JHEP03(2019)160 Springer March 15, 2019 March 26, 2019 October 6, 2018 : : : is a free Lie super- α , Q Received Accepted Published d Published for SISSA by https://doi.org/10.1007/JHEP03(2019)160 and Jakob Palmkvist b, [email protected] , . 3 Axel Kleinschmidt 1809.09171 a The Authors. Global , M-Theory, Space-Time Symmetries, Supersymmetric c

We study systematically various extensions of the Poincar´esuperalgebra. The , [email protected] SE-412 96 Gothenburg, Sweden E-mail: [email protected] Max-Planck-Institut f¨urGravitationsphysik (Albert-Einstein-Institut), Am 1, M¨uhlenberg DE-14476 Potsdam, Germany International Solvay Institutes, ULB-Campus Plaine CP231, BE-1050 Brussels,Division Belgium for , Department of Physics, Chalmers University of Technology, Departament i de and Qu`antica Astrof´ısica F´ısica Institut deUniversitat Ci`enciesdel Cosmos de (ICCUB), Barcelona, Mart´ıi Franqu`es,ES-08028 Barcelona, Spain b c d a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: that we discuss insuperalgebra detail. gives rise We explain to how manypreviously this other in universal various extension as places of quotients,can in the some be the Poincar´e of literature. very which neatly havesome In appeared related new particular, to angles we on Borcherds show exotic . how some and The quotients extended ideas put forward in also M-theory. offer Abstract: most general starting from a set of spinorial supercharges Joaquim Gomis, Symmetries of M-theory and free Lie superalgebras JHEP03(2019)160 (1.1) 10 18 3 15 ] and the study of such deformations , 3 – 12 12 is associated with the constant electro- ab 1 ] Z 3 ab [ 24 11 Z ] = 26 13 b 8 19 8 – 1 – = ,P a representation? ab P 1 [ 8 13 Z ` 28 2 and 6 = 2 supersymmetry > = 4 11 E N N D = 1 supersymmetry = 1 supersymmetry = 1 supersymmetry = 1 supersymmetry N N N N = 4 and = 4 and 1 27 D D in the global symmetry Poincar´espace-time algebra commute among them- a ] = 0. This is no longer true when one considers translations in the presence ]. The most general deformation one can allow for (based on b P 6 = 11 and = 4 and = 4 and extended supersymmetry = 11 and = 4 and = 4 and extended supersymmetry – ,P 3 a D D 3.3.1 Free3.3.2 Lie Relation to D 3.1.2 ComparisonD to Maxwell superalgebra in3.2.1 the literature 3.2.2 D D 3.1.1 Free Lie superalgebra P 4.2 4.3 4.1 3.2 3.3 2.1 Maxwell algebra2.2 and free Lie Free Lie algebras superalgebras 3.1 cohomology) is where the anti-symmetric generator magnetic background field and transforms as a under Lorentz transformations. In Translations selves: [ of a constant electro-magneticof background field the [ abelianalgebras algebra [ of translations has led to the investigation of so-called Maxwell 1 Introduction 5 Conclusion A Gamma matrices in 4 Free Lie superalgebras and Borcherds superalgebras 3 Instances of free Lie superalgebras Contents 1 Introduction 2 Review of Maxwell and free Lie (super)algebras JHEP03(2019)160 ]. ]. 6 13 . Any a P and subse- } ] and we shall β 12 ,Q ] of some of these α 10 Q one in this way has { a ], the anti-commutator x ], a dynamical realisation 9 , 11 8 and iterate the procedure [ but also additional non-central a ] identified quotients that could a 7 P P with ab and so on. The dynamical model based Z with a coordinate ab there, generated by the translations a ] (called ‘M-algebra’ there) where non-trivial Z P ∞ 10 – 2 – for ]. In particular [ 7 ab θ in [ has the most general generators in ∞ α Q -extended versions can be obtained from a free Lie superalgebra. an appropriate spinor index). The infinite-dimensional free Lie N α algebra describes the dynamical object together with the background ] that the most general structure one obtains in this way is the infinite- 7 Lie algebra, called Maxwell ∞ (with α . Associating the translations free Q ∞ should yield not only the translation generators } β ] as it arises as a particular finite-dimensional quotient. In [ Our paper focusses almost exclusively on the algebraic aspects unifyingAn the interesting observation, various that also opens up connections to other algebraic structures Studying the free Lie superalgebra in four space-time dimensions we are also able to In the present article, we shall extend the picture of the free Lie algebra generalisation ,Q 11 α Q algebras in terms of freerealising Lie these superalgebras. symmetries but We it shall should be not possible consider to particle construct or these alonginvestigated the models in lines the of [ literature in connection with potential symmetries of M-theory, is that in [ of the supersymmetricwhere Maxwell again algebra additional was coordinates foundinterpretations are in in introduced terms for of terms a the ofMaxwell Maxwell new algebra superfield a for generators background. extended massless and Similar supersymmetry superparticle thesealso generalisations have of also have exhibit the been how investigated these [ interpret the infinitely many newcuss generators other in the possible free connections Lieand to superalgebra. extended proposals space-times. We of will also infinite-dimensional dis- M-theory symmetries make contact with the supersymmetric extension of the Maxwell algebra that was studied superalgebra as we willpersymmetry show algebra explicitly. were also Further studied extensionsmulti-commutators in of arise. [ the As eleven-dimensionalnatural we su- home shall in the describe free ingenerators Lie more superalgebra. as detail superpartners The interpretation of below, given the these in [ bosonic again have charges a might give some idea of how to quent multi-(anti-)commutators. Again, it admitsdiscuss. many non-trivial quotients Of that we particularsupersymmetry shall interest algebra. to us As{ has will been be proved quotients forterms related that example to serve as in the sources [ for eleven-dimensional branes. These additional terms arise naturally in the free Lie of the Poincar´ealgebra to agebra. free The Lie basic superalgebra buildinggenerators blocks generalisation in of this the construction Poincar´esuperal- superalgebra are generated the by (global) the odd supersymmetry field in an ‘unfolded’ treatment ofintroducing the additional electro-magnetic coordinates field strength. forof This the was additional Maxwell achieved generators by contained inan the extended quotient space withon coordinates the Maxwell it moves in. It was shown in [ dimensional free Lie algebra admits manyof Lie which algebra were ideals studied and for associatedbe Maxwell quotient used Lie in algebras, a some particle model to describe motion in a general, non-constant, electro-magnetic a next step one can analyse the commutator of JHEP03(2019)160 ] 19 = 4 – (2.2) (2.1c) (2.1a) (2.1b) D 17 and the ab space-time contains a M D 4 ). The fundamental , we then discuss in ba ], Borcherds [ 3 M 16 ] and then to free Lie – 7 − 14 ab M , ( 1 2 c ] a = M ] b [ form a vector representation of the ab d [ a η M 2 P 1) of the anti-symmetric − ab − d , Z ] ] a . We use the mostly plus convention for the flat a ,D } M P 1 ] = b b (1 b [ [ c c − – 3 – . η η so ,P a P [ ] = 2 ] = 2 ] = 0 ) is the standard property of the Poincar´ealgebra that ,...,D b c cd 0 ]. This is due to the definition of these algebras as being = 11. We also explain how our construction relates to ,P ,P . a 2.1c 23 ∈ { ,M ab P – D 4 [ ab M [ 20 a, b M [ ] based on Lie algebra cohomology, the most general deformation 6 1 contains some concluding remarks and speculations. ) is 5 As shown in [ The paper is structured as follows. We first review and motivate the free Lie algebra 2.1c Our anti-symmetrisation convention is of strength one: 1 Lorentz indices lieMinkowski in metric. the range Lorentz algebra. The third linethe ( translations commute.and This its is free the Lie algebra relation generalisation. that is deformedof in ( the Maxwell algebra The first line issecond just line the expresses that Lorentz the algebra translation generators 2.1 Maxwell algebra andThe starting free point Lie for all algebras bosonicdimensions, constructions here is the in Poincar´ealgebra 2 Review of Maxwell andIn free this section, Lie we (super)algebras reviewin some order basic to aspects motivate ofsuperalgebras. our the generalisations conventional bosonic first Maxwell to algebra free Lie algebras [ discussion of cases where quotients ofBorcherds the superalgebras and free how Lie this superalgebra in can turnSection be can be -expressed recast through in terms of Kac-Moody algebras. construction in relation tosuperalgebra the that is bosonic the Maxwell maindetail object algebra various of and examples study in that then this areand introduce paper. related simple the In to supersymmetry free section simple in Lie andvarious extended algebras supersymmetry in in the literature and, potentially, to exotic branes. Section or tensor hierarchy algebras [ constructed as quotients of freeof algebras. these conjectured A algebraic noteworthygenerators structures difference that in is, are however, M-theory not that present alsois in many have our discussed ‘negative free in level’ Lie detail symmetry superalgebra in constructions. section This connection many quotients of the free Lie superalgebra embed in Kac-Moody [ JHEP03(2019)160 ) a P P ( ) or f (2.3) D ( gl ]. each level ) does not 6 . Choosing with trivial a – in a light-cone 0 D 3 } P ( g ij ab gl free Lie algebra M ,Z ]. We will refer to ) which is defined a 7 and P i ) consists of Lorentz . As a representation ,P ( + f P ab ] acting on ( M ba,c f , ). It has a natural graded 7 M 0 Y , also transform under the − { g P 6 = 1, to their commutator − + ( ` f ` M . = = 0. The comma notation for ) (2.4) ] P ; the total of generators ( 0 ab,c f ab,c ab,c g Y [ Y ⊕ Y = 2 ) at higher levels. As . 1) ] =: ,c will then act on all the positive levels as D ] c ( − 0 ab [ ,P gl g = 3 then is irreducible while it is reducible Y ,D ab ` . This semi-direct sum is the infinite general- Z (1 as positive level at ab as transforming in the fundamental of – 4 – and we denote it by so a a M a appears in a multi-commutator. In this grading, 0 of the free Lie algebra ] = [ P = , but that the transformation property of the ab,c P c P a a 2) algebra with generators Y = 1 transform under the Lorentz algebra (as vectors) ∞ P ]. The algebra spanned by P ,P has the property − ` b ] = 0 algebra ` > -dimensional representation. This representation need b D ` = 0. . For yet different choices of ,P 3 ] = 0 is the Maxwell algebra introduced in [ D ab,c . ` a ,P = 2 and so on. . This is also the commutator in the free Lie algebra based 2 0 Y ] a cd P ` g relate to non-relativistic systems with a Galiliean symmetry or very P ab [ 0 ,Z [[ Maxwell on level g Z ab a with a generators Z = P 0 ab D g ), one obtains of 1) as level Z ] = [ ]. . D ( − bc 25 1) Lorentz generators of the Poincar´eand Maxwell algebra introduced in [ gl ab,c ,Z ] where only the Sim( Y 1), see footnote − ,D a ∞ = bc 24 P η − (1 = 0 is not something that is fixed by the free Lie algebra construction. In 0 ,D ] as positive level g is kept [ b ` so (1 } ,D by giving a name to [ i ,P so (1 a a , x P P  so x { At this point it is important to remark that the construction of a free Lie algebra The basic generators This construction can be iterated to obtain all possible multiple commutators of the From the Lie algebra cohomology perspective, it is possible to deform also these trivial = [ We note that this is notOther an interesting irreducible choices representation of of the Lorentz algebra as one can define the up to the Jacobi identity (and anti-symmetry). This is by definition the 2 3 arranges differently into irreducible representations of ab a independently of the choice of non-trivial trace special relativity [ basis for instance have an invariant trace, the generator under ` on each level of course does not change as this is determined by under level fact, we could also viewany the other generators Lie algebra not even be irreducible. Thiswell level and we could arrange the higher levels in irreducible representations of with the isation Maxwell the algebra only requires a set of and consequently, we canLorentz make algebra. the Thus, generators everytensors and at level we any can level form the semi-direct sum P generated by the translation generators structure given by howwe many refer times toZ the translation generators For this to formimplies that a the consistent Lorentz tensor Liethe indices algebra, reflects that the theof tensor Jacobi also the identity satisfies symmetric has it to has be the Young obeyed. tableau This on the commutators [ commutation relations. The next most general commutator is [ with a new generator JHEP03(2019)160 ] a 1) P 26 − (2.5) (2.6c) (2.6e) (2.6a) (2.6b) (2.6d) ,D (1 consists of . Then the , which is the so k 1 X 0 T = with the basic g 5 0 ∧ is spanned by all g X spanned by these 3

i ) generates an ideal, T 1 ) is X T P h ( 3 f 2.4 in the multi-commutator ∧ a in ]. The generating ⊗ ]]]. If the set 7 P 2 . X T 1 P,X ), leading to the tensor product. tT ⊕ [ ) plays a special role since it fully − 1 2.3 1 T T factors of for some fixed finite level consists of all commutators of the -th anti-symmetric tensor power of a ⊗ = 1 P,..., k k 2 [ ). In fact, any ideal generated by a set 2 # +1 T T ` denotes the vector space of generators at , P k and 2 P, , as in ( ( T 1 T 2 ` f ∧ k i T T T 1 4 ∧ ⊕ ∧ k` − 1 – 5 – t T ) can be computed from a generating function [ k ⊕ from the tensor product. 2 P transform. The choice in ( 1 ] that was reviewed in [ 1) 1 ∧ ( denotes the T f a T − 2 27 3 : ⊗ ( P t ) can be derived from the denominator formula for a R ∧ ,...,D 3 ∧ k =0 ) with more than ∞ T ⊗ k M 2.5 ( ∧ P = 0 " 2 ( f

T a ] related to particle models in electro-magnetic backgrounds ) ) may be equal to | =1 ∞ 7 2 `

O a P T ) ( , P 2 f h with the elements in we denote the removal of representations from the module. The 1 ⊗ T T 1 2 of 3 3 =

T ∧ T i ∧ 1 in X ⊕ ⊕ T h

is a formal parameter and 1 1 a 1 t T T P , T , ). The identity ( is the same as the ideal generated by the subspace 1 ⊗ ⊗ i , that is, the ideal is spanned by [ P ⊗ T a a 3 4 ( X 2 f 2 T P T P T h ∧ = ( = = = = ( ··· = 0 algebra under which the , such that 5 1 2 3 4 and therefore is the anti-symmetric product. The vector space ` ` T T T T T = 3 and so one has to remove The subspace Free Lie algebras admit many ideals and associated quotient Lie algebras. Several The tensor products here can be of as either just vector space operations or The structure of a free Lie algebra 1 ` T are generated by aof set elements which is aelements. subspace of ideal consists of all elements in examples were discussed in [ and previous examples in thespanned literature. by Any all set multi-commutators ofgenerators obtained elements by acting ononly the one elements element, then this is called a principal ideal. One can also consider ideals that at –and this will be morelevel useful to us– as tensorand products the of tensor representations of products would be those of Lorentz algebra representations. lowest levels are simple to understandand as the representing Jacobi the anti-symmetry identity ofin of the the Lie bracket free Liecommutators of algebra: The totally anti-symmetric Jacobi identity has to be satisfied by the triple commutators The notation here isrepresentation. such With that In this formula, level determines Borcherds algebra. Expandingidentities out at the the generating first few identity powers yields of the following recursive or explicitly using the Wittidentity formula governing [ the free Lie algebra is JHEP03(2019)160 2) , (1 . The so 1 ]. Since A need not of finitely 28 i s n s transforming . = 2 gives the 2 a ⊕ k P ... ⊕ 1 , and then the quotient s +1 c = k d c T ˜ ] and also the non-relativistic Y Z X 32 . For example, the truncation abc abd 2 a k  a  ] where a realisation in terms of . . . P S 1 7 ab,c,d = 3 translations = = S . These representations D ab generator 0 g ab,c Z Y , and the quotient is a finite-dimensional . Using standard mathematical Dynkin 1 ` > k [6] while the generators represent the associated – 6 – are generated in this way. Even more generally, ⊕ [4] may be a subspace of 1 4 . . . ⊕ with i A [2] [4] [2] [2] = 3 space-time dimensions. Representations are labelled ]. ]. A four-dimensional gravity system with Maxwell ` X 2) whose complexification is of Cartan type ⊕ T [2] h can also be obtained by dimensional from , 34 D 33 1 in the quotient. (1 [2] representation ) in so ) serves as a very universal object to construct kinematical ab algebra that admit an invariant and non-degenerate bilinear P P Z ( = ( f ∞ f ` 4 3 2 1 0 . . . g T T T T T and a ]. The three-dimensional case is of particular interest since one can P may be decomposed into a direct sum 7 i X ) including only generators up to level h ] with a supersymmetric version studied in [ P ( f 31 – 28 . The free Lie algebra = 1 recovers the standard Poincar´ealgebra while the truncation to The generators listed in table As an example of a free Lie algebra that has not appeared in the literature before to More generally, the vector space k symmetry has been investigated in [ 2.2 Free Lie superalgebras The bosonic free Lie algebra algebras relevant in various contexts. It is natural to consider the corresponding Lie su- find quotients of the Maxwell form that then can be usedthree-dimensional to Chern-Simons construct theory is three-dimensional related Chern-Simons to theoryof gravity, [ this gravity [ leads to a Maxwelllimit version has been studied in this case [ under the Lorentz algebra corresponding lowest levels arelabels, the listed translation in generators table are denoted by [2]. the general result in [ all be on thesuch same an level. infinite-dimensional Again, quotient theparticles was quotient in discussed may general be in ‘unfolded’ infinite-dimensional. [ electro-magnetic An backgrounds was example achieved. of the best of our knowledge, we consider the case of Maxwell algebra with may be infinite-dimensional. The SerreKac-Moody ideals appearing algebras in discussed the in construction section ofthe Borcherds vector or space or infinitely many subspaces, or representations of tensors. For mixed symmetry, these are reducible in the way describedor, in equivalently, the footnote direct sum oftruncation all of to Table 1 by the complexified Lorentz algebra of type JHEP03(2019)160 α Q ) by even (2.8) (2.7) (2.9c) (2.9e) (2.9a) (2.9b) (2.9d) Q 2 ( f Z products , 1 on = 0 algebra R 5 ` ` gives . ∨ , the generators 1 ` 0

g . tR 1 symmetric − R 3 ∨ ] = 0 β = 1 ⊗ # are by definition odd while 2 ,Q ` R α } R ) is given by α is totally symmetric as well. k Q ⊕ are interchanged for symmetric 3 ∧ ) 2.5 ,Q dimensions or could also contain 1 R k` k γ sense while even t R ∧ k D Q is given by all 2 { ⊗ 1) Z = 1 and we denote the generators on 2 2 − ` R ( R transform under some level ] + [ 1) in , nature of the generators introduces some 2 α =0 1 ∞ α ∧ 2 k − M by treating the tensor products as products R Q Z " ,Q 4 ⊕ 0 } g ∨ ,D 1 – 7 – γ R even (1 ⊕ O 2 ` ,Q 1 so ∨ β ⊗ R is even. One can again define a level Q 2 ⊗ # { ` ∨ 3 } ) is given by all possible graded multi-commutators of β R R Q ⊗ k ( ( ] + [ f 2 ∨ ,Q γ

α -grading of superalgebras; the R that occur in a general graded multi-commutator. For odd k` ) 2 t Q 2 ,Q k

Z α { } R ) , 1) Q β 2 1 ⊗ − R ( R 3 2 ,Q 3 α and the Jacobi identity relevant for ∧ R =0 ∞ ∨ k M Q denotes the generators on level α ⊕ ⊕ ), the anti-symmetric tensor powers { " , in a few places. For instance

[ i Q k 1 1 1 α 2.6 ∨ R R odd , , R Q O and subjected to the graded Jacobi identity ` h 1 . Expanding out the generating identity one obtains for the first few levels ⊗ ⊗ i ` ⊗ α α R 3 4 = of the R 2 2 Q labels the supersymmetry generators and we here think of them as spinorial R R Q 1 R h ∨ } is again a formal parameter and the β α by R t = = ( = ( = = ··· modules. ` 3 4 5 1 2 Just as free Lie algebras, free Lie superalgebras admit many quotients and one can again We can again consider the case when the Similar to the bosonic case one can describe the structure of the free Lie superalgebra The free Lie superalgebra We shall consider the free Lie superalgebra generated by odd supertranslations ,Q 0 can be arranged into representations of α g R R R R R . This could be the Lorentz algebra one obtains generators that are also odd in the ` 0 Q possible R-symmetries. We shall consider manyR examples below. Given a of distinguish the case of finite-dimensional and infinite-dimensional quotients. Examples of Compared to ( ones, denoted { g signs. level in terms of a generating function. The generalisation of ( Here, their graded commutator counting the number of ` generators such that the two gradings are consistent. section is independent of these details. the basic ‘Graded’ here refers to the examples will be discussed in the next section. where generators. The exactwhether range one consider depends simple on or the extended number supersymmetry. of The space-time general dimensions discussion and of on this peralgebra and in this section we give some general properties of its construction. Explicit JHEP03(2019)160 and (3.2) (3.1) α 4 here 1], the 3), the , Q free Lie , [0 (1 ) at lowest ,..., ⊕ so Q involving the ( 0] f dimensions, or = , = 1 ab 0 [1 Γ D g ). = 4. Keeping this α C ↔ N D ( = α su . The first non-trivial T Q ) , 2 ab ab Γ . P gives a real representation of C 4 1 αβ ) A + + +) and below we shall also . ab . Gamma matrices with multiple α and ( ⊕ − Γ β ab 1 a ) C η ( Γ A ab = ( 1 2 C (Γ η = 2 + β = = 4. The spinor index } a Q b T P 1 2 ) Γ that we define to be D a ) and construct the free Lie superalgebra gen- are balanced, i.e., either of the symmetric form , αβ – 8 – a Γ α ) 1 ] to compute the tensor product decompositions in ] = Γ a 2.9 C is either the Lorentz algebra in Q A { α Γ 35 α 0 etc. The gamma matrices that we use are given C g Q ,Q ] . In this table we have labelled the representations of b ab Γ = ( 2 ) is obtained by taking all possible (graded) commutators a [ M } [ Q ]. β 3) in terms of Dynkin labels of the corresponding complex ( f , is given by . ,Q = Γ p, p (1 i α C α so Q supersymmetry ab Q . We use the signature { h = A 0 = = 1 g , such that a Majorana spinor decomposes as 1 1 R N A ) generated by the ] of directly [ ⊕ Q 1 ) this leads to table in various cases where ( 1 f and , p α A obeying only graded anti-symmetry and the Jacobi identity. Following the 2.9 2 . Generally, a complex representation of Q p 1 [ α = 4 A Q ⊕ D ] ⊕ 3) if the representations of the two 2 1 Let us also give the commutation relations for the free Lie superalgebra The free Lie superalgebra , , p A (1 1 p levels, using thecommutator names is for that of the the generators supercharges introduced in table familiar decomposition into a left-handedin and mind, right-handed we spinor can in of use the LiE softwareso [ [ charge conjugation matrix of the ( the Lorentz algebra Lie algebra The real gamma matricesindices here are satisfy definedexplicitly by in Γ appendix encounter the symmetric combinations ( Following the philosophy outlinedsuperalgebra in the previous section,labels we the now four construct independent the transformation components of of a Majorana spinor. Taking We begin with thethen free Lie compare superalgebra the generated resultsappeared by in to supersymmetry the generators various literature. extensions of the3.1.1 Poincar´esuperalgebras that have Free Lie superalgebra In this section, weerated apply the by algorithm ( the direct sum of the Lorentz algebra3.1 and an R-symmetry algebra are finite-dimensional quotients andthere we are shall cases illustrate where this thesesuperalgebras in quotients and examples can Dynkin below. be diagrams Moreover, described as conveniently we in discuss terms in of section 3 Borcherds Instances of free Lie superalgebras extensions of the standard that have been discussed in the literature JHEP03(2019)160 ] ) γ 3.2 ,Q (3.3) (3.4a) (3.4b) } β as ,Q α ab Q . P { } β and ,Q a α P ). The resulting most ; this corresponds to the Q commutators in eleven { 2 } 2.7 ] ] . αβ with represents a tensor structure ab ab [ ) [ α 1 4 α P Z β ]. It does not appear in the Q, Q α a α α − . As mentioned in the caption ) R . . . a α Q ] { B ˜ P = = b Σ Σ Q 38 C Σ – a, b ab (Γ ab ab vectorial two-form generator 36 P Z (Γ a β , [ 1 4 Σ a α − 3 4 0] 1] , , = + = 4 space-time dimensions for a single four- [2 [1 + Σ 1] 2] 2] 1] 1] 1] 2] α ab , , , , , , , α ⊕ ⊕ 1] β D β ) , [0 [0 [0 [0 [0 [2 [0 ) 2] 1] 0] 3] – 9 – ab a . . . [1 , , , , ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ,P ) in [0 [1 [0 [1 (Γ (Γ × } 0] 0] 0] 0] 0] 2] 0] β β Q , , , , , , , 3 β ⊕ ( ⊕ ˜ f Σ [1 [2 [2 [1 [1 [1 [2 2] 1] representation ,Q , , α ] = Σ ] = [2 [3 a Q ab { ,P . We have grouped representations together that form a nice rep- ,P has already appeared in [ α αβ α α ` 1 2 3 4 ) ] . . . Q 1 Q [ Q R R R R R ab [ − [ ] in a way similar to the complete P C a 36 = (Γ 1 4 3). All tensor-spinors are gamma traceless, but we retain the tensor traces, ab , = 4 [ P = (1 a D so ]. We shall come back to this interpretation below. We also note that ( = 1 supersymmetry algebra and it is not central as it does transform non- P 9 = 3 that have to be subjected to the Jacobi identity ( N ` . The free Lie superalgebra where one can still take the Lorentz trace as discussed in footnote In the free Lie superalgebra, one can then form the triple commutators [ 1] representation listed there. Some representations occur with a non-trivial multiplicity and we , ab,c trivially under the Lorentz algebra. Itmembrane can in be interpreted as adimensions source for [ the supersymmetric implies The anti-symmetrisation only refers to the vector indices [1 have not given names to generators that do not appear elsewherewhere in the this article. term standard at level general expression can be split up into the commutators of Table 2 component spinor generator resentation of meaning that for exampleY the first non-trivial Young tableau in JHEP03(2019)160 × and i (3.7) (3.5) (3.6) (3.8c) (3.8e) (3.8a) (3.8b) (3.8d) . For the 2 = 3 generators with the ... ` + B 4 = 12. We also note that the αβ − . ) 4 , 5 α a Γ β × ) P C , , ab ] ..., αβ a ) + ( (Γ P + a = 0 b β [ Γ ab c β ab Q , . ) the left-hand sides have superficially 4 η Z C α ) for the moment and go back to the Poincar´e Z 1 2 ) ]. Q a αβ 3.4 ( ) = ( – 10 – f ). We shall see below how a quotient relates to ] = 11 (Γ ] = 2 ] = ] = 0 ] = 0 ab } b b c α α β Γ a α A.4 ,P ,P ,P C Σ ,Q ,Q a a ( ,Q a ab P α P ab 1 4 [ [ P [ M Q [ M = { [ = 1 supersymmetry): } β N ] we have slightly changed conventions by removing some factors of Σ , 11 = 4 we have to commute the level α ` Q { ) through the introduction of a new (non-central) Majorana spinor = 1. We shall not give the full commutations at this level but restrict = 4 (with It was shown there that one can in particular deform the commuta- ` 3.8e D 4 ]. 11 ) and ( on level , the tensor spinors we use are gamma traceless which means that 6 = 40 elements while the right-hand side has 4 + 12 + 4 = 20 elements and the α 2 × Q 3.8d The Poincar´esuperalgebra has non-trivial cohomology and it can be extended as was Continuing to level We note that compared to [ 4 studied in [ tors ( rescaling some generators. Let us now leave the freesuperalgebra Lie in superalgebra showing the relation betweenIn the the free two Lie superalgebracrucial and in above the the the free bosonic dotscomparison Lie free denote in superalgebra. Lie the additional Their algebra. next generators tensor section type we that do has are not been3.1.2 present require listed the and in precise table Comparison form to of these Maxwell terms. superalgebra in the literature such that the Maxwell superalgebra introduced in [ basic ourselves to defining a part that is relevant to the comparison below, 4 + 4 discrepancy is due toof the the Jacobi 40 identity, possible meaningof that commutators the there that free are vanish Lienon-trivial due 20 superalgebra gamma linear to commutation matrix combinations the relations identities Jacobi with ( identity. the The Jacobi consistency identity requires the and accordingly they span aJacobi subspace identities of can dimension imply 4 does that not a naive always counting work. of generators For in instance, commutation in relations ( of table JHEP03(2019)160 . ) ], α to β 23 ) ]. Q – (3.9) 5 3.12 . We (3.13) (3.10) (3.11) (3.12) 40 . From 20 4 , (Γ ] to pair α β 39 β 11 Σ ) 5 − ]. The algebra (Γ 13 β ] = Q α Σ allows for the -symmetry [ , ] = κ 5 α B 2.2 ; extending this beyond ,Q ` . In the previous section, B. 5 0 5 − together with the free Lie B g ), etc.; in the algebra ( ) B B. β 0 + αβ g Q 2.2 } ) 5 ab and 4 ] = 0 and [ ,B αβ Γ that was introduced in [ a =4 4 is the Lorentz spinor index while {z Z ` ` ab C . C α ab B | ,P 3) (possibly extended by a chirality α ,Z 5 , satisfying [ β M + ( ) B α ,..., 5 = 0 algebra (1 1 2 + Σ a =3 ab = 10 dimensions had been introduced previously Σ ` ab ` B β |{z} -brane models and their so Z − (Γ Z , p = 1 D Σ β ) described in section β = a αβ α αβ =2 Q ] = Σ ) -extended supersymmetry. The corresponding ` 0 b ( |{z} C f g ab ,P αβ N α – 11 – ] = Σ Γ ,P α C a a ] form a subset of Q =1 C α ( ` where where according to ( P ( |{z} [ 1 2 ,P 11 ,Q Q 1 4 ) for I α } α 5 } + Q ab N ] then consists of → Q = [ ( ,B a 0 β } g 11 {z ,Z ab su P = 1 supersymmetric extension of the Maxwell algebra that β ) admits an invariant Casimir of the form a Σ a Σ ⊕ M | P αβ ,P , N ) pairs generators on levels 1 2 α 3) 3.12 C ab , α Q = M { (1 Q 3.13 { 2 5 ) would require generators on negative levels that pair with gener- so ), C = 4 Lorentz algebra 5 = 4. Such a structure is provided by a tensor hierarchy algebra [ D 3.12 0 BB g ) defined above and is consistent with the commutation relations. It is in + Q ` > ( by letting is the R-symmetry index. f B and extended supersymmetry 5 α N B ( = 4 1 2 D ,..., → ] has also been extended to study supersymmetric A similar extension of the supersymmetry algebra in 13 B 5 = 1 5 consider the case basic generators will be denoted by I in order to write theof Wess-Zumino [ term of the Green-Schwarz string as an invariant term [ The construction of a freebe Lie superalgebra odd generators transforming underwe some considered level the operator). In thisthere section, is we a shall non-trivial consider R-symmetry the acting case on of the extended supersymmetry supersymmetry generators. where That is, we tension, the CasimirB would also bethis made is more not manifestly necessary as graded these symmetric generators3.2 by commute up letting to central terms. The quadratic Casimir ( the generators in ( ators on level yet another algebra that can be defined from the Poincar´esuperalgebra. In such an ex- superalgebra fact a quotient ofin the terms free of Lie a superalgebraalso Borcherds and note superalgebra the that as quotient the can, we algebra shall moreover, ( be explain described in more detail in section the commutation relations above one thenThe deduces total [ algebra considered in [ We thus see that the generators of [ one also has to impose by the Jacobi identity that In the relation above we have alsowith included the a bosonic generator Lorentz scalar chirality operator In order to obtain aincludes minimal the generators generator Σ JHEP03(2019)160 = 2 = 2 2 by (3.15) (3.14) N N 2. > , N ) leads to = 1 where each I 1 3.14 A ) that we have a fundamental ⊕ I N 5 ( 1 P have representation A su αβ I α ⊕ with ) ] 5 1 Q ab IJ I α [ Γ 2 and we treat this case A P Q ) C . ( > IJ = = 4 dimension for ( IJ a ab IJ 5 ) = 2 generator (  N P P P I α D a . . . IJ = 2 by Poincar´esuperalgebra an ( , + P ab Q = αβ N 2 due to a proliferation of terms. ) , P P P N ab IJ a generator = Γ αβ ` > P 5 1; 1] C = 1 generators IJ Γ , ab -extended supersymmetric version of the ` IJ ] from a contraction of a superconformal P [0 C  -type rather than the ) to the ( N with 12 ⊕ 1 2 so + ` = 4 space-time dimensions for extended 2.9 a R + D P 1; 1] 2; 2] 0; 1] – 12 – , , IJ , a αβ ) [0 0; 0] [0 P ) in a , . . . 1; 0] 1; 2] ⊕ ⊕ Q Γ αβ = [1 [0 , , ( ) 5 f a 1 Γ [1 [1 × Γ R C 0; 1] 2 0; 2] ( C , , representation [1 [2 IJ -extended supersymmetry in (2). Then the level  + ( supersymmetry supersymmetry N representation. The fundamental R-symmetry index is su = 2 1 ` 1 2

. . . A J β R R R > = 2 = 4 was introduced in [ of Maxwell algebras. An (2) R-symmetry representation is given by the last Dynkin label while the ⊕ ,Q 1 N N ab D su I α A Z Q  and and ] = b ]. ,P = 4 = 4 37 a , P 9 D D . The free Lie superalgebra , where we do not give the result for 3 The can be carried out to higher levels and will contain the hallmark The commutation relation in the free Lie superalgebra representing the first two levels is as a complex structure. This is no longer the case for denotes a complexified 1 IJ in particular a raising and lowering of R-symmetry3.2.2 indices. One can similarly consider letting the generating elements of the free Lie superalgebra be charges [ relation [ Maxwell algebra in algebra. Their algebra hasassumed an here R-symmetry and of therefore the resulting commutation relations are quite different, allowing The first line corresponds to theelectric central extension and of magnetic the charge. The second line introduces additional string and membrane  on its own below.table Applying the algorithm ( labels so that the last labelis is special the as R-symmetry the label, R-symmetry separated representation with [1] a can semicolon. be treated The as case a real representation using first two specify the 3.2.1 Complexifying the algebra as beforeA we are therefore considering Table 3 supersymmetry. The JHEP03(2019)160 ) ) 2 ). ). = N N > 0 ˙ N ( ( BIJ ( = 11 g ˙ 3.17 A (3.19) (3.16) (3.17) su su N ¯ su (3.18c) (3.18a) P (3.18b) D that yields and I α Q IJ AB , ] P IJ [ 1] of the complexified P ]. Writing the , 0 = , J 42 0 I , a , , ˜ IJ P 0 1] ˙ 41 , A , which is twice the number of 0 A ) a , N ˙ 2). The important point here is = [0 σ , 1 ˙ IJ 1 ,..., ¯ R P + ( ˙ IJ = ) = 4 B . a 1; 0 ) properties in a way similar to ( ˙ 1 P A ,P , ∗ ˙ P  ) A ) ˙ R N A [0 ( ( I A AB + IJ A C  ( ) AB ⊕ Q su a P + 0] σ ˙ ( = ( 2 and BIJ = = 8 [ I and the corresponding complex representation is J ˙ to relate the two representations, but for – 13 – , IJ AB A ) δ 1 ¯ ˙ P P one then obtains the (anti-)commutation relations AI N IJ ,..., AB  IJ ¯ = 1 ˙ 0 ( Q = 2 = = N − A , P a it is necessary to use 2-component Weyl spinors that A A

A σ a J B = ⊕ 0; 1 ˙ ˙ P , 1 AJ BJ ,Q ¯ ¯ A IJ Q AB Q I A , , (with ⊕ = [1 I A -extended supersymmetry. A real slice through the complex Q 1 ˙ 1 supersymmetry  AI Q A ˙ N ) subalgebra one can make the transition between the complex AI R ¯  Q ¯ Q N  ( = 1 10) and form the representation , so 3) but this obscures their are real 32-component Majorana spinors that are irreducible under the N , and (1 ,P α (1 I A so Q so ) is of type Q ). In order to write an anti-commutator of the elementary = 0 and I N N I ( a ( ˜ P su su = 11 ⊕ = 2 we could use the invariant D . This space has complex dimension dim 3) 1 In terms of the Pauli matrices , N (1 N − space-time dimensions as this is the case3.3.1 relevant to M-theory. Free LieIn superalgebra this case the Lorentz algebra with symmetric pairs of spinorLorentz indices group combine into an antisymmetric two-form under3.3 the The last example we discuss in some detail is given by minimal supersymmetry in where with similar relations for the conjugate generators. The generators Weyl spinors. and the R-symmetry transformationsindex. in a This real happensin basis for terms will instance of therefore for itsWeyl also real and affect a the real spinor invariance. Majorana We shall basis not carry manifest, out but this at here the but instead sake work of with giving the (complex) up manifest real supertranslations of representation is picked by a reality condition of the form We note that the conjugation affects the R-symmetry index and spinor index at the same so illustrating the that theA two Weyl spinors transform in conjugate representations under standard translation generators we denote by that the two Weyl spinorsFor transform in conjugate (fundamental) representationsthere of is no corresponding invariant tensor to achieve this. The complexification of index of JHEP03(2019)160 ) are (3.20) Q Dynkin ( f 5 B . 5 ...a 1 a P αβ ) ] 5 5 a Dynkin diagram. Carrying ··· ...a ] 1 1 5 a ab a [ [ B Γ P P C α a . . . ( P ... = = Q 1 5! 5 spinors ab generator + P ...a 1 vector-spinors a ab two-form spinor four-form spinor = 2 in the free Lie superalgebra three-form spinor P P = 11 space-time dimensions for a single 32- ` αβ ) D 1] 1] 2] 0] 0] 2] 0] 2] 0] 0] 0] 2] 0] 0] 0] ab , , , , , , , , , , , , , , , 1] 0] 0] 2] 1] 1] 1] 2] 0] 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 Γ , , , , , , , , , – 14 – , , , , , , , , , , , , , , , C 0 0 0 0 0 0 1 1 1 10) representations are labelled by their ) in 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 ( , , , , , , , , , , , , , , , , , , , , , , , , , . Q . . . 0 0 0 0 0 1 0 0 1 1 2 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 ( (1 , , , , , , , , , f , , , , , , , , , , , , , , , 4 0 0 1 0 1 0 0 0 0 so + [0 [1 [0 [0 [0 [0 [0 [0 [0 [0 [1 [1 [1 [1 [1 , , , , , , , , , a [0 [1 [0 [0 [0 [0 [0 [0 [0 × × × × × × × × × × × × × × × P representation 2 2 6 4 4 2 5 2 3 2 3 3 4 4 2 αβ . All ) α a Γ Q ` 1 2 3 4 . . . C R R R R R ) leads to table = ( } 2.9 β ,Q α . The last label refers to the spinor node of the Q 5 { B . The free Lie superalgebra The commutation relations leading to level Lie algebra out the algorithm ( Table 4 component spinor generator labels in BourbakiSome numbering. representations occur Tensor-spinors with are multiplicity gamma as shown. traceless but we retain tensor traces. JHEP03(2019)160 ] as 55 = 2 = 4 5 ` ` space- ...a 1 1 in the a = 11 su- P D ` > and = 11 Maxwell ]. = 2 additionally ab D ]. ` P 10 , ]). From the point 10 a )) and on level P 53 , mixed symmetry 3.9 ] contained more brane 52 10 -forms but represented by ] it has been argued that the p ]. Exotic branes in the sense 44 ) is the standard Poincar´esu- ]. , Q 49 , ( = 11 Poincar´esuperalgebra were 48 43 f , , 15 9 [ (similar to ( D 47 α 11 beyond the usual translation generator E 5 a ]). In [ ··· ) and another one is the truncation to 1 10 5 a P ]. Including also generators from higher levels ...a 9 1 – 15 – a seems to capture all the corresponding ‘electric’ P and = 11. This extension of the Poincar´esuperalgebra 11 ]. ab D E and P 57 representation? – captures all the relevant mixed symmetry potentials [ ab 1 ), their (electric) coupling to mixed symmetry potentials ` 54 P ]. This algebra is a quotient of the free Lie superalgebra by 11 Q ( E f 46 , . Other extensions of the and = 3 only a single spinor Σ 45 ab ], and is nowadays called the M-algebra (but should not be confused ` Z 11 8 E = 1 in ` ; on 5 representation of at = 11 (without ...a 1 ]. With electric charges we mean that while the M2-brane of 1 α ` a D Q P 58 , ] are extended objects of co-dimension at most two and they tend to be very non- 49 51 , Even though exotic branes can typically be expressed as supersymmetric solutions of We also note that very recently a supersymmetric extension of the One possible quotient of the free Lie superalgebra but not are the ones that correspond to the M2-brane and M5-brane of eleven-dimensional 50 ab a free Lie superalgebra.the The adjoint representation motivation of fromand this that investigation comes the fromcharges [ the fact that pergravity couples naturally to the 3-form potential, the M5-brane couples more naturally all such supersymmetric branesbackgrounds has have been been studied classified [ and their relationusual to supergravity, non-geometric meaning that theyated preserve by some the of thesuggests usual to also gener- investigate whether there can be any relation to higher levels string coupling to a poweroriginally that using is larger U-duality than inof 1. low view These of dimensions exotic an (see branes electrictime have for space-time been potentials, instance coupling discovered meaning they [ potentials typicallyYoung couple that tableaux to are of not more necessarily complicated type than a single column. Using this language It is temptingsupersymmetric to objects speculate as onby they a approaches appear relation based in between on supergravityof the the [ and Kac-Moody free M-theory algebra Lie andperturbative superalgebra are when and captured analysed exotic in , meaning their mass scales with the inverse P only a single two-form studied in the context of free differential algebras3.3.2 in [ Relation to which then reproduces theone M-algebra may of construct [ quotients that agree with the algebraalgebra considered was in proposed [ in [ retaining the following generators besides the Poincar´esuperalgebra: on supergravity. The evensources more and general even fermionic algebra onesforms constructed that in were in . obtained [ by Thesome treating immediate were the interpretation used of for studying all the these Wess-Zumino charges term was of notperalgebra the clear in M5-brane but [ supersymmetry transformations in was first considered in [ with the generalisation thereof constructed in(non-central [ under Lorentz) terms P The right-hand side contains the most general combination that can be written for two JHEP03(2019)160 ]. ), ]. ξ ( 43 a = 11 59 , i.e., , (3.21) in the p X branes, D 4 , ...a 58 , 1 p R the charge a a all ]. The rela- 49 p Z Q dX 58 , representation of + 2)-form flux in 49 [ and the content of p 1 appear naturally in ` 11 levels of the free Lie ∧ · · · ∧ on the world- 11 5 ). In order to identify E 1 E representation has also p Q a ...a a ( ⊂ 1 1 can be integrated to give f a ` even under supersymmetry [ P dX ξ dX +2) (11) p gauge potentials [ p ( d +1) gl and F representation cover p 11 ( Z 1 E p and the C ab ` ∧ · · · ∧ P Q . 1 11 , a 11 ) is related to the M2-brane while the ∼ a E p P E dX ...a . The table also lists their decomposition 3.20 1 ]. This can be also understood through the a 5 Z in ( 44 , space-time dimensions the transverse space has -form contains all the mixed symmetry ‘charges’ that 9 ab p – 16 – is conjectured to contain all the mixed symmetry should contain a (non-central) term P D 11 11 E +1 where E p representation arise already on the first two bosonic , ...a 1 + 1)-form gauge field , the generators 10) representations predicted by 1 p ` and in , a p 11 ...a C (1 1 E ... . a ), we reproduce the lowest levels of the so + 4 Z Q to a ( ( f αβ ) +1) p p -brane the closed +1) ( p p . As can be seen from that column, most of the mixed symmetry ( ...a representation of 1 dC 5 C a 1 Σ ` ) we look for them in the free Lie superalgebra Γ of the free Lie superalgebra; it is possible that the question marks will = (11) representations in table R at infinity over which the dual of the flux C gl 4 2 ( representation when it is decomposed under of the extended object. From this we see that the supersymmetry algebra is related to the M5-brane [ ! R +2) 3.20 − 1 1 p p p p 5 ` ( − Q F = ...a D 10) representations. This should be compared to the and 1 } , S a β 2 P (1 R ,Q -brane coupling to a form so in terms of In order to compare the The comparison has been carried out at the level of representations up to From the point of view of As exotic branes couple to mixed symmetry potentials one might therefore by extension α p Q 11 { last column of table representations contained in the levels be contained in the higherthe levels. analysis As above includes the charges standard in and the exotic branes. the free Lie superalgebra E into superalgebras listed in table tion between space-time potentials inbeen the discussed adjoint in of these references.space-time As potentials that (supersymmetric)servation branes that can the couple to,can it be is obtained an by removing important one ob- index from any of the supersymmetry algebra to which theysuperalgebra couple. ( As there isthem no we room use in the the following standard tentative connection to the so-called the charge for a form with one index less. wonder whether there are mixed-symmetry ‘charges’ sitting somewhere in an extended of the brane. With5-form this the 2-form gauge-field that the brane couplesspace-time to as follows.a The sphere brane requires a ( Zumino coupling The contribution to the supersymmetry algebra is of the form with the integral over the non-trivial (spatial) cycle that the brane wraps and tric coupling is always such thataction the is corresponding simply Wess-Zumino term an infor integral the over a world-volume (the supersymmetric pull-back of)Σ contributes the to corresponding the potential.wraps supersymmetry Roughly, algebra a if topologically the non-trivial brane, cycle. embedded via This the is maps due to the quasi-invariance of the Wess- ‘electrically’ to its dual 6-form rather than ‘magnetically’ to the original 3-form. The elec- JHEP03(2019)160 ) Q ( f proposal along the as levels 0 11 1 12 ` 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 E . . . E ? ? ? ? ). This can be R R R R R R R R R R R R R R R ⊕ 11 2.2 E . Level here refers to 11 Occurrence in E subalgebra. We shall make 11 0] 0] E , , 0] 0] 2] 2] 2] 0] 0] 0] 2] 0] 0] 0] 0] 0] 0] 0] 0] 0 0 , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 , , , , , , , , , , , , , , , , , , , . . . 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 2 0 0 1 , , , , , , , , , , , , , , , , , , , 0 1 0 0 0 1 0 0 0 0 2 0 1 1 0 0 0 [2 [0 , , , , , , , , , , , , , , , , , no longer commute, see ( [1 [0 [0 [0 [0 [2 [1 [1 [1 [1 [0 [0 [0 [0 [0 [0 [0 × × 10) representation a , 2 2 P (1 – 17 – so under its obvious 12 lowest weight representation of 1] E 1 , . The last column lists a possible match with the even levels ` 0] 0] 0] 1] 0] 2] 0] 0] 0] representation as odd such that they satisfy non-trivial 0 11 , , , , , , , , , , 1 0 1 0 0 0 0 1 0 0 0 E ` , , , , , , , , , , 0 0 0 0 0 0 0 1 0 0 ⊂ . , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 4 , , , , , , , , , , commute: they form a generalised abelian translation algebra of 0 0 1 0 0 0 0 0 0 0 (11) . . . , , , , , , , , , 1 , gl ` 0 0 0 0 0 0 0 0 0 ]. This embedding embeds the semi-direct sum 0 , , , , , , , , , , 0 0 0 1 0 0 0 0 0 0 64 , , , , , , , , , , 0 0 0 0 1 0 0 1 0 0 -covariant by allowing non-trivial commutation relations for the , , , , , , , , , , representation. One possibility would be an embedding in 0 0 0 0 0 1 1 0 0 (11) representation 11 1 [1 ]. One of the hallmarks of the Maxwell algebra and free Lie algebra ap- , , , , , , , , , ` gl E [1 [0 [0 [0 [0 [0 [0 [0 [0 × 63 – 2 60 ], see also [ 58 . The first few levels of the . . . 3 2 5 2 7 2 9 2 2 11 ], all generators in We wish to stress that this comparison is speculative and we have only analysed the Level 49 elements of the lines of [ and 1 of amore graded decomposition comments of on suchto level consider decompositions the in elements in the the next section. Another possibility is representations. The algebraic structureof [ is quite different,a however. generalised space-time In in the ageometry way similar [ to the generalisedproach diffeomorphisms pursued of here exceptional is thatachieved the in translations an the natural grading under of the free Lie algebra in table Table 5 JHEP03(2019)160 ij A can for a (4.1) i lines. (4.2c) (4.2a) (4.2b) f ij A ` > k and − i . Any finite- , , . nodes, where e are odd if and r i 2.1 of the index set = 0 = 0 , f ] = 0 j } i . j S i f e . α determine the Serre ij , h , f is semisimple one can Q i i A ij f 0 h − = 0 [ = 2 and gray (depicted { g 1 A ) ii i ii f A A ⇔ ⇔ (ad are connected with , S S j among which ], or the tensor hierarchy algebra i i ∈ h ∈ 65 , ij , f i / δ i and in [ i at level zero, such that the positive level = 0 = , h i ) if 0 j K 11 e g j e ,

B ij , f i A e = 0 − J with two subalgebras at positive and negative 1 } ) – 18 – is defined from a symmetric Cartan matrix i i 0 ]. As we will show, when e g , r , e i j 67 generators e , f (ad ij r modulo the relations 66 , i A i to denote the graded commutator since the − , f i = 2 K · ⇒ { ⇒ , ii ] = , h · i j J A e , f S i are even. The Borcherds superalgebra is now defined as the Lie ⇔ = 0, and where two nodes h = 0 ∈ i [ S h i / ii ii ) is white (depicted with , ∈ A A j i / e and ). The adjoint action there is also by the graded commutator. ⇔ ) with non-positive off-diagonal entries and a subset ij ]. We refrain from speculating further in this direction, but we will in the , . . . , r j A S 23 2 , 4.2b , and all 6= labelling the rows and columns. Here we will only consider the case when the ∈ ] = S i i , . . . , r } j 1 otherwise constitutes an ideal, as described in the end of section = 1 2 ∈ where the basic objects are the supersymmetry generators , , e i ≥ i i ( ) if 3 h k [ i ⊗ = 1 , . . . , r The Dynkin diagram that we associate to the Cartan matrix consists of A Borcherds superalgebra of rank 2 , 1 i, j be odd or even.relations of The ( off-diagonal entries of thenode Cartan matrix with We have used the notation To the Cartan matrix we associateonly 3 if superalgebra generated by additional Cartan generators at level zero. ( { Cartan matrix is non-degenerate, -valued and satisfies the Borcherds superalgebras thatcontragredient Lie we (super)algebras will [ describeextend next, its Cartan and matrix or most Dynkin generally, diagramto and to obtain apply the so-called a generalised construction corresponding inlevels, order extension respectively, of which are quotients of two (isomorphic) free Lie (super)algebras, and given dimensional complex semisimple Lie algebra hasat this level structure, zero with and the root Cartannegative subalgebra levels. vectors corresponding The to construction of positivecan the and be algebra negative from generalised roots its in at Cartan different positive matrix and ways, or leading Dynkin diagram to infinite-dimensional Kac-Moody algebras, 4 Free Lie superalgebrasWe and have Borcherds seen superalgebras that apositive free levels, and Lie that (super)algebra we can decomposessubspaces add into form a a Lie representations algebra of direct it. sumto In of continue order to subspaces to negative at obtain levels a since simple the Lie subspace (super)algebra spanned by we all need elements at levels bedded in the Borcherds superalgebraconsidered denoted in [ next section apply thesection construction of a Borcherds superalgebra to the case considered in anti-commutation relations rather than commutation relations. This structure can be em- JHEP03(2019)160 i ) 0 f g 1]. 0]. , , (4.3) 4.2c [0 [1 = 1 if ⊕ ⊕ i through v 1] i 0] , e , [1 ].) ↔ , then we get the α 68 i = 0 subalgebra Q . ` 1 ). In other words, the -grading: odd elements 2 = 0 the complexification Z 4.2c ` = 1 the spinors [0 ` = 1 for all 1, respectively. i and its representations at levels v ) and ( . − to have an arbitrary non-negative 0 i g = e      = 4 correspond to nodes 1 and 4 of 1 ` . If 4.2b i − v D − 2 0 1 2 = 0 for the others. By setting − − i v 1 0 0 2 0 , a lower-triangular part generated by the = 4 and at level i = 1 and − − h ` – 19 – D 1 0 2 0 0 0 − , and i = 0 subalgebra      ` 3) in -grading consistent with the , = to have level Z i (1 1 2 3 4 A f so as required by the fundamental spinor of the removed nodes. In this level decomposition, the Serre i 1 h A supersymmetry we get a = 4 supersymmetry. S D = 1 ∈ i / = 1 only for some N forming the Lorentz algebra in i v ) as the subalgebra at non-negative levels of a Borcherds superalgebra. In 1 ) determine the level A and = 0 if i 3.12 4.2b v . Dynkin diagram of a Borcherds Lie superalgebra that is related to a quotient of the = 4 of the Lorentz algebra , and the corresponding 1 i 1 (which will be dual to each other), whereas the additional Serre relations ( D and v A  The two As noted above, Borcherds superalgebras have a triangular decomposition: they consist S ⊕ = 1 ∈ This can also be depicted in terms of athe Dynkin Dynkin diagram diagram as and in therepresentations figure gray of nodes the are two fermionic and represent the two fundamental One way to arrange this is by taking the Cartan matrix 4.1 We will now try tosidered reconstruct in an ( infinite-dimensional extensionagreement of with the the superalgebra discussionA con- above we should have at level with the Cartan generators relations ( ` define the ideal that is factoredlevels out generated from by the the free subspaces Lie superalgebras at at level positive and negative choose to have i appear at oddis levels, then and a even semisimple elementsgray Kac-Moody at nodes algebra even from with levels. the a Dynkin Dynkin The diagram diagram level of given the by Borcherds removing superalgebra, the centrally extended a free Lie superalgebraSerre modulo relations the define Serre an ideal relationscan in ( refine a the free observation Lie by superalgebralevel allowing that each has generator to begrading quotiented described out. above, We with only the Cartan subalgebra at level zero. But we can also for superalgebras as their are odd Weyl reflections thatof can the be Cartan applied subalgebra [ through spanned multi-commutation by and the multi-commutation. an Important upper-triangular for part us is generated the by observation the that the upper-triangular part is Figure 1 free Lie superalgebra of (We note that the relation between Cartan matrices and Dynkin diagrams is not one-to-one JHEP03(2019)160 2 h . (4.5) (4.6) (4.4c) (4.7c) (4.7a) (4.4a) (4.7b) (4.7d) (4.4b) 3) (no A do not here is , ` 3 (1 , h so . Note that 3 1 . This is the , f , , 2 3 , then we would e ) ) 4 4 4 , e that also appears f f 2 5 , h . 4 3) is a non-split real − + , h B 0] by acting with the , 2 4 4 , , f , e e 4 (1 , f 2 − − transforms as a Majorana , e so = 1. Looking for example e ]: , ]] = 0 1 1 1 2 2 α ) ` f f 4 ]] = 0 , h , e Q , e ]. In order to get a Majorana h 2 1 1 1 4 − + concerned. e e , e [ 1 1 + , f ) in the case where the Cartan , e 1 here. The single lines connecting 1 , e e 1 3 1 4 e A e = 0 to e , , [ } − − h 4.2 h = 1 is spanned (over the complex ] ] , ( ( ( 3 [ ` 4 4 1 , 1 2 1 2 1 2 ` e = 1 associated with 1] representation. The level 2 appear in a multi-commutator. Thus , e , e , { and [ 3 3 ` = = = 3 e e 4 [ [ . = e i e 01 31 03 3 , . Note also that − S e 1 − but [ or 3 ] ] 2 is )). This shows that A − 2 ie 2 and the chirality operator e } 2 , ), which state that , e ⊕ – 20 – , e 4 + 1 d ie 1 , e 1 2.1a 2 [ e 3.1 3 i e − A 6= 0 , ] 2 ,M = [ = = = 2 , ) ,M ) although we consider the real Lie algebra 1 1 2 3 4 4 , , e ) { f 1 4 Q Q Q Q ) indeed follow from ( 4.4 e (2) of f [ − . One can verify the Dynkin labels [1 sl 4 − 1 on the highest weight vector [ e ]] = 1 2.1a , 4 ⊕ A 2 4 e + ), or equivalently, by the Dynkin diagram in figure h ,M ⊕ ] and the other one by , e 1 − ) 2 (2) 1 1 6= 0 f 4 e 1 4.3 ), and the explicit form of the real gamma matrices given in sl [ 2 A , e h 4) from them, we have to take the complex linear combinations f , + and 1 e , 1 appears in ( e 1 3 4.2 − 1 + h ). , e i [ 1 h 1 of the index set 2 − e h , ( ( ( S and [ . If we instead just took the real span of 3.12 i i i 2 2 2 1 2 ) since we have removed the gray nodes. However, the Cartan generators = 1 e A = = = α is given by ( 4.4 ⊕ ( 02 23 21 1 Chevalley generator, the Serre relation implies ij α M M M do belong to level zero as well and by taking appropriate linear combinations of A 2 A Q e 3 h We now put back the gray nodes and go from level Let us first see how to get the Lorentz algebra by removing the gray nodes. If we set 0] representation of , appears in the commutation relations ( Indeed, now the commutation relations ( spinor, follow from ( one of the twonumbers) by 2-dimensional Weyl spinorsspinor at level A similar reasoning forgiven nodes by 3 the and sum 4 of yields the the number [0 of times so that there are[1 two non-zero elementsCartan at generators level and them, one can get thein dilatation the ( at the matrix the imaginary unit i from of get the split realappear form in ( the commutation relations ( the gray nodes to nodesto 1 a and two-dimensional 4 fundamental imply representation that of the the corresponding Chevalley generators belong Thus the subset JHEP03(2019)160 } 23 2 A , e (4.9) 2 (4.8c) (4.8a) (4.10) (4.8d) (4.8b) e { ), which = 0. In 6 ). } 2 3.8a 4.2 , e 2 2], respectively, ) of the free Lie e , implies the same { = 2, but any such ) corresponding to 3.2 ` 23 4.9 A . ] and the corresponding nodes ] 0] and [0 } , 69 3 , , , . ) is a free Lie superalgebra on , e ) and the sign in front of the } } } } ) by also allowing negative diagonal ] ) does not apply and therefore there ] 2 3 3 4 4 e 4.3 4.1 { , e , e ). Also the relation ( , e 4.2c , e 4.2c , . 2 2 3 3 2 e e e e does, however, lead to inequivalent e 4.2 [ { { [ [ i i , , = 0 2 2 23 − e − e } ] ) is a consequence of ] ] + appear once each. A multi-commutator A { 3 3 i } } . In order to arrange for this, one has to consider ] = 0 (4.11) ] ] 3 − { , e } , e 4 4 ab e − 3 ] 3 ) from ( 4.11 P } ] e } = 2. , e , e 2 } 3 , e { 3 3 . This Jacobi identity therefore implies ` 3 2 ) would also belong to level e e , e 2 – 21 – [ [ , e 3.8c e and 2 = 2 e , , e , 2 { e e 2 2 2 2 e } , { e e e e 2 2 { 1] modulo the Serre relations ( { { { , e , [ , , , 1 , e 1 1 1 2 e [0 [ ] = [ e e e e [ [ [ } { ⊕ − i i i and no . The value of 3 2] at level 3 , 0] 23 = = = = , e e , [0 2 A 2 3 0 1 e P P P P ⊕ { = 2 and set , we have chosen a double line between nodes 2 and 3, correspond- 2 0] ` . Rather we get ( e , 1 [ ab (or two 2. For the Serre relations drawing a single or multiple P 3 , that appeared in the anti-commutation relation ( − e ab = P 2. The defining relations can for instance be found in [ − 32 transforms as vector under the Lorentz algebra, follow from ( as well. As long as one is only interested in the upper triangular part, = A are lowest weight vectors of the representation [2 a representations [1 i and no ii P f } 1 = 2 A 3 e A 23 , e ⊕ 3 A e 1 { In drawing diagram We proceed to level A At this point, we note that there is the possibility of writing down a Borcherds algebra whose positive 6 levels agrees with the freemore general Lie Cartan superalgebra matrices including thanentries, the e.g. ones we haveare introduced sometimes in called ( ‘black’ nodes.are no For Serre black relations nodes, atsuperalgebra. the positive relation levels ( associated with the black nodes, in other words they form a free Lie the free Lie superalgebra,triangular this part relation of is the not Borcherdsthe superalgebra imposed. defined We by thus ( concludean that ideal the generated upper- by [2 independent of the valueBorcherds of superalgebras when onegenerators considers the action ofdoes the not lower-triangular matter. Chevalley As we have just shown, ( The first term onsecond the term right-hand follows side from the vanishes oddness due of to ( states that ing to lines between the nodesideal. does This not is matter, due as to any the negative Jacobi choice identity of This means that superalgebra is not presentand in the Borcherds superalgebrawhich we together consider form here, since Note that in eachwith multi-commutator, two element is zero because of the Serre relations JHEP03(2019)160 ). 2] , is a 4.3 [0 d ⊕ (4.12c) (4.12a) (4.12b) (4.12d) 0] , = [2 ]. 2 S 11 , 4 S . 6

1 acts as chirality while ] B 5 4 ab [ B ∨ Z and the notation is in that table. 5 α α a ab α aα . . . ⊕ d 2 B P B = Σ Q X 2 M X = 0. B ab ] for the case of only one gray node ` generator ) is the presence of a certain represen- Z ⊗ 26 1 2.9 B 2 1] 2] 1] 1] 2] 1] 1] 2] , , , , , , , , , ∨ 3 [3 [0 [2 [0 [0 [0 [0 [0 S

1] 1] 1] 0] 1] 0] 0] . . . , , , , , , , ) – 22 – ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2 [1 [1 [1 [0 [1 [0 [0 1 3] 0] 2] 0] 0] 0] 0] 0] B B , , , , , , , , 2 ]. This is displayed in table 3 ∧ [1 [2 [1 [1 [2 [1 [1 [2 , ` ∨ , representation [0 ⊕ 2

S 1 ⊕ 1 B

` , 6 5 4 3 2 1 0 B 0] . . . 1 ⊗ i B B B B B B B B = 4 this is the Maxwell superalgebra studied in [ ⊗ `, α B 3 ` 2 2 Q B = [ h ∨ B ` = = = = ( ··· S 1 2 3 4 B B B B can be derived from the representation content of a corresponding Kac-Moody k at each level that has to be removed. We have already seen that ` . The non-negative levels of the Borcherds superalgebra defined by the Cartan matrix ( S Alternatively, the representation content of the Borcherds superalgebra up to an arbi- The representation content of the positive levels of the Borcherds superalgebra can be tation and an analysis similaryields to that the in general one carried out in [ trary level where the difference to the free Lie superalgebra ( constructed recursively using aBorcherds similar superalgebra algorithm one as has for the free Lie superalgebra. In the This is a quotientWe of have the also free included Liescaling the superalgebra operator. Cartan shown Up generators in to table from level level Table 6 JHEP03(2019)160 . - ]. k 7 70 , 2, and 19 and − , and for = 3, the = 3, this ] with an + 6 k = 6 k 11 E 32 = 4 its affine representation A and the Cartan k ). Furthermore, 1 . For 1 k = appearing in the A ( 2 A ` sl 23 ⊕ ⊕ comes together with , for B 1 1 A 6 as its eigenvalue. 1 A A + 3 E ` A k ⊕ subdiagram. For . A level decomposition of 1 , the 1 k A − ++ + 2 k 6 ) or with the Dynkin diagram in ≤ +3 corresponding to the Borcherds k A E k ` = 4 its affine extension 4.3 = 5 by comparing tables k k with split real form = 0 have not only , for 1 6 ` − E k by the ‘Borcherds-Kac-Moody correspondence’. A 1 1, forming an – 23 – − ]) and then generalised to other cases in [ k 20 ), each representation of ) is precisely the representation k k . ( 5 6 ( -th antisymmetrised tensor power of the fundamental sl sl k ++ 6 that can be chosen to have the level (see also [ E d = 5 its hyperbolic extension . The method is only directly applicable if 11 6 k E E ). It turns out that, at each level k 1 2 3 4 , but also ( 3 sl h , and for and + 2 6 ]. Furthermore, since we now at level h subdiagram. The resulting Dynkin algebra is shown in figure E 71 are not known. This ‘Borcherds-Kac-Moody correspondence’ was first explained . Dynkin diagram of the Kac-Moody algebra of rank 1 , and it reveals a remarkable connection between the Maxwell superalgebra and the − ` k S ] in the context of 1 A 1 white nodes, each connected to the previous one with a single line, so that they The Dynkin diagram of the corresponding Kac-Moody algebra is obtained by replacing 18 − = 5 its hyperbolic extension Borcherds superalgebra. ThisUsing either can tool be we verifiedadditional obtain scaling for the operator supersymmetric Maxwell algebra studied in [ SimpLie [ generators a representation of that comes together withdimensional the representation of corresponding Kac-Moody algebra isextension the exceptional Liethis algebra Kac-Moody algebra canas be performed for with the respect originalthan to Borcherds nodes a 2 superalgebra, Lie and and superalgebra, 3 since in the the we representations same now can way have be a computed Lie easily algebra using rather the the gray nodes withk ordinary white ones,from removing the the Dynkin double diagram of line,the the and new Lie adding nodes algebra another 2this and 3 replacing the gray ones should be connected to the first node in exceptional Lie algebra this is the reasonvalue why of we these have made entries thislevel in decomposition. choice, the although, Cartan as matrix we gives have the seen, same any representation negative contents in the algebra. This can for exampletions be more useful thanin the method [ above when theHere representa- we will justcase briefly of describe the Borcherds the superalgebra consequence withfigure Cartan of matrix the ( general procedure in the special superalgebra with the DynkinThe diagram number in figure of nodesKac-Moody in algebra is the the lowerk exceptional line Lie is algebra Figure 2 JHEP03(2019)160 are 3 e = 0, but . We have ` and 2 2 e = 4). ` ), dual to each with a semicolon. N ( 1], respectively, as 1 in table , A ` su 0 ) with Dynkin diagram 5 on the generators in B after the two Dynkin 5 ⊕ 4 6 5 4 1 1 E ...I I ...I ...I 1 5 ab IJ ...I A 1 1 α 1) Dynkin labels. As in ,..., . . . I IJK d ,..., 1 I a I B N − I α M Q α 1 white nodes between the P ab aα A Σ B 1; 0 X Z = 1 X , generator subalgebra obtained by removing . The generators N − 4 3 N − A 1] 0] 0] 1] 0] 0] 0] ⊕ , , , , , , , 1 representations 0 0 0 0 1 0 0 I, J, . . . , , , , , , , A 1 0] and [0 0 0 0 0 0 0 0 A ⊕ , , , , , , , 1 ⊕ in addition to the (complexified) Lorentz A 0] 1] 0] 1] 0] 0] 1; 1 1; 0 1; 0 2; 0 1; 0 1; 1 2; 0 1 , , , , , , ,..., , , , , , , , 1 1 0 0 0 0 0 A 0 , , , , , , , [0 [2 [0 [0 [0 [0 [0 0 0 1 0 0 0 representation N − . . . -th Dynkin label is equal to one and the others to , , , , , , ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ – 24 – ` 4 0; 1 A (the hyperbolic extension of , 1] 0] 0] 1] 0] 0] 0] A 0; 1 0; 0 1; 0 0; 1 0; 0 0; 0 , , , , , , , with the Cartan generators. are separated from those of , , , , , , 0 0 0 0 1 0 0 -dimensional representations of ⊕ ++ 3 4 , , , , , , , [0 [0 [1 [0 [0 [0 6 1 e 0 0 0 0 0 0 0 N A E , , , , , , , A , separated by a semicolon. The Dynkin labels of the two ⊕ and 1 0; 1 2; 0 0; 0 0; 0 0; 0 0; 1 0; 0 1 , , , , , , , 2 A A e [1 [1 [1 [2 [1 [1 [2 = 5) we obtain the ⊕ fully antisymmetric indices ` 1 ` = 1 are then [1 ` A ` . . . 5 4 3 2 1 0 level = 0 (commuting with the Lorentz algebra), and the two Weyl spinors = 0 and representations where the , the representations now have 1 + 1 + ( ` 1 ` 4 = 5, decomposed with respect to the A = 5 (which coincide with those of the Maxwell superalgebra up to A and extended supersymmetry k ` ⊕ we have put the Dynkin labels corresponding to 1 = 1 transforming in two for = 4 A ` . The non-negative levels of 2 3.2 up to D ) at level 2 N ( section labels corresponding to representations at level can be verified by acting on other (‘fundamental’ and ‘anti-fundamental’),two we gray nodes insert in thethen Dynkin still diagram, lowest as weight shown vectorssince in of figure two this representations subalgebra of now thealgebra subalgebra also at contains level 4.2 The Borcherds superalgebras considered inway the corresponding preceding to subsection extended cansu supersymmetry. be extended In in order a at to level get an R-symmetry algebra Restricting to zero (all zero for illustrated this by putting table Table 7 in figure node 2 and 3. The Dynkin labels of JHEP03(2019)160 ) ]. |N = 2 ) R- 73 2 , (4.14) (4.13) , N N ( 68 (2 su [ 1, respec- su 3 − 1), which has a N − ]) it is straightforward , ] one can obtain equiva- (3 72 } 2 and off-diagonal entries A 68 as before while the two gray − = 4 as a real form. 1 -extended supersymmetry. The 1 A D N ⊕ {z N − 1 A , . ) in A in the middle of the diagram.             |N 1 2 1 1 2 } | , A − − − (2 1 1 1 corresponding to the complexified . The fact that the Dynkin diagram is dis- su 1 − − 1 4 1 2 0 1 21 0 0 2 {z N − N − A − − − A – 25 – 1 1 0 0 0 1 0 0 0 ; it is − − − 3 | 1 0 0 0 1 1 0 0 0 = 2, this gives the corresponding Cartan matrix 2 0 00 0 0 0 1 1 0 2 0 00 0 0 0 − − N             , this means that if the white nodes and lines on the left hand , with 3 4 1 2 3 4 1), which has the . Dynkin diagram of the Borcherds superalgebra for . Distinguished Dynkin diagram of the finite-dimensional contragredient Lie superalgebra generators are still odd elements, and the corresponding diagonal entries are still f = 4. This is a real form of the complex Lie superalgebra N − At this point we note a certain similarity to the superconformal algebra We also give the Cartan matrix corresponding to the simplest extended case , D and (3 zero, but on one sideopposite of signs. the gray For node figure theside corresponding of entries the in gray the nodetively, Cartan then correspond matrix those to have on diagonal the entries1, right 2 respectively. hand and For side off-diagonal figure have entries diagonal entries lent non-distinguished Dynkin diagrams ofgray the node. same Following Lie the superalgebra, rules forto with odd more show reflections than (see that one for one exampleHowever, [ the of meaning the of possible gray nodese diagrams in look this context the is same slightly different. as The the corresponding one in figure Row and column 5 correspond to the R-symmetry in distinguished Dynkin diagram shown intinguished figure means that there isrelaxing only this one condition gray and node, performing and so-called this odd condition reflections makes [ it unique. By A for which there are five nodes in figure nodes correspond to the two spinors. Figure 4 Figure 3 middle part issymmetry. the The Dynkin outer diagram nodes of labelled type 1 and 4 correspond to JHEP03(2019)160 ). 4.13 10). To , (1 so and the notation is as . 4 8 ] ab [ Z α a ab α ab aα . . . d P = Σ . The level decomposition of the Q P Σ M 5 ab generator Z 0] , 0] 0] 2] 0] 0] 0] 0] 1] 0] 1] 0] 0] 1] 0 , , , , , , , , , , , , , , from which we see that the Borcherds con- which is the complexification of 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , 5 4 . . . 0 1 0 0 1 0 0 0 0 0 0 0 0 – 26 – 1 , , , , , , , , , , , , , B , 0 0 0 1 0 0 1 0 1 0 0 0 0 = 1 one can attach a gray node to the ‘spinor node’ [0 , , , , , , , , , , , , , ` representation [1 [1 [0 [1 [0 [0 [0 [0 [0 [1 [1 [1 [0 × 5 2 B 10) representation listed in table , (1 ` 0 1 2 3 4 so . . . B B B B B B supersymmetry 1 2 3 4 5 6 = 11 supersymmetry. D = 1 N 1) is not a Borcherds superalgebra but a contragredient Lie superalge- , (3 and A should be compared to table 8 diagram with a single line. This leads to figure . This is a quotient of the free Lie superalgebra shown in table . Dynkin diagram of a Borcherds Lie superalgebra that is related to a quotient of the = 11 5 . The non-negative levels of the Borcherds superalgebra defined by the Dynkin diagram 1), where the entries in row or column 5 have opposite sign compared to ( 5 ], and this change of signs makes it finite-dimensional. , B D 67 (3 Table A of the Borcherds algebra defined by thisabove and diagram this can leads be to computed the using the methodsstruction described contains considerably fewer generators than the free Lie superalgebra. This is 4.3 We can also apply apoint Borcherds for construction the to Dynkinobtain the diagram a eleven-dimensional is spinor case. representation that at of The level starting in that table. of Accordingly, bra [ Table 8 of figure Figure 5 free Lie superalgebra of JHEP03(2019)160 (5.1) ]. 10) acts , 7 of the free 40 (1 , ` 7 so = 2 in a Borcherds ` that α , Q  at level ... 5 for a black node such that the )) in the context of West’s + 5 11 ...a 1 bd E a ( η ) can be seen as a representation P C ac -symmetry of the particle action. Q I ( η κ f 10), the eleven-dimensional Lorentz cd j , then leads to each level L (1 10). The general 0 ab i denotes the components of the Cartan- , g so L (1 1 2 ... then is a lowest weight vector of the representation so it is worth noting that the Lorentz algebra is + = 2, but it extends it beyond that. + } 6 ` ab 11 ab – 27 – , e η θ 6 E b j i e . In the discussion of the free Lie superalgebra for ) (or sometimes ∂ L { 0 . Demanding that these basic generators transform a i 11 g α = L = 0 algebra E ] up to  Q ( ) contains ` ab i 46 ij 11 K L the construction of a free Lie (super-)algebra starts from E γγ ( 2 − K and √ σ 2 ... we considered the case d , and + Z 11 a x E = 2 associated with the M5-brane term in the superalgebra. Thus it 3.3.1 i , we note that it is in fact possible to get also ∂ ) can be viewed as universal structure that encompasses all these proposals. ` ] and we expect a similar universal behaviour of a superparticle moving in a 6 ) transforming under Q = ( Q 11 f , ( a i f 7 ) can be made to act on the same 32 supertranslations L 11 ]. In this way, the whole free Lie superalgebra E 2]. on level ( = 0 and gives rise to a 5-form since , 75 0 , 5 } , K 6 Let us also point out a few other possible applications or connections to other work. 0 74 As in footnote , ...a = 11 in section , e 7 1 0 6 , a e on [ superalgebra, but then onecorresponding has diagonal entry to in interchange the{ the Cartan matrix gray is node not 6 zero[0 but in negative. figure This removes the Serre relation generalised to an algebraconjectures. called This is an infinite-dimensionalon Lie (the algebra that split is real) fixedof by an this involution algebra acting hasthat not been classified as it is not of standard type. However, it is known a set of basicin generators, a e.g., representation the ofLie some algebra level D algebra. In view of possible relations to where the Maurer forms of a string moving on the groupAs manifold was of emphasised the in superalgebra section [ Extensions to string and branegeneralisation models of moving the in Polyakov-type backgroundscan term can be also for written be the as constructed. string A invariant under the superalgebra We have focussed indynamical this model realising paper the on symmetrythe lines but the of this [ algebraic can aspects besupersymmetric Maxwell done and background for if have the a not point suitablethat ‘unfolded’ particle worked will quotient along out is deserve applied. detailed any An study aspect in this context is the 5 Conclusion In this paper, weextensions have of put the forward Poincar´esuperalgebra. thecomprise structure generalisations The of existing various asuperalgebra in examples free the that Lie literature we superalgebra have for by presented studying quotienting and thus the free Lie free Lie superalgebra by thealgebra Serre ideal. and the One noteworthy freeP difference Lie between the superalgebra Borcherds isagrees that with the the algebra former given does in not [ contain the generator of course expected since the positive levels of the Borcherds algebra are a quotient of the JHEP03(2019)160 ] discusses the ), however, the 80 P ( f ) forms an irreducible Q ( ) albeit still of the same f 11 E (     1 K = 2 of −     ` ] attempt to also derive the equa- 1 0 0 0 79 , 0 0 10 0 1 0 0 0 0 0 − 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0         78 , = = 15 3 1 Γ Γ – 28 – ]. matrices can be written as a 77 , ) symmetry. ]. In vector supersymmetry, the basic supersymmetry 76 11 = 4 E 77 ( ,         but transform in a vector and a scalar representation of the K D 76 ) one has for instance that α 11 1 0 1 0 Q E − − ( K 1 0 01 0 0 0 0 0 00 0 0 1 0 1 0 0 0 10 0 0 0 0 0 0 1 − −         = = 0 2 Γ Γ ) providing an infinite-dimensional realisation of 11 E ( In the physical realisations of Maxwell symmetries studied to date, the background Yet another possibility of applying the free Lie superalgebra is in the context of so- K An explicit real representation of the Γ by the Swedish Research Council, projectfor no. 2015-04268. Fundamental It Physics began at andsupported the in Mitchell Astronomy Institute part at by NSF the grant Texas PHY-1214344. A&M University and wasA then Gamma matrices in discussions. AK and JPwould thank also the like University to of thank Barcelonafor the for hospitality. University its of JG hospitality. Brussels has JGConsolider and been CPAN and and the by AK supported International the in Solvay Spanish Institutes goverment2014-0369 part (MINECO/FEDER) of under by ICCUB project (Unidad MINECO MDM- de Excelencia FPA2016-76005-C2-1-P Maeztu). Mar´ıade and The work of JP is supported Acknowledgments We would like to thank Carlo Meneghelli, Hermann Nicolai and Ergin Sezgin for helpful tions of a gravitational backgroundunfolding from of the an Maxwell algebraic formulationon for while algebras the [ that gauge are fieldators sometimes itself. and called These higher Ogievetskii constructions rank algebras are cousins thatcommutation based of include relations the translation appear type gener- to containedto differ in us free from how Lie those to algebras of reconcile the these free two Lie pictures algebras. at the It moment. is not clear generators are not spinors Lorentz group. A particle realisationto of two this uncoupled vector Dirac supersymmetry equations leads after [ quantisation field was non-dynamical. The proposals of [ unfaithful type. Under 528-dimensional representation. This perspectiveparticle opens or up brane the models possibility with of constructing called vector supersymmetry [ of JHEP03(2019)160 . ) a , A.2 (A.1) (A.2) (A.3) (A.4c) (A.4a) (A.4b) , abcd Γ for its inverse . and that C 1 − αβ     ]. − C = C 1 0 T − = ) SPIRE 1 0 0 C IN . We also note the cyclic abcd − − [ β , , . β Γ γ 0 1 00 0 00 0 0 0 1 γ ) = C ) 4. The first relation in ( a     ( b ) is crucial for supersymmet- = 0 = 0 = 0 T (Γ 3) are symmetric. They satisfy anti-commutes with all the Γ δ δ δ = (Γ C , ) ) ) , γ 5 3 γ γ γ 2 αγ ,..., α ) ) ) , A.4b Γ ) b abc C a (1972) 701 2 ab a = 4 are Γ Γ Γ Γ = 1 Γ = = 1 1 C C C 20 D ( C α ( i Γ ( − 0 αβ 10 are important for the construction of = (Γ ) αβ , αβ ( ( αβ a = β 6 ) ( ) , α . Γ (for ) a = Γ T ]. ) ab 5 i ) b C Γ ab 5 Γ Γ ], whereas ( Γ Γ = 3 Γ C abc a C C – 29 – ( C 1). One also has Γ 82 ]. Group-theoretical analysis of elementary particles in Group-theoretical analysis of elementary particles in − ( D , , (Γ SPIRE C + ( 1 ( IN = , Fortsch. Phys. 81 δ ) etc. The matrix Γ ) [ 1 , T a γ , where ( , , ) SPIRE ⇒ 1 Γ C 5 b βα IN ab − Γ ) Γ ), which permits any use, distribution and reproduction in αβ [ Γ a ( C     β ) Γ − C α a ) b C Γ 1 0 a = (1970) 289 Γ C a − and ( = diag( T ( (Γ = ( ) (Γ 1 . The spinor index range is ab ab 1 2 β (1970) 267 α αβ 1 0 0 0 η − A 70 Γ ) δ CC-BY 4.0 0 00 0 0 1 0 1 0 0 a − = C = ) and similar ones in ]. In the context of free Lie superalgebras they appear in the Jacobi = Γ ( for the anti-symmetric charge conjugation matrix and ] This article is distributed under the terms of the Creative Commons     2 b C The Maxwell group and the quantum theory of particles in classical ) with 36 A 67 Γ γβ 5 , αβ = is anti-symmetric while the Γ a [ a A.4a C C ab 0 0 Γ η ). αγ C = Γ C = Γ Nuovo Cim. 2.7 = 2 = , ab C

T b ) Γ an external electromagnetic field.field 2. the nonrelativistic particle in a constant and uniform homogeneous electromagnetic fields an external electromagnetic field.Nuovo 1. Cim. the relativistic particle in a constant and uniform field a We use the following explicit spinor index notation on the gamma matrices, following , H. Bacry, P. Combe and J.L. Richard, R. Schrader, H. Bacry, P. Combe and J.L. Richard, Γ a [2] [3] [1] Γ C ( References identity ( Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. The identity ( supersymmetric Yang-Mills theories [ ric membranes [ satisfying reads in indices ( identity that is satisfied by the gamma matrices in the SW-NE contraction convention, together with where Γ It squares to (Γ The matrix Γ JHEP03(2019)160 , , , 31 05 , , ] , B 607 ] ]. = 3 B 223 JHEP , D , 11 SPIRE = 2 . IN ][ D Phys. Lett. (2012) 066 Maxwell Algebra ]. Phys. Lett. , hep-th/9507048 , Beyond E ]. Class. Quant. 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