, but in D = 9it has recently been found in that a 9-form predicted by E is not detected by thetensor hierarchy [33]. In D = 10 (type IIA and IIB) the embedding tensor formalismleading to a tensor hierarchy is not applicable, but the spectrum of p -form fields thatcan be introduced consistently with supersymmetry has been shown to agree with thepredictions from the corresponding Borcherds algebras and E [34–37]. Acknowledgments
I would like to thank Martin Cederwall, Marc Henneaux, Hermann Nicolai, BengtE.W. Nilsson, Teake Nutma, Daniel Persson, Henning Samtleben, and especially AxelKleinschmidt for discussions and correspondence. The work is supported by IISN –Belgium (conventions 4.4511.06 and 4.4514.08), by the Belgian Federal Science PolicyOffice through the Interuniversity Attraction Pole P6/11.
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