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This content was downloaded from IP address 128.194.86.35 on 02/10/2018 at 21:13 JHEP11(1998)016 4). | (8 =8AdS OSp N November 19, 1998 = 8 AdS supergravity N Accepted: = 8 anti de Sitter superalgebra ,N = 8 supersingletons yields an infinite tower of November 12, 1998, supergravity N =4 † D =8 Received: N = 8 singleton field theory yields the dynamics of the 1 correspond to generators of Vasiliev’s super higher spin algebra and Per Sundell N Space-Time Symmetries, Supergravity Models, M-Theory. ∗ The product of two ≥ s [email protected] 4) which contains the | (8 HYPER VERSION Research supported by The Natural Science Research Council of Sweden (NFR) Research supported in part by NSF Grant PHY-9722090 E † ∗ Center for Theoretical Physics, Texas77843, A&M USA University, College Station,E-mail: Texas shs Abstract: Higher spin Ergin Sezgin massless states of higher spinwith in spin four dimensional anti de Sitter space. All the states supergravity in the bulk,phase of including M-theory. all higher spin massless fields,Keywords: in an unbroken Gauging the higher spin algebra andrepresentation introducing leads a to matter a multiplet in consistent a andsometime quasi-adjoint fully ago nonlinear by equations Vasiliev. of motion We as show shown the embedding of the equations of motion incations the for full the system embedding atthe of the the boundary linearized interacting level theory. and We furthermore discuss speculate the that impli- JHEP11(1998)016 4 53 52 4) | (8 4) | )space 23 E 1 (8 equations of motion 40 4) E | shs 3 2 x, Z (8 ≥ shs E s shs AdS supergravity 45 =8 1 sector 44 4) connection 10 N | ≤ subalgebra of (8 s E 4) | shs (8 OSp 3.1 The 6.3 The spin 3.2 The quasi-adjoint representation4.1 General discussion4.2 Extension of4.3 the higher spin The superalgebra extended4.4 field content The equations of motion in ( 14 19 21 17 4.5 Integrability of the higher spin field equations5.1 The anti5.2 de Sitter vacuum Perturbative solution expansion5.3 The linearized curvature constraints6.1 Elimination of6.2 auxiliary fields The analysis of the spin 25 27 32 29 36 7. The linearized D. Symplectic differentiation and integration formula 55 Contents 1. Introduction2. The higher spin superalgebra 2 3. The field content 10 8. DiscussionA. Spinor conventionsB. The 51 48 4. The higher spin field equations 17 C. Oscillator realization of 5. Expansion around the anti de Sitter vacuum6. Spectral analysis 27 36 JHEP11(1998)016 × 2). 4 , =11 (3 8) was AdS | D SO 8) in four (4 | (4 boundary of OSp 1 S group OSp × = 8 supersingleton 2 AdS S N = 11 supermembranes in [15, 16, 17]. However, for 7 D S × 4 AdS 8) singleton multiplet occurring in the | (4 = 11 supergravity compactified on 2 D OSp =8singletonfieldtheoryon N 1) [6]. , (8 4) will be spelled out in section 3. It was argued in [11, 12, 13] that all | , an attempt was made to obtain the singleton field theory from the 7 (8 SO S did not emerge fully as the critical mass term for the boson was lacking. This [18]. × 4 4 4 Interestingly enough, in a related development Fradkin and Vasiliev [19, 20] In a later development, in the course studying the Notwithstanding this state of affairs, in [11, 12, 13] the After the eleven dimensional supermembrane was discovered [7], it was specu- OSp [1]. This is the ultra short representation of the AdS supergroup 7 AdS reasons explained in [16] the S Sometime ago, the singleton representation of the super AdS group 1. Introduction supermembrane action by expanding around problem has been recently circumvented by embeddingtarget the space membrane in worldvolume in such afield way that theory the on resulting a worldvolume threeAdS field dimensional theory is Minkowski a space conformal that serves as the boundary of dimensions, consisting of 8described bosonic in terms and of 8Dirac local fermionic [4] fields states in in [2, the the context 3], bulk of of which the AdS, cannot singleton as be representation was of discovered the long ago by encountered in the spectrum analysis of field theory was assumed toand be its a quantization was quantum studied. consistentconsidered. theory In In of particular, that the context, the a supermembrane, spectrumwas well known of used, and massless namely a states remarkable the property was factan of the that infinite singletons the tower symmetric of massless productfor higher of spin two supersingletons states [14]. yields The detailed form of this result AdS of these states shouldthe arise occurrence of in the the infinitelyof many quantum massless infinitely supermembrane higher many theory! spin fieldscoordinate (local) implies Furthermore, and the gauge local existence supersymmetries symmetries associated with analogous spin to 1, 2 the and 3/2, Yang-Mills, respectively. general Indeed, it has been shown [5] that the lated in [8] thatconjectured singletons in may [9] playgravity and theories a [10] may role be that in closelyThe a related its field whole to description. various theories class singleton/doubleton forperconformal of field Soon theories. them field AdS after, were theories, compactifications it constructed buttested. of was on the super- the main boundary thrust of of the AdS conjecture as remained free to su- be Kaluza-Klein spectrum is puremodes, gauge. the singletons are It needed to hasalgebra fill the also representations been of the shown spectrum that, generating though gauge JHEP11(1998)016 4), the resulting | (8 OSp 4 supersingletons! [11] | (8 OSp fields coincide with the massless states 1 2 3 ≤ s could indeed play a role in the description of bulk 4 AdS = 8 supergravity in four dimensions [34, 35], can be de- N e limit. 4)algebra [36, 37]. (The details of this algebra will be discussed in the next | higher spin gauge theory and simplified the construction considerably. In [23, 24] (8 The purpose of this paper is to address a more modest aspect of this problem Remarkably, applying the formalism of Vasiliev to a suitable higher spin algebra In a series of papers Vasiliev [22]-[29] pursued the program of constructing the With recent observation of Maldacena [31] that there is a correspondence between E = 8 de Wit-Nicolai gauged supergravity [32, 33], which is the gauged version physics has again arisen.the past, we As still far expectstates as all and the the the massless massive singletonAn higher states field interesting spin to theory question states arise to to is inmight which arise concerned, the we arise as product have as in two no singleton of M-theory. in answerthat three However, at they reviving and this will more the time arise conjecture singletons. is as of how a [11, such new 12, states phase 13] ofby we M-theory examining expect that the is Vasiliev yet theoryclass to of of be higher uncovered. higher spin spin fieldsN (which superalgebras) is and applicable determine toof the a the precise wide manner Cremmer-Julia inscribed which within the this framework.the higher We will spin indeed AdSshs show supergravity how based this on embedding the works higher in spin superalgebra known as resulting from the symmetric product of two the spin 0 and 1/2differential fields algebras. were introduced The toform theory the in system was [25, within furthermore 26, the 27, cast frameworkspinorial 28] into of variables. by free an extending the elegant higher geometrical spin algebra to include newthat auxiliary contains the maximally extended super AdS algebra AdS were in the course offor developing references a to higher spin earlier gaugefield work). theory theories These in for its authors higher own succeededconstructing spin right in higher (see fields. spin constructing [19] theories It interacting canspace was be observed and bypassed that to by the consider formulatingspin previous the an algebras difficulties theory infinite based in in on tower AdS certain ofIn infinite gauge particular, dimensional the fields extensions AdS of controlled super radiusoccurred by AdS could various in algebras. not higher be the taken higherPoincar´ to spin infinity since interactions its and positive therefore powers one could not take a naive spectrum of gauge fields and spin section). In our opinion, thisthe constitutes M-theoretic a origin positive of step the towards theresults, massless understanding higher we of spin will gauge comment theory. on After further presenting aspects our of this issue in the concluding section. the physics in AdSaffirmatively space in and a atwhether number its the of boundary, singletons which examples of has with been very successfully interesting tested results, the issue of JHEP11(1998)016 - γ =8 (2.1) These N 1 , (8) vector. 8 SO ,..., 2 is the space of fully sym- , =1 A 4) =1 | . ˙ α (8 forming an A can be replaced by Grassmann even E i = 8 supersymmetry is explained in 4)algebra are given in Appendices C i θ | θ N (8 shs E ,α, with Lie bracket given by the ordinary 4 A shs 2); our spinor conventions are given in Ap- , ]=0 4) is a Lie superalgebra which we shall define ˙ | β (3 ¯ y (8 , ˙ E α , SO y shs ]=[¯ (which together form a four-component Majorana spinor ]=0 β i ˙ α ,y ,θ y ˙ ˙ α α ,i y y =0 (8) Clifford algebra. ]=[¯ ]=[¯ } i β j SO 4) and discuss its infinite dimensional UIR’s based on the ,θ ,y | ,θ = 8 AdS supergravity. Section 8 is devoted to a further discussion α i α y θ (8 y [ ) of the AdS group [ { N ˙ E α ¯ y of the , i shs α y Since the techniques for constructing higher spin theories that we shall employ The organization of the paper is as follows: In section 2 we define the gauge In an equivalent formulation, the Grassmann odd =( 1 , its complex conjugate ¯ α α metrized functions of the two-component,y complex, Grassmann even spinor elements Y and some useful lemmas are collected in Appendix D. pendix A) and 8 real, Grassmann odd elements supersingleton. In sectionform 3 of and the 4 nonlinear we highertheory spin give around field the the equations. anti field de Thesection content Sitter perturbative 5, of vacuum treatment with where the of we theory the analyze also and the derive the linear the dynamics linearizedcase in equations of more of detail the motion. and in In section section 7 6 we we treat the particular of our results and someAppendices speculations. A and Our B. conventions Some and details notation of the are collected in generating elements therefore obey the “classical” commutation relations The higher spin gauge algebra 2. The higher spin superalgebra in this paper arecontained not as all possible too by well-knownspin including we theories. some have basic chosen However, features to weis of make would to the this like argue construction paper to that of as stressM-theory higher higher self- that and spin the in supergravity particular main plays that purpose AdSing a bulk/boundary of physical the could the role be subject paper within instrumental of foraddressed the bring- higher context than spin of previously. theories Therefore,allows to while for a the quite formulation point general of extended wheregravity, higher we algebras more spin without believe nontrivial any theories that issues obvioussupergravity the are truncation - extended to which super- we higher shall spininterest. refer theories to as that higher allow spin truncation supergravities to - are of particular algebra as a subspace of an associative algebra commutator based on the associative product of matrices Γ JHEP11(1998)016 2 × ;see 2) (2.8) (2.7) (2.6) (2.2) (2.3) (2.4) where ˆ ··· y , m ··· + (3 → α n + ··· β y 1 SO , n m α β α Y ), in which case m q ) 2) invariance. ˙ α , β θ 1 , ( ¯ y,θ − ··· n , ˙ m 1 n β (3 ˙ n β y, α β ¯ ˙ y ( ··· β 1 , ¯ y ··· , ··· − ··· 1 F k +1 ? SO i 1 k α m ˙ ··· β + ··· − = α n F 1 ¯ 1 y ˙ n ··· n ˙ α ) by replacing i † β β β m y θ i ··· 2 y )) ( m m + 1 1 ) +1 − ˙ α 1 α α − F k n j m ¯ ··· y β ¯ 2 − m i y,θ i 1 j 1 m ··· := ¯ ··· . α δ y, 1 − α m, n i ( y β n +1 ( p n 1 θ 1 m 2 β k j j F k i n β j ) (2.5) k i ˙ − k i − β ··· δ ··· ··· ··· = − 1 m ··· 3 k m ··· (8) vector indices. It is important 1 ··· n j i 1 j β α n 1 will be suppressed when there is no 1 β † i i β 2 j 1 ˙ ¯ δ y, θ β ) ··· n i − F − ··· ··· ˙ ¯ 1 y β F SO n 1 2 θ ?y m m y, α j ! j is defined by the following i β ( ··· ( α 1 y 1 k k 1 ··· ··· m ˙ ··· − β − 1 F α , obtained from 1 A i +1 m 8 m m 5 α 1) i ··· k θ , α j α 2) invariant monomials αβ 1 ··· are Grassmann numbers carrying fully sym- , of k α , ··· ··· 1 − i , 0 m i − 1 1 y ):= 1) ˙ (3 k α ? α k α α i m n i ,..., y i 3 y ≥ y ( ··· F − θ X ··· 1 SO = ! ! i n 1 =¯ Y,θ imn y n ]=2 i n k ( n n ··· =0 ( ˙ 2! mn θ β † θ ··· β 1) ˆ q 1 y 1 ) ! + F ˙ n 1 k ··· m, n ! , ) is isomorphic to the algebra of fully symmetrized functions β α i α + 1 ! n − y α ˙ k n m 2! ··· y θ y β ( 1) β y n 1 ( . k ˙ j m can thus be written as a formal power series m β ··· F θ m 1 α . The argument of ¯ − ··· )= y !
( := := β 1 p ··· ! A j A k β 1 m m ):= k m k m m dependence of (2.3) is not restricted by i m α α ··· ( i ¯
y, θ α k 1 2 ··· y ··· F ! ··· i i 1 β ··· 1 i 1 m 1 k y, y a i m, n θ ( α + + y θ ( + + y k F and ¯ i ? = = ··· y 1 i n n j β n F ˙ ··· α ··· 1 1 j ··· β 1 ˙ α product of functions 2) invariance requires a reality condition, such as ( ?θ ¯ ) are real functions of y ?y , ? θ ) can be expanded in terms of m ( m (3 m i α m α ··· ··· α ¯ y,θ A general element of The associative algebra product 1 ··· 1 SO The i (8) invariant ”contraction rules”: ) of the Heisenberg algebra [ˆ ··· 1 2 3 α θ 1 y, y α y ( (ˆ α y metrized spinor indices and antisymmetrized wherewehaveusedthenotation where the coefficients For convenience we shall use the shorter notation and the reality conditions Appendix C. to note that the F ambiguity in notation. for arbitrary elements in F F SO JHEP11(1998)016 .In (2.9) η (2.17) (2.15) (2.14) (2.12) (2.13) (2.11) (2.16) (2.10) to be an , 1) spinor . algebra , and τ F (8) Clifford ˙ α ? (3 ξ ¯ v = ˙ α SO algebra in (2.8) SO u ? , ?F +¯ , . ˙ β is equivalent to the i ˙ α 2) invariant formula α i , v :all θ i iθ α ∈A (3 u + : )= i SO bd i obeying the conditions [37] ¯ A y θ . ˙ ( α A k y i . , P, ··· )exp in † 1 =¯ − ) i . αβ ˙ θ V β j ,F,G P F = ¯ y θ ,τ ( ) iC 1) i + † ˙ ? α θ − ? 2 F ¯ k + y ˙ ( † α ( Y + ) k ( i β is then defined by declaring ¯ 2) covariant form y , Y ?τ G ij G 1) α A δ 6 ) (3 ) Y )= − elements 4) as the Lie superalgebra given by the sub- for Grassmann odd quantities ˙ G =( | α = U ( , y = † P, P † SO j τ (8 (¯ =( ) ξ + β − αβ E † † even ?θ η FG Y i i k ?Y ( i θ shs 1) )= = α + ··· F?G 1 − ,τ ( Y P i β † ( VF (anti-)symmetrization convention θ ) y α 4 τ α is a graded anti-involution of η ξη iy )=( τ Ud = 4 product rules (2.7)–(2.6) for commuting spinor elements can d β )= ? α Z y F?G ?y ( ( α τ τ (8) vector indices are automatically subject to unit strength anti- on a general element in y )= τ Y SO ( spanned by Grassmann product rule (2.8) for the anti-commuting elements ?G ? A ) Y ( We are now ready to define The F where the indices in the exponent indicate Grassmann parities. where the normalization of the integration measure is such that 1 particular The action of We next introduce the map The hermitian conjugation acts as an anti-linear, anti-involution of the provided that we define ( where we use theindices following of thetion, same and type all are automatically subject to unit strength symmetriza- decomposition rule for generalized Dirac matrices of the (associative) which can be written in a manifestly It is also straightforward tospecial verify cases the of (2.7) associativity we of have (2.9). We also notice that as space of symmetrization. The be summarized more concisely by the following manifestly algebra involution of the classical productand (2.1). (2.9) From it the definition follows of that the JHEP11(1998)016 + 8 i (2.19) (2.20) ··· .The +2)is (2.21) (2.23) (2.18) (2.22) 1 i k θ +2 ) k (4 4 )ofdegree P P m, n , ( Y,θ 8 ( i ! ··· +2 1 7 k i i 4 + P ··· representations (the ,... , P + +2)where 1 1 2) invariance of the map 8! i 2 6 , − k i θ , 5 ) 1 i ··· (3 + (4 1 , ··· i 4 P i 1 θ i SO +35 k ) ··· θ m, n 1 =0 ) + i ( P 7 θ i ) → . ··· m, n commutator of two elements in 1 ( , ) , m, n i 6 m ( i ? k k P 5 i 4) is closed under the bracket. The L m, n i ··· | Q?P. 1 | L ,k ( 7! ··· 1 l m, n ··· i 4 1 ) ( i (8 1 i − l − i P =0 + k ∞ ··· P E m + | M k 1 P ˙ 1 6! k α M i = . Also note the Y,θ 1 5! +2 ··· ijk α ( P m k 1 + shs θ 7 Y ˙ 1 4! α ) + +2 =4 P?Q n ij 4) = i k n | β θ ]= 4 θ + ) l ··· = ) P (8 )+ 1 m m, n ? β E ( ] +2 ,L P P k m, n k ijk 4 m, n ( m, n . shs ( i 1) λ L ( P i ij [ − 1 2 3! P P P k P,Q ( [ 1 2! k )= ) mod 2. Hence the bosonic (fermionic) fields carry an +3 +1 +2 4 n is the vector space spanned by the polynomials 1) k k k is Grassmann even, the k k − + =4 =4 =4 =4 L X X X P n n n n (8), that is, the 1, 28 and 70 = 35 λY, λθ X + + + m + ( =( byafactorof m m m m † SO + + + +2 α is ( k
Y k 4 i k P i ··· 1 =0 ∞ ··· i X k 1 n i ˙ i ’th “level” β n 1 ˙ 2 k β ··· 1 ··· ˙ is a complex number. Hence β 1 ˙ 4) receives contributions only from odd numbers of contractions between the β m | λ )= α m + 2)’th order homogeneous polynomial in (8 α ··· Notice that since From (2.13) and (2.16) it follows that The expansion (2.19) is a sum of homogeneous polynomials 1 The reality condition in (2.17) implies that the expansion (2.19) can be expressed in terms of E k ··· +2,thatis (see footnote on p. 5). For example Y,θ 4 α 1 commutator has the following schematical structure: ( α P k α , which multiplies where 4 8 and 56 representations). ? shs where we haveP used the notation defined in (2.4). The Grassmann parity of and with Lie bracket defined by where the generators. general solution to (2.17) is P Y even (odd) number of spinorrepresentations indices of and even-rank (odd-rank) antisymmetric tensor τ a(4 JHEP11(1998)016 1, m , 776. , (2.28) (2.24) (2.27) (2.26) (2.25) =0 .(The k 8 n -invariant =11 1thereal =8super- =1super- subalgebra 1 entries of τ 4). In fact f 2 | . − k ≥ 4) containing N ≥ +3 3 4) irreducible N | (8 | m s k E (8) (8 = (8 ,... . E b 2 2 shs SO , N (Γ)=Γ OSp shs × τ + =1 = 0 multiplet, which is = (8). . k (8) † of homogeneous degree , i ! , 4) contains the gauge fields θ 792 and SO 8 | (Γ) SO , ! θ ). Hence, for (8 6 forms an +1 ,k n 8 8 E 2 − ) 12 k n =1 k 2 4 2 )and L k M , f 1 ! ¯ y shs forms an irreducible representations spanned by fermionic generators is and real dimension n N ! y, n k n =1 2 m θ = L 2 m b 1 (5 + 16 Γ 3 3+2 = Y N k 8 3 2+2 s − θ, ... , Y 3 Γ), where Γ is the hermitian and 256 k − +1 4 ± , k k 2) generator 4 =
form a closed subalgebra of 4 , (1 f k
0 1 2 4 (3 =0 =0 2) spin spanned by bosonic generators is given by X L N n ,Y ? , 3 ? =0 k X n } ij , (2.23) shows that SO = = (3 +2 L i θ k k b b k k = 4 ,θ SO N N f k Y Γ 2, the 512 physical states described by the gauge fields form N { is the diagonal subalgebra of the 4) the physical states coming from the gauge fields (see sec- ≥ | )= 0 k while the truncation of the theory to the gauged 2 states are needed to form supermultiplets. The k (8 L ) (see Appendix B for conventions and normalizations). The L / − E ( 1 , (8) with ,R R 8 (8) ≤ shs θ (4 s SO dim SO 4) is the maximal finite dimensional subalgebra of denotes a hermitian monomial of ( Sp ··· 1 2 | × 1 n × (8 4) with generators θ θ = 0 and arbitrary ' | 2) + m l , (8 Y 2) respectively. The space spanned by (3 E (8) OSp Γ= , n In gauging Thus the higher spin gauge theory based on Equation (2.23) shows that For (8) chirality operator (3 (8) subalgebra of ' = 8 supermultiplets (consisting of two irreducible 128+128 sub-multiplets related SO SO shs 0 dimension of the subspace of SO multiplet also has a total of 516 physical states, while the CPT-conjugate, contains a total of 256 physical states. The dimension grows rapidly; for example spin is the eigenvalue of the of As expected of of L SO given by and the real dimension of the subspace of tion 6 for a detailedtable spectral 1. analysis) are At given any [11, level 12, 36, 37] in the N by CPT conjugation such that the full multiplet is CPT invariant). At level however, spin representation spanned schematically by the polynomials where and gravity theory contains the gauge fields of the diagonal SO JHEP11(1998)016 , , ··· ··· ··· k 28 28 − =8 ⊗ ⊗ (2.29) +2 N +35 ) ) s + 2 4) which | ,s ,s (8 2 3 4) singleton E | +1 +1 + s s s (8 4) gauge fields. ( ( shs | representation: D D (8 4). In particular, c 4)isbasedonthe +1 2 | | E OSp ,... ,... s 5 5 2 , (8 , , (8 3 3 , , E i E shs X X 2 1 c is the minimal energy =1 =1 + 8 56 28 8 1 4) actually requires that s s s 0 | shs shs 2 in the 8 ⊗ E (8 ) − ]+ ]+ E 1 2 Di − + , s 1 1 8 28 56 35 +35 2 35 35 shs (1 + 35 D ⊗ ⊗ h is the maximum eigenvalue of the ) ) ··· ··· ··· ··· ⊕ s ,s ,s 4) gauge field (that will be analyzed in i − | s 2) for which 8 +1 +1 (8 , 4 +35 9 E s s ⊗ (3 + ( ( ,and D D 0) 04 shs 4) supermultiplets labeled by the level number , | SO 7 2 1 2 1 span the spectrum of the M (8 ( (8) and a singleton = = = 1+ 1+ ≥ 3 S D S S h s ⊗ ⊗ SO OSp i i i 2 5 ) ) 56 28 8 1 s c c = 8 8 8 ,s ,s 4) [36]; see end of subsection 5.1. Not all higher spin (8) content of the symmetric tensor product of two − | ⊗ ⊗ ⊗ Di (8 1 8 28 56 35 2 ) ) +1 +1 +35 SO E 1 2 1 2 0) ⊕ + s s , , , × ( ( 1 2 shs ( (1 (1 D D 2) [ [ 3 2 Rac in the lowest energy sector. From the oscillator representation , D D D ,... ,... 4). Therefore also the symmetric and the anti-symmetric tensor (3 | 1 4 4 12 28 828 56 35 1 , , 2 2 ⊗ ⊗ ⊗ , , (8 X X ) denotes an UIR of M SO 2 1 representation of ) ) ) 56 E s c s =0 =0 s ,s s s 8 8 8 0 − 4) given in Appendix C it follows that the space (2.29) actually forms shs The E | = = ⊗ ⊗ ⊗ ( 0 +35 ) (8 1+1 8 1 2 0) 0) + D E , , , 35 2 1 1 2 in the 8 Physical consistency of a gauge theory built on ( (1 ( 1 s shs ...... s D D D 0 1 2 +1 − \ s s k spin operator fact that the particle spectrum of the detail in section 6) fits into the symmetric tensor product of two decomposes into infinitely many defined in (2.22). The states with the complete particle spectrumdimensional forms algebra a unitary representation of the full, infinite of products of two copies of the space (2.29) form UIR’s of where h h h Table 1: singletons. This product is a unitary irreducible representation (UIR) of eigenvalue of the energy operator the decomposition of the symmetric tensor product algebras are admissible in this sense. The admissibility of aUIRof supermultiplets [11, 36, 37].Rac Each singleton supermultiplet consists of a singleton JHEP11(1998)016 (3.2) (3.3) (3.4) 4): | 4) given in (8 | E . The latter is (8 5 shs µν,αβ r OSp = 2 sector one must contains contributions , s µν , ? R ] µν,αβ ν R ) (2.30) v dx ε, R ∧ 56 ω. µ 4)-valued function. ⊗ 4)valued gauge transformations , while ) (3.1) | − =[ | ) dx (8 1 2 (8 = R ,s µ,αβ E Y,θ E † ε ( ω ) µ ω +1 := shs ω ( shs s µ with the Riemann tensor ( 4) leads to the UIR’s of dx ,δ | D 10 ? ] (8 + µν,αβ E 4)-valued 2-form is defined by )= v R | ω, 8 − (8 1) gauge field ω, ε shs , Y,θ ⊗ E [ ( R?ω, ) (3 )= − ω − ω?ω, R ,s shs is an arbitrary ω SO ( 4) to be a suitable algebra for a supergauge theory of dε ε − | τ . +1 1 2 (8 = s E dω ω?R ( is the ≤ ω D ε s = = ω ( δ shs connection 1) valued curvature R ,... , 5 2 dR is the exterior derivative and the wedge products are suppressed. 4) subalgebra of , 4) | (3 supergravity. However, the determination of the particle content of | 3 2 µ , X (8 ∂ 1 2 (8 SO µ E = s AdS 4) gauge theory requires a lot more analysis; there are auxiliary gauge dx | OSp shs = that are bilinear in gauge fields corresponding to generators whose commutators contain (8 = E d 1) generators. , ω?ω The curvature of We thus expect shs We shall temporarily set the gauge coupling equal to one. We shall discuss the gauge coupling (3 5 where not confuse the The one-form master gauge connection where the gauge parameter is by definition Grassmann even and takes its values in the algebra under the 3.1 The 3. The field content The curvature transforms covariantly under and the gravitational coupling in section 7. Also note that in the spin fields which must befields eliminated and algebraically table in 1 favorset makes of of it fields the clear with true that spin dynamical one gauge has to couple the gauge fields to a finite higher spin the from SO by definition the curvature of the table 1. ThereAnother are derivation a is number given in of Appendix ways C to (see derive (C.9) this and result. (C.10)). See, for example, [11]. JHEP11(1998)016 + q − 2 + (3.5) p 8 and i (3.6) ··· , = 1 , +35 i denotes αβ ) s θ 2) y + ) )andits ◦ ,n − αβ , 2 ) . m, n 1) ab − m, n ( ( σ 8 1) + i ( − m, n k m i ( 1 2 ( )ofspin ··· − k 7 ··· 1 k i i i 1 i ,n + ··· ··· 1 ω ··· - p, q 1 + 1 ,n µ,i 1 i i 1 ( i 8! 6 ω η i θ − k ξ i 5 ) ··· i + +1 ··· 0) := 1 m m , represents a component i ··· 1 , 4 i ( 1 )+ i θ )+ i m k η ) i ··· (2 ( m, n θ 1 ( ,... ) k ··· i i ab 7 1 2 according to the following rule: i θ i m, n ··· , σ m, n ) ζ ( 1 ··· m, n ( 1 i ω 1 k ( , k , i i ζ m, n i 6 ...... ˙ α ( i ω ··· ··· 5 ¯ y 1 ··· 1 i m, n 1 i 7! i 1 α ( =0 i ··· η ...... ξ 4 y ♦ × 1 i ω i ˙ +1)+ + α ··· 1 ω α 6! 1 +1)+ ...... ) i )+ ♦ 1 ×× 5! ijk ,n a denotes a generalized vierbein; a m, n ω + θ 1 σ ,n ,n ) 1 4! ...... + 11 +2)+ ♦ ♦ ij • ××× − i θ θ ) +2 +1 ) ...... m ♦ ♦ )+ ×××× m, n ( ( m, n m m k i ( ( ( ...... m, n ♦ ♦ has the expansion k ijk ××××× ×× × k k 1) := ( ··· m, n i ( i i m, n 1 , ( ω i ( ··· i ··· ··· ij ω ζ 1 1 ...... 1 1 3! ♦ ♦ i (1 ×××××× × × × i i ω ω ω products: a + ζ ξ η 1 2! σ d ? ...... ♦ ♦ denotes an auxiliary generalized Lorentz connection and the × × × × +3 +1 +2 k k k k ♦ d d t , we can decompose the gauge field ):= ):= ):= ...... =4 ) into irreducible Lorentz tensors ♦ ˙ × × × × × =4 =4 4) valued connection one form β X X X | n ˙ n n = 1 component; a =4 (8), while the fermionic fields are always in 8 and 56 rep- α n + + + ¯ d d t 6 y (8 ...... s X ? ♦ ˙ × × × × × × + m m, n m, n m, n β m m E m, n ˙ SO ( ( ( α m n ( + + + k k k ) d d t i i i k i
shs ab ··· ··· ··· ··· 1 1 1 σ 1 d d t (¯ =0 ∞ a, i 1 2 X k ab, i ab, i µν,i Each entry of the integer grid, i d d t 1 R 2 ?ω ?R ?R )ofthe d d t 1) θ 2) 0) , ; 2) := )= denote the spin , , , d d (1 t Note that the bosonic gauge fields are always in the 1, 28 and 35 From (2.19) it follows that ? a (0 (2 (0 Y,θ m, n σ ab ab ’s denote the remaining auxiliary connections. ( ( ab d t ¯ σ σ ¯ ω σ curvature a generalized gravitino; a × the resentations. By defining by expanding the following representations of Figure 1: ω JHEP11(1998)016 ) ) + dy- m then ,... (3.8) (3.9) (3.7) ( ,s,θ 1) are 2 k n i 2 , − 3 2 ··· 1 6= − , i ζ ,m s m It is impor- ( 1 a =1 − ω s m , while the cor- ( by means of the k k ) defined in (3.6). )( i a s ··· +1)and 2 1 i , ,... ζ 2 ,m (0 , 1 are dynamical, as will 1 k by (B.7)). If ,... , 3 2 i ,... , 7 2 ˙ α 4 , ··· − , 1 , 2) are related by hermitian i , |≤ 5 2 3 µ,α η m , 2 n , ( =1 ± ω ’s in figure 1) are 3 2 +1) and k i generalized Lorentz connections s − ♦ , in terms of the remaining, ··· =2 = |≥ 1 ,m i m )( n | ζ s m, m 0) and 2 +1 − , = s, m ’s and ’s in figure 1): ’s in figure 1): m ( (0 1, grows rapidly with p | ,s ,s k k (2 × • ◦ i i ) k ) )with i ≥ )( ··· ··· ), while 12 1 1 ··· i i ,θ ,θ 1 k k i ζ η . 1 2 1 , ξ m, p ) ( − − m, n, θ k i ( then ,s ,s ··· ω 1 1 3 2 n i 0) and η (denoted by auxiliary gauge fields − (denoted by m, n, θ − = s, ( s ’s in figure 1). a s ( (2 ( has been converted into a flat index m ω ♦ a a k i ω ω µ )and ··· 1 i ξ p, m ( , as well as to yield dynamical field equations for the latter. (The k i ··· 1 i ξ generalized Weyl tensors (which is given in terms of the gauge field ) and their hermitian conjugates as the . In particular we shall refer to the auxiliary gauge fields µ,a 3 2 generalized gravitini generalized vierbeins ω ,... 3 1) and ≤ , which are real, and their hermitian conjugates, and s − As for the dynamical gauge fields, we categorize them as follows: The auxiliary gauge fields (denoted by )the )the gauge fields, as given by (2.28) for i =2 ii k ,m s Auxiliary and dynamical gauge fields and gauge symmetries. are called the tant to notice that theL number of algebraically independent components of the level ( (and these are denoted by As a result only the gauge fields conjugation. The chiral components 1 and their hermitian conjugates.spin Notice that there are no auxiliary gauge fields with real or purely imaginary (depending on vierbein where the curved index these irreps are complex. If be shown in subsection 6.1; see also figure 1. responding supermultiplets in tablesymmetry 1 each alone contain cannot 256 remove +some 256 all (curvature) states. the constraints In that unphysicalof fact, serve states. gauge the to gauge determine One fields, algebraically alsonamical known a gauge has fields as certain to subset impose curvature constraints also incorporate thegroup). spacetime diffeomorphisms The into the constraints gauge actuallyeralized involve Weyl tensors all curvature components except the gen- JHEP11(1998)016 (3.13) (3.12) (3.15) (3.14) (3.10) in figure 1 ): ,... , ,... , ,... , ? . 5 2 7 2 2 ,... , , , , 4 (3.11) 0) 3 2 5 2 3 , , =1 = = (0 The auxiliary gauge symmetries |≥ 6 i =2 n ··· 6 1 − i with parameters ω m ,s ,s ,s | with parameters ) ) ) ,s with parameters ) 13 with parameters ,θ ,θ ,θ does not act on any of the dynamical gauge 1 2 1 2 1 , 0) and , ) − + − ,s,θ (0 2 ,s ,s ,s ij 1 5 2 3 2 − ω s − − − ( m, n, θ ( ε s s s (8) gauge fields (denoted by ( ε ( ( ε by definition has a nontrivial action on a dynamical gauge ε ε SO =1 s auxiliary gauge symmetry local fermionic transformations generalized reparametrizations generalized Lorentz transformations generalized local supersymmetries which are real and their hermitian conjugates and their hermitian conjugates, and and their hermitian conjugates, The auxiliary gauge symmetries have parameters We also differentiate between dynamical and auxiliary gauge symmetries. A )the )the )the )the ) the two spin By definition the commutator algebra of dynamical gauge transformations closes on the dy- i 6 ii iv iii iii generate algebraic shifts innot the determined Lorentz irreps in of termsThe the of auxiliary auxiliary the gauge gauge dynamical symmetries fieldswhere gauge that can the fields are therefore undetermined by be irrepsfields solving fixed are are the uniquely set given constraints. uniquely equal by in fixing to terms a zero of such gauge the dynamical that gauge the fields. auxiliary gauge and hermitian conjugates. As(3.8)–(3.10), for we the separate dynamical them gauge into symmetries, in analogy with fields. The dynamicalalgebra gauge of symmetries the therefore dynamical constitute equations the of local motion. symmetry dynamical gauge symmetry field while an namical gauge fields. JHEP11(1998)016 4) | (8 (3.21) (3.18) (3.16) (3.17) (3.19) (3.20) (3.22) 0). E , (0 ) [28, 27, shs 6 i 4): | ··· 1 Y,θ i (8 ( ε are involutions E φ π . , . i i shs i θ θ θ , and ¯ 0) and − θ is given by the gauge , π )= )= , algebra: i i (0 φ θ θ Γ . ij ? )= ( ( π, π i . ε ) ? ¯ θ π ) ε ( sector shown in Table 1. To ) ( θ ω , φ 1 2 ( ¯ π ( ) idem ¯ ¯ π π ε ≤ , ( φ? ˙ α s = ,π ¯ π ω φ? ¯ ,π y ˙ † α ˙ , D α ) − y + ) y φ? φ . − ( G − A ( φ )= )=¯ 14 )=¯ ω ˙ ˙ α α ˙ ω?φ α y y ?π y (¯ (¯ , (¯ ) − ε?φ θ ¯ π ) ε?D F 4) on the master field φ = | ( dφ ( = π φ (8 π = ε E φ ,π δ φ ω ,π )= α ω , α )=¯ D y shs y α D ε φ − y δ ( τ )= F?G α ( 4)-valued function. The covariant derivative )= )= | y α α π ( y y θ (8 (8) chirality operator given in (2.24) and does not quite transform in the adjoint representation due to the ( ( π E (8) gauge symmetries with parameters -operation in (3.20). For this reason we shall refer to the ¯ π π φ are arbitrary elements of π SO SO shs G 4) requires the inclusion of the spin | and is an (8 E ε F The local fermionicgeneralized transformations Lorentz (3.14) transformations (3.12). are Thearriving the role at of fermionic the all correct analogs these number symmetriesin of of in section degrees the 6. of freedom will be analyzed in detail The maps in (3.17)–(3.18) are also involutions of the In addition, it is useful to define the classical algebra involution The representation of Notice that )thetwo shs v transforms as 3.2 The quasi-adjoint representation As explained at the end ofon section 2, the construction of a unitary gauge theory based where this end, one introduces a Grassmann even, zero-form master field where Γ is the where transformation presence of the ¯ 26, 25, 24] in the following infinite dimensional representation of of the classical algebra product (2.1) defined by JHEP11(1998)016 1 2 ¯ π τ 8 i ≤ ··· 1 s in the (3.24) (3.25) (3.23) i . Hence, θ π ) τ 7 , i and ¯ m, n 6 ··· ( i 1 8 i π i ··· The closure of θ 1 5 ··· i ) i 1 θ i ··· 7 ) 1 C i θ 1 8! m, n ) ( 2). As a consequence, the 7 m, n i + , 4); see the discussion at the ( | 6 ··· 4 i (3 i 1 m, n (8 i ··· ( ··· E 1 1 5 C i SO i i 1 7! θ C ··· ) shs 1 1 i 6! + φ. C , representation. + ? 1 5! ] m, n Γ ijk 1 ( ij θ ? 4 + ) θ ,ε i ) ) i 2 ··· † θ 1 [ i ) C δ m, n ( C ( m, n 1 = 4! ( π 15 2) symmetry of m, n ijk ij states that arise from the two-singleton states , φ ( + ] i C C 1 2 (3 2 )+ quasi-adjoint 1 1 2) invariant as opposed to the anti-involution 3! 2! C ε C , ≤ SO . This reality condition is necessary to obtain the (3 ,δ = s φ m, n 1 φ ε ( SO +3 δ +1 +2 C 0 [ 0 0 as the k k k ≥ ≥ ≥ φ k in terms of a Majorana spinor of =4 =4 =4 0 X X X φ are not n n n =4 ≥ π m, n m, n m, n − n − − X 8 and ¯ − m m m has the expansion m, n π m + + + follows from (3.19): C φ =0 ∞ X k 4) on , we cannot express | )= ω (8 The general solution to the condition (3.16) is Let us emphasize that the main reason for the introduction of Γ, We note that the reality condition stated in (3.16) differs from the reality condition used in [28]. The involutions E Y,θ 8 7 ( C shs where the field tabulated in table 1 andunitarity that requirement must discussed be at included(3.16), which in the involves the end Γ theory in of in a order section crucial to 2. way, satisfy is the engineered The to reality be condition consistent with in the invariance condition imposed on sector correctly produces the spin representation carried by definition (3.16) of the quasi-adjoint representation is to ensure that its spin correct field content. unlike end of subsection 4.3. full theory does not possess the external JHEP11(1998)016 C (8) and y (3.30) (3.27) (3.26) (3.28) (3.29) )= SU (8) and C obeys ( )isgiven θ and ¯ ). In order SO C , contains all components φ Γ ¯ y ππ ( m, n θ π ? real scalars of Γ) C ( π k , ? i − ) † ··· Γ 1 )=¯ i C ? which is eliminated ( φ )and φ † +35 n, n m
V
φ ( D =0 ]) 2 + )= SO ]) , respectively, we have Z and ¯ ¯ φ z V,Y
ω?φ φ Φ ˙ α α =0 and f Φ[ id ¯ − y ) that projects onto anti-chiral (i.e. z V Φ[ ω,φ | , represent an expansion in ( , ] )= ] φ Z φ W = , ( z, y product contractions between the aux- and ) ( ... = G ω,φ ω,φ )=0and κ =0 ω,φ ? [ (Φ) + A [ ( ) W ( Z ω =0 19 φ | V f ( ¯ W π U 2 α Z W ( . Hence the obtained FDA of the form (4.1) | ( R with the auxiliary, commuting spinors ? V + φ W Φ by , idz Φ is by definition obtained from ?κ f f A ... , f Φ b ˆ + − f A U, Y =0 , and the internal spinor variables )+ ˙ Φ )of )= )= α ? ] with respect to + ] z space is symplectic which implies the existence of a φ i and ¯ Z y, θ ω?ω, f ω,φ ) result from the leading, classical terms in the A? A?A ω,φ ω,φ ( ( A Z ,θ ( ( ˆ y, space is isomorphic to the space of the commuting spinors f and ˙ φ F ω W α = = )= z f f ω,φ f α Z Z Φ A ( z ˆ ˆ f Φ f ]andΦ[ =[¯ f ω,φ )= ( )= ? ] W ω,φ i [ f ω,φ ( ,θ )and Z, Y W ω ( α f z and defining the associative, manifestly one finds [ allows one to define nontrivial ω,φ † ( ?G Φ ˙ ) is some function consistent with the extended integrability condition (4.5). α ˆ α f W y ) z V f then leads to spacetime constraints of the form (3.31). The fact that (4.7) leads to a FDA of the form (4.1) with a nontrivial This corresponds to and and ¯ := ( Z, Y [27, 28] V = 0 yield nonlinear expressions in independent) components, i.e. Φ ( ˙ A α α b ˆ ¯ from the higher order contractions of thederivatives auxiliary of spinor variables. The latter involve ing the set of generators ( z Secondly, the fact that y Z represents a nontrivial deformation of allows one to construct a special function relies on the fact that the where A F invariance can be used to fix the gauge of the auxiliary spinor variables, while the gauge symmetry). Denoting the solutions for spacetime components of iliary spinor variables where 4.2 Extension of the higherThe spin extended superalgebra associative algebra of y product of the auxiliary spinor variables. Using this f JHEP11(1998)016 Z d and (4.14) (4.16) (4.13) (4.15) (4.17) (4.12) (4.18) (4.19) . and the y?z 9 αβ by setting , 2) ¯ z , b , iC and we have A ˙ -space, which α 2), obey (3 − ¯ z p z? ∂ of one-forms in Z , τ. β , , ∂ ,¯ ˙ ˙ α ˙ (3 α Z α α , , SO , ¯ z ¯ , z α ¯ ˙ ˙ z ˙ β β β d ˙ iz i ˙ ˙ is used. ˙ . Y d α α α and α SO z?z ¯ z q † − − α i i + i π = d z : B , 0 and − β α † Z + − − )of = S ) )= )= ∂ ˙ ˙ ˙ ˙ α α = 0 β β ˙ β † α α ∂z ¯ z ¯ ?d ¯ ?Z z ) ¯ y ˙ z S α z z idem ¯ ˙ α ˙ ˙ α dz α α α p z (¯ ( α ,d y z z A α dz dz =( p ( of one-forms in =¯ =¯ =¯ ˙ α dz ,Y ˙ ˙ ˙ ˙ ¯ , z 1) β β β α := ˙ ¯ ¯ ¯ z α y z ¯ z − )) αβ ¯ ,τ z =( d ? ? ? Z 2) covariant form ˙ d α ˙ ,τ ˙ ˙ α , α α α iC ¯ z +( ¯ ¯ ¯ y z z α F, + q (3 − + z α dZ and α β Z, Y, θ z 20 ( Y α dZ α ?B SO ,d α )= )= ) space form of total degree F , , , p p ˙ α dz α ( dz Z = . Therefore, if we set F z z A Z . Notice that the hermitian conjugation acts as b (¯ αβ αβ αβ ( α Z Z d A = = x, Z product imply the Leibniz’ rule F ( ¯ ¯ π d π i i i id α β π ? 2 = Z − + − 1) for each contraction of type − α ?Y . Note that (4.16) has the correct hermicity properties )= β β β − q α α F?dZ z z y , = ?F )) = dZ α α α α dz 0 , z y z = defined in (2.15) and (3.17) are extended to ?B z ˙ = α ∧ p ,Z − z = = = π µ 0 ?S 4) valued ( | . As a particular case of (4.11) we find (2.12) and A S ?F p β β β Z, Y, θ ( b ) is a purely imaginary Majorana spinor of αβ dx ( A (8 α F ˙ Z α )= )=¯ p − E ?z ?z and ¯ d ?y ¯ F z ˙ iC α α ( dZ α α 1) α z z − π = π − ( (¯ z y z d , ( shs − , β µ α ( π π Z τ z Z d dx α − p Z ∧ is an := ( α is an arbitrary element of = p ?F α dz β F 0 Z F S space can be generate by the inner, adjoint action of . See Appendix D for further details. ?Z By definition the basis elements This leads to contraction rules analogous to (2.7) with the only difference that then from the contraction rules (D.1) it follows that the exterior derivative The maps ¯ Our conventions differ from [28] where the reality condition ¯ y α Z 9 -space by Z ¯ z? where where form a purely imaginary, Majorana spinor there is an additional factor ( normalization is such that 1 an anti-involution of and that the associativity of the in where Z which can be written on the manifestly and declaring these maps todefine be involutions the of action the extended of classical these algebra. maps We also on the basis elements defined JHEP11(1998)016 ) µ W ,R µ (4 (4.21) (4.26) (4.25) (4.23) (4.20) (4.24) (4.22) dx given by Sp 4) gener- | b A )= (8 E ) factor com- ,R shs . Z, Y, θ (4 ( )) Sp ¯ y W 4) of | , , κ, z, ) ) (¯ , (8 θ θ κ ; ; ) where the extra , Γ ( , , ¯ ¯ z y ) ) ) ? .This τ ˙ , θ θ ,R β OSp − ˆ ) gauge symmetry which does z, ; ; P. ¯ z κ?F? W ¯ ¯ ˙ z y (4 ( α − y, , − )= W, (Φ ) ,R z ; ; have inner actions in ˙ =1 ¯ π α z, y, ¯ ¯ z y Sp − )) π = ; ; (4 ¯ )= y ? π κ ¯ ¯ z ˙ y − † α y, × )as = = ) F and ¯ ¯ Sp z is set equal to zero. z, y − y, z, z, ( † ˆ † ( 4) i 4) is defined by the extensions of (2.17) P ˙ κ? ( ( ( ( β | | ( ¯ and ¯ π κ z, y Φ Z ¯ z − ( ( (8 α (8 Φ+Φ 4) is Grassmann even and it furthermore π κ τ κF κF κF κF | z E 21 , (8 = = β , W?W, OSp E z shs )= W? )= )= ˆ . κ α =exp( P, α θ θ − b z ? ?κ − † ; ; A y d − shs W, W (Φ ) ¯ ¯ y y ) ) α element )) Φ − π 4) of θ θ | in d dW y, y, ; ; b iz A )= ; ; ¯ ¯ , y y (8 z, y ˆ ¯ ¯ ˆ z z = P ( P κ?F?κ, ) are defined by E )= y, y, ( κ z, z, ; ; Φ= R τ ( ( =1 ¯ ¯ z z W (Φ ) = ¯ d shs D ( )= τ z, z, τ ( ( Z, Y, θ F ):=( ):=exp( ( 4)-valued spacetime connection one-form F F κ?F κ?F ¯ y κ?κ | π z, z, y (8 4) and generates a fictitious | (¯ ( E κ κ (8 d shs OSp is an arbitrary element in and Φ are Grassmann even. The extended curvature 2-form and covariant F W Now the enlargement The extension of the finite dimensional subalgebra Using (4.11) we find We next define the special which in turn implies that the involutions exterior derivative such that the initialthat conditions defined in (4.6) indeed obey (3.2) and (3.16). Note obeys and Weyl zero-form Φ( ated by quadratic elements is given by and (2.18). Thus an element where mutes with not arise in the spacetime FDA (4.1) where 4.3 The extended field content The extended factor is generated by the elements JHEP11(1998)016 W and (4.28) (4.34) (4.33) (4.29) (4.32) (4.30) (4.31) (4.27) W represent the = 0 there are κ. Z ... space connection . Combining with ) expansions of α ¯ z to transform in the Z S ... , z, S S, − ... , κ?V ? + = + − φ † )= ω φ α ε δ D S Γ= ( , V. θ contractions of terms in ) = = . 4) valued function and the vari- − ? ε | , ? ¯ ππ (ˆ ] =0 ˆ =0 ε. ? π (8 = ] S, S ˆ ε ¯ π Z Z | | E ˆ † ε S?S. ?V ?κ − ? 1 4 Φ Φ S? ˆ ε Γ )= V, Φ d D shs α [ = − W, )= S − [ ( − S 2 22 ( ˆ . Φ indeed transforms as a ε − τ ? ] Z is an ˆ ε?S ε 1 V, V d (in particular there are nontrivial cross terms V ˆ ε V?V ε? d ε , κ, κ − 2 =ˆ as follows: z space the Grassmann even connection one-form ε = − [ˆ ... , δ ... , = δ S ) introduced in (4.3) by ˆ Z ε V ˙ V + )= α Φ=ˆ ˆ ε δ W Z and ¯ )+ ˆ δ ε ¯ 4)-valued gauge parameter and the ω V 4), that is ]= ˆ iV , τ ε V d | δ S?S ε | ω κ?S? , 2 ( z δ ( δ ˆ ε α (8 − (8 τ δ R V +2 1 E E = ˆ ε contractions): 0 it follows that δ = S space “covariant derivative” =( y =0 Γ= α shs d -¯ shs Z α =0 z | = Z iS Z V and Φ were given in terms of the initial data (4.6), then setting S W space one-form defined in (4.15). We take − ˆ ε R| is an and ¯ δ W ?S?κ Z y )= Γ - =0 α z κ Z | S ( ε τ is the Φ equal to zero would yield a nonlinear FDA of the form (4.1). =ˆ 0 D ε S From We also define a We proceed by defining the It is important to notice that when evaluating (4.26)–(4.27) at From (4.16) it then follows that and with components and Φ. Thus, if R contributions from the quadratic terms thatΦ involve which are higher order in where (4.22), we find ations obey the algebra [ transform covariantly under the extended gauge transformation contributions from the higher order Taylor coefficients in the ( V whose curvature is related to coming from adjoint representation where where the transformation parameter ˆ one-form JHEP11(1998)016 (4.42) (4.39) (4.40) (4.35) (4.41) (4.36) (4.37) (4.38) and its ,where ¯ k k κ and ¯ κk , ) ¯ κ ? , (1 + Φ ) 2 . We shall show the integrability ?S, † ε ¯ z , ) (ˆ Γ 2 ? , respectively. The addition of ] id ¯ π κ κ?S, κ , 2 2 ˆ ε dz ? ¯ z ) ) space of the higher spin field theory dz d W. , Φ ρ space =( W i ? Γ)+ W, ] = ( − ˙ Γand¯ [ ) α = 0 and (4.34). Finally, (4.37) follows from 23 x, Z ¯ ¯ κ z π κ Γ= Φ γ ?κ − d )= ? ? ˆ ρ ε x, Z ε, S ∧ dz , ) ?κ ( ( Φ d ε? S?W, ˙ ˆ α ε ) ) S ∧ -transformations by ¯ z ( − − S ε S = =[ˆ β d ( ( ¯ π (1 + Φ Φ 4)-valued gauge parameter. The equations are also 2 dz ¯ Φ=ˆ π ¯ | π = 2 ¯ z δS 2 δ ∧ ? δW d 2 (8 ¯ z α dz E W?W, W? W?S idz d dz = = = d shs ) and by making use of (4.22) and and Φ= Φ=Φ dS d α dW to the set of generators of the associative algebra, however, would give rise S?S dz S? S?S ¯ k ( ∧ α S? dz = = is incorporated into the ˆ ) chirality operators in higher spin algebras have been discussed in [37]. 2 is an arbitrary µ ?S 10 N ε ρ dz ) (2 = The full set of field equations (4.35)–(4.39) can be derived from (4.39) and either are Kleinian operator [21], instead of our In the original formalism [27, 28] the analog of (4.39) is written by using SO µ ¯ k 10 S?S in section 4.4. Apartare from that the they integrability preserve the the representation crucialthey properties properties (4.24)–(4.25) are and of invariant (4.30) under these and the that equations internal gauge transformations hermitian conjugate k, 4.4 The equations of motion in (4.36) (or vice versa) by(4.39). combining (4.36) with the covariant spacetime derivative of which in turn follow from where manifestly invariant under spacetime diffeomorphisms,ing since only they spacetime are differential formulated forms. us- δx In fact, the general coordinate transformation where ˆ to new dynamical gauge fieldsan which are unwanted in attempting to reproduce table 1. The use of (4.36) or (4.37).for To (4.37). begin with Next (4.35) (4.38)( is follows the from integrability (4.39) condition by exploiting for the (4.36) associativity and property The integrable equations of motion in ( are [28] JHEP11(1998)016 2) , )are (3 (4.45) (4.44) (4.43) (4.46) K SO ± (1 1 2 . Also notice is an 0 ? ] as functions of ϕ = A , A ? ± ϕ ] have been inserted P 0 )= .Thustheresultof ε, ϕ ϕ Φ κ S ( 2) spinors and vectors and τ , ε, product (4.11) invariant =[ˆ ϕ = (3 ? A . The reality condition on , † =[ˆ ,where ϕ 0 Γand ϕ 3 0 dZ , ˆ ε K SO κ , Φ Φ 2 ˆ ε ? ). For example, the linearized A = , 1 ϕ ω 1) invariant constraint † − , ( A , ) which leads to a constraint of the ,δ K R Γ 2) vector. eq. (4.43) is manifestly ¯ iP y (3 i ? , , ] =0 − -invariance and gauge invariance. As ,δ while the internal transformations leave + 0 (3 τ (0 SO )= α = Φ φ ϕ 0 ε, ϕ K . This shows that the real and imaginary SO 24 2) Γ-matrices, = dZ ( , τ K 1+ K? , , =[ˆ and anticommutes with =0 (3 0 0 Z 0 = ¯ ϕ | Z is an ˆ ε 0† 4) curvature 1of SO ,a | id ?κ and ImΦ Φ A The extra factors of ± (8 0 ϕ = E and Φ =ImΦ =0 =ReΦ Φ =1and 5 a ε ,δ ? 2 ϕ , ϕ ϕ ,ˆ shs ? )= 0 ] + K S?S A P W 5) are the 1) invariant constraint expressing ϕ , 2) transformations that rotate , ( 3 = , ε, S τ , (3 )=Φ 0 )involvesΦ 0 2 (3 , ω = =[ˆ ( 1 (Φ SO , † SO τ R S ) ). The external transformations leave the ˆ ε A Γ, obeying α δ =0 ϕ κ each contain half the number of degrees of freedom of Φ κ dZ A 0 1) invariant, quasi-adjoint master field Φ. 1) invariant constraint (4.45), the associativity and (4.37) is the following ( obeys , , 0 commutes with A := (3 (3 K invariant (as can be seen from (4.14)). K These requirements do not, however, fix the right side of (4.39) uniquely. In For example, (4.39) is the result of the The interaction ambiguity amounts to the degrees of freedom associated with SO SO implies that ReΦ α 0 where Φ form (3.31). order to describe the interaction ambiguity we start from the identity that on the right handalready mentioned, side these of factors also (4.39)set play to a of crucial ensure role constraints in on obtaining the the appropriate the projectors onto the eigenvalues parts of Φ dZ invariant under both “internal” gauge transformations (including and “external” where Γ An interaction ambiguity. and they cannot besince the incorporated external into transformations rotate the group of internal gauge transformations and thechoiceofan contribution to the Φ scalar and ( JHEP11(1998)016 5 V 2) , + (3 (4.49) (4.47) (4.50) (4.52) (4.48) (4.53) (4.51) W SO = A ) space can be x, Z . ) 4) on the fields will ε (see (4.4)). One then | (ˆ (8 ¯ Z π d ? dZ . , 2) invariant except the Γ + , Φ 0 OSp d κ (3 − ? = Φ ImΦ Φ SO ) connection one-form ˆ 5 d 2 , Γ . ¯ ε? , z i . ? d ) Γ) x, Z ] ) space and its Bianchi identity are Γ + A 0 ( A?A, Φ=ˆ ?κ ?W, ¯ π Γ+ ?κ 0 − x, Z ? A, F Φ 25 (Φ matrices but in such combinations that the ?κ Φ ˆ V 5 dA − =Φ Φ =[ 0 0 ,δ − = 2 1+ReΦ = Φ ? Φ 0 ] Φ ˆ dz ?S, F dF ˆ Φ ε ¯ Z 0 function. i 4 id W? A? ? A, 4) algebra is not violated. [ | = = =Φ = 0 0 (8 − 1) invariant. The situation is analogous to the one in Φ= F ˆ , Φ Φ ε ˆ ˆ d d d 1) invariant equations (3 , OSp S?S = S? (3 SO δA SO 1), the representation of the internal , (3 SO is a regular, complex V All quantities in this equation are manifestly Notice that (4.51) follows from inserting (4.50) into (4.52) and using (4.42). The The interaction ambiguity amounts to the fact this relation can be replaced by Now, eqs. (4.47) coincide with (4.36), (4.38) and (4.39) provided that we set = 8 AdS supergravity theory [30]; both equations of motion and supersymmetry where 4.5 Integrability of the higherThe spin integrability field of the equations higher spin field equations (4.35)–(4.39) in ( where the total curvature two-form in ( given by term. As a consequence of breaking the manifest external invariance from set of manifestly down to gauge symmetry of (4.50)–(4.51) is given by finds that (4.35)–(4.39) can be cast into the form closure of the internal (see (4.3)) and using the total exterior derivative made manifest by writing them in terms of the ( the more general one only be manifestly transformation rules contain explicit Γ N JHEP11(1998)016 )- ¯ z z, (4.55) (4.56) (4.58) (4.54) (4.57) 4) gauge | dependence (8 refers to the E x q dependence of 1)-components , d x shs 0 2)-components of , is a fixed point in , , ) 1 p , κ , ? ) 2) components yield , p κ 0 ? , space equations” obtained (where 0) and (1 Φ , ) space where into the two remaining field , p 2 0)-components of the Bianchi Z | , 1 ¯ z (1 + Φ 1) and (0 , 0) components of (4.50) yield , S ) S , , d 2 2 =0 x, Z 2 , 0 ¯ z V , ˙ , ( α V?W, α ¯ id π 1) and (0 Γ+ and , + F ) refers to the form degree in ( , ? 1 p | , Γ ?κ = Φ r, s ,Φand κ, Γ)+ Φ ˙ − ?κ ? αµ 4) gauge transformation. Thus the space 2 W dependence thus obtaining an FDA of the | 26 F W?V Φ )-forms in ( Φ Φ 0), (0 ?κ ˙ (8 dz s β , Z ( ˙ p = . αβ α = E 2 i 4 + ) , V? i i space and could be a promising starting point for 2 2 p αµ r d shs S = dV − F − Z ( + + ¯ (1 + Φ π q Φ= = = = 2 Z ? ˙ β d W ˙ p µν αβ α V?V Z F idz F F )of( d − 4) symmetry of the AdS vacuum. = =Φ | V p p (8 Z q, r,s Φ d E ?S ? p p shs S -space and the bi-grading ( S x is determined by an from (4.35)–(4.37) in terms of Φ p S These equations are invariant under spacetime independent As already discussed in subsection 4.1, the integrability of the higher spin field In order to discuss the equivalence of (4.50)–(4.51) and (4.35)–(4.39), we intro- The curvature constraint (4.50) can be written in components as ,Φand (4.52) are equivalent to the equation yield which is equivalent to (4.37). The (0 identity (4.52) we read off (4.36). Finally, the (0 obtaining other classical solutions ofas the solutions theory with than nontrivial the higher AdSbreak spin vacuum the solution, background global such fields or solutions that partially spacetime). In fact, the solutionaway obtained from is pure gauge such that the (4.35). Using (4.16) and (4.30) we find that the (1 equations (4.38)–(4.39): by inserting the pure gauge solution for of gauge inequivalent solutions toto the full the set space of equations of (4.35)–(4.39) gauge is equivalent inequivalent solutions of the “ which is equivalent (4.39) using (4.33). From the (1 transformations which are local in form (4.1) from which the dynamicalauxiliary spacetime equations fields. follow upon the Another eliminating possibility is to begin by solving for the which yields (4.38). equations can be used to solve for the duce the tri-grading ( space regarded as a complex space. Thus the (2 W form degree in JHEP11(1998)016 2) of (5.4) (5.3) (5.2) (5.1) , (3 , SO ) Y ( 0 by a := Ω i , ˙ β a ¯ y ,µ . α 0 radius ˙ ˙ δ ˙ δ e y β 1) subgroup of , ˙ β δ α , β µ 0 0 0 α e e ˙ (3 e 0 β dx e defines a AdS-covariant deriva- AdS y ∧ ∧ ∧ ˙ α 0 ˙ ˙ ˙ = δ SO δ α ¯ y ˙ α δ ˙ β +2 β a 0 which acts irreducibly on Lorentz 0 0 ˙ 0 ˙ α 0 β e e a ω ω y 1)-valued connection one-form ˙ , α + + D +¯ +¯ ¯ y ˙ γ γ ˙ (3 ˙ β ˙ β β β β ,e γ α y 0 0 a 0 0 0˙ α 27 ¯ e ω ω e SO y ω ˙ , α ∧ ∧ ∧ αβ α ˙ 0 +¯ α ˙ γ ) ¯ z ω αγ α αγ β a ˙ are the vacuum Lorentz connection and vierbein α 0 0˙ 0 y σ ¯ z 0 ( α ω ω ω i e d y 1 λ 4 =¯ = = space equations (4.37)–(4.39) and the elimination of 1 2 + αβ ˙ ˙ is given in terms of the 0 β β = α − and α α αβ Z ω z λ 0 0˙ 0 0 h 0 α = ω ¯ , ω i ω de 1 ˙ α d dω 4 dz ,¯ ,α 0 0 ω e 2)-valued connection one-form Ω )=0 )= )= , (3 = 8 supergravity theory. As already discussed in subsection 4.1 this geometry can be identified as the vacuum solution given by [28] = 8 supergravity model. Having established the physical consistency Z, Y, θ Z, Y, θ Z, Y, θ N SO 4 ( ( ( 0 0 0 N S Φ W AdS The To prove that (5.1) is a solution of the higher spin equationsThe (4.35)–(4.39) flat, one where the mass parameter which defines a Lorentz-covariant derivative of anti-de Sitter spacetime: tive which mixes irreducible tensors of the Lorentz tensors, and the vierbein 5.1 The anti de SitterThe vacuum field solution equations (4.35)–(4.39) constitutebut an internally it consistent set remains ofcontact to equations, with establish the physical relevance of the equations and to make 5. Expansion around the anti de Sitter vacuum requires the solving of the at the linearized level, onethe then equations has (4.35)–(4.39) an which interacting is higher tractable spin in field classical theory perturbation based theory. on where the one-forms first observes that (4.36)(5.2). and Using (4.16) (4.38) one are easily trivially verifies (4.37) satisfied and while (4.39). (4.35) reduces to the same spin. It splits into the curved, the auxiliary fields fromfollowing the two algebraic sections equations we containedspin shall in dynamics in (4.35)–(4.36). verify the that Inlinearized AdS this the vacuum, and yields in the section correct 7 free we shall field make higher contact with the JHEP11(1998)016 . ˙ r α η 4) | (5.6) (5.5) (5.9) (5.7) (8 (5.11) (5.10) : E 4) sym- and the | ˙ α... , (8 λ shs 8 E a...α... ··· T shs ,..., . + 4). The last fact | 4 ˙ =1 γ... (8 E ,..., . c...δ... shs T δ =1 rβ , η ˙ . ab,γ α ,... , k ) 0 β r 2 bc 1 2 0 , , , η e obeying the following generalized ad + ,r = η , . =1 0 r ··· =0 ˙ ˙ β ε r α η ··· ? ˙ 4) and their hermitian conjugates ¯ − αβ α ¯ ] + η ) ) . 0 = + bd ˆ cd cd ε ˙ δ... † η 1 a ) σ σ , acting on a Lorentz tensor ˙ ac ,..., γ... r ( (¯ 0 28 ] ,m = η η b ( ( n c...γ... 2 i ,D λ [Ω 2) connection Ω D ab,cd ab,cd =1 T ˙ =0 (5.8) obey (5.8). By making use of Leibniz rule it is λ ··· a , β d...γ... ˙ r r δ [ 1 − r 0 ˙ r i 1 4 1 4 r γ T − (3 ˆ ¯ ε D η d θ 0 η ˙ ) , β ˆ ab, Z ε )( = = = ˙ α m r = 1 and return to the relation between d α d x r ab,c SO ˙ 1 2 0 δ ¯ y ( r ˙ e γ λ ˙ r α r α 1 2 + ?η η η ab,cd ab, ab,γδ = r r r = +¯ r α 4) symmetry. More explicitly, the solution (5.1) is invariant ··· | α ? ˙ γ... y (8 Dη 1 r α r η ( η c...γ... = OSp T ] r 1)-valued Riemann curvature b η , := D (3 a [ n i D SO ··· 1 i m r ··· 1 In the following we shall need To construct the Killing parameters explicitly we first introduce the four com- We then define the hermitian, Grassmann even, commuting elements From (5.9) it follows that the We shall temporarily set r η where the Killing equations follows from the flatness of the metry and not just under gauge transformations with parameters ˆ dimensionful coupling of the theory in sectionGeneralized 7. Killing symmetries and admissibility criterion for obeying the AdS covariant Killing spinor equations muting AdS Killing spinors whose solution space forms a superalgebra isomorphic to easy to verify that Let us emphasize that the AdS vacuum solution (5.1) exhibits the full JHEP11(1998)016 (5.17) (5.12) (5.14) (5.16) (5.15) (5.13) ,... 2 , is integrable Z =1 ), the 32 unbroken ) is straightforward ij ) is isomorphic to the λ , the real dimension of . ( ,n , rs n 0 k j λ n i i
denotes the traceless symmetric part. In deriving (7.9) one has to make = + } 2 L ab 1 { i,j,... − are the 35 + 35 scalars obeying (3.30) and the fermions are Weyl. The complex e In summary, the linearized equations of motion of the level We find that (7.10) is in perfect agreement with (7.11) provided that we make The linearized, gauged of the Riemann tensor in the right side of in the two first equations in (7.8) are ijkl cd projected onto (anti-)selfdualselfdual components. and anti-selfdual Eqs. components of(7.5) the and (7.9) constraints (7.9) then (7.1) yields and follow (7.2). by Combining adding up use of the self-duality properties (A.6) and notice that the pairs of indices where of motion for the spin cosmological constant. given by the identifications and dimension 1 to the vector fields. Thus the Lagrangian as follows from (5.43) and (B.7). The constraints (7.1) and (7.2) then yield where we have reintroduced the inverse conjugation changes chirality, and consequently both chiralitiesas occur for well the as gravitini thehave spin assigned 1/2 (energy) fields. dimension Thus, the theory is vector like. In writing (7.11) we action [32] where φ JHEP11(1998)016 − m have ,the ( )and (7.14) (7.13) κ a 7 θ i ω ; ··· .These 1 i α ,m ˜ C 2 gW − , → and m ( 0) 0) a , , ijk W α ω ˜ C = 8 AdS supergravity. ,... 6=(0 1 , N )=(0 ) = 8 supergravity multiplet. = 8 AdS supergravity can =0 m, n m, n N ( ( , ,N ) θ one then defines component fields ; =4 λ . 2 ,m,n D m, n λ ) ( ) is equal to 1 this means that the gener- 1 2 θ µ , which is the same as the dimension of the ; , 48 ω = ) | have dimension 0. Thus, in particular, (7.14) +2 2 n ). Finally the fermions 2 2 m θ − 2 ,θ g κ m, n m, n, θ 0 m ( ijkl | ( 0; , ˜ φ µ C ) has dimension 1. The generalized Weyl tensor 1+ (0 ω n m, µ − . The gauge coupling is introduced into the full set = 8 AdS supergravity seem to be unique due to the + 2 ( ,θ λ ω λ 0 and = 8 AdS supergravity equations at the linearized level, m αβ , N λ ˜ φ ˜ R N (0 ) has mass dimension 0, the generalized gravitino ˜ µ ) with canonical mass dimensions as follows [20]: and the generalized Lorentz connection ˜ ω and the inverse AdS radius: θ )= )= θ ; radius ; 1 2 θ θ g ; ; 1) has dimension m, n m, m AdS ( ( ≥ m, n m, n a ˜ ( ( C µ ω ˜ m C ˜ ω and the scalars )( θ 2 1 )and 0; θ , ; (8) vector fields ˜ ) has dimension highest possible supersymmetry in the theory, θ +2 ; SO Since the mass dimension of The free parameters of the higher spin theory are therefore the gauge coupling and Φ. Using the dimensionful coupling m, n (8) gauge coupling m ( and the inverse ( (c) the interactions in the (a) the full higher spin equations of motion are consistent, (b) they yield the correct µ ,m ˜ ˜ C pure curvature component dimension 1 yields the correct canonical dimensions of fields of the be embedded into aconsistent but higher we spin have shown gaugenext the step theory. embedding in at The this thefields linearized fully program in level. is nonlinear the The to equations important equationsstudied study are before of the but interactions, motion. they have starting not Various with been compared aspects the to quadratic of those of such interactions have been 8. Discussion The results of this paper suggest that the and identify the following importantSO relation between the Newton’s constant the g Given the facts that: alized vierbein ˜ equations are consistent withW the assignment of dimension 0 to the master fields ω of higher spin equations (4.35)–(4.39) and (4.40) by replacing JHEP11(1998)016 =8 =8 =8 AdS N N N fields. If 5 2 -theoretical ≥ M s spacetime serves (the inverse λ AdS would the be introduced in the (8) coset structure of Cremmer- g /SU 7 E 49 = 8 supergravity [32, 33]. N 4) symmetry in which the massive multiplets are symmetry. Recalling the uniqueness of the | 7 (8 spacetime. It is tempting to believe that the singletons E E shs AdS = 8 AdS supergravity is required to settle this question. 4) symmetry algebra as charges of conserved currents in the | N . (8 5 E . Higher spin supercurrents have also been constructed [46], also in 5 AdS shs = 8 supersingleton propagating at the boundary of AdS = 8 AdS supergravity, along with the sector for the spin N N e [34, 35] as well as the AdS Relevant to the problem of finding the hidden symmetries in the theory is the Since the boundary theory involves massless as well as massive composite states, The A desirable formal consequence of bulk/boundary duality would be to ultimately = 8 supersingleton theory. In that case we would expect that the nonlinearities Poincar´ so, then one would also expect to uncover the AdS supergravity, we expectprovided that that there we should insist on be the no consistent ambiguity truncation in of the the theory interactions to the pure Julia [34, 35] which plays an important role in the descriptionquestion of of both how the uniquebeen is the addressed higher in spin subsectiondiscussed AdS 4.3. supergravity. in The This that full issue sectionin has significance is the already of search not an for clear interaction the to ambiguity hidden us at present. It may as well play a role one may expectfledged that the higher spin equations at hand already contain the full bulk/boundary duality would also yieldmassless higher and spin massive bulk “matter” interactions sectorsless including (allowing sector). a both consistent Inclusion truncation of to massivebreaking the sectors mass- [27, could generate 11] mechanisms for of spontaneous the the context of boundary theory by a rescalingWhile of this the program composite byspin states and describing currents large the has remains gauge been tocontext fields). recently be of investigated realized, [45], the at construction least of for higher low lying spins, in the N of the bulk theorysingleton would states be (the reproduced dimensionless by gauge the coupling interactions between the composite could play a morefrom the fundamental bulk/boundary role duality intheory prescription and the of boundary derivation [31, theory of 38, has 39]. the the same effective Notice dimensionful that bulk coupling both action bulk AdS supergravity. A carefulspin comparison theory of and the the first interaction terms in theas higher a spectrum generatinggating representation in for the the bulk of massless the higher spin theory propa- radius). Since thethe (massless) essential spectra test of of the therepresent bulk bulk/boundary the and duality the is boundary therefore whether theories it agree, is possible to framework. “eaten” by the masslesssince we gauge do multiplets. not presently know This how to point fit is massless higher of spins great into an physical interest JHEP11(1998)016 D and W and the 6 5 case is that it higher spin AdS D AdS , respectively [40]. D 7 spaces of dimension AdS = 4. Furthermore, it is AdS D boundary theory is a more tractable D 50 2 sector of the theory does not fit in a natural = 11 supergravity theory alone, Kaluza-Klein / 1 D = 11 origin of the AdS higher spin supergravity ≤ D s ) space field equations (4.35)–(4.39) using background x, Z ) in five dimensions (the definition of traces of higher spin W ∧ R ∧ R ( tr 2. Therefore, one is led to speculate about the existence of either a new 2. On the other hand, the fact that there is a gauge master field type actions have been considered before [19, 20, 41, 23], but one drawback of / 3 s> It would, of course, be desirable to find an action which yields the consistent, The bulk/boundary duality of the type discussed above may also exist for the The construction of higher spin superalgebras and free field equations of motion R higher spin theory described in [29]. In this case, the study of bulk/boundary doubletons (vector multiplet) propagating at the boundary of = 5 origin in which the matter master field may emerge as the fifth component ≥ ∧ D D of the gauge master field upon dimensional reduction to a matter masterD field Φ in the Vasiliev formalism studiedencouraging here that there is exists suggestivethe the of form possibility a of a Chern-Simons type Lagrangian of and geometrical way intos the part of the action that describes the fields with spin these actions is that the spin may be possible to constructsuperalgebra, an combining action the for elements a of theorythe [42] based that work on involve the described a 3 in Chern-Simons action [29]. and fully nonlinear equations ofR motion of Vasiliev that we have studied here. Indeed, The possibility of constructing higher spin interactions in duality is expected to beconformal simpler because field the theory. 2 Another reason for the tractability of the 3 doubletons (tensor multiplet) propagating at the boundary of reduce the rather cumbersomearound calculations the implied AdS by vacuumthe the free outlined boundary perturbative in supersingleton expansion subsection fieldit theory. 5.2 is An possible to interesting related to calculations issue accommodatederive entirely is auxiliary the whether within spinor higher variables in spinfield the ( methods. boundary theory and 4 for higher spin fieldssions becomes [41] rapidly and very understandably complicatedsometime. in not higher However, much than the progressmotivation four recent has for dimen- exciting been exploring developments made the in in M-theory this provide front ample for algebras is explained in [29]). studied here. Startingcompactification from gives rise the to masslesscorners and of massive M-theory fields of where maximumgive it spin rise can two. to be The infinite treated towersspin of perturbatively, on higher spin the fields, other but hand, all can of these fields are massive for corner of M-theory, or supermembrane theory, which may give rise to the higher spin larger that four3 has been investigated in [41]. A more tractable example is the JHEP11(1998)016 3) , 2 , (A.2) (A.1) (A.3) (A.4) 1 .Using , γ β δ =11AdS =0 = a . D ( αγ ˙ α ˙ β β α ) = 11 limit which ) αβ a a σ , D σ ˙ γ δ =( ) a obeys σ ˙ αβ ( , , ˙ ) β ˙ a ˙ β β α βδ ˙ , , α α σ , ˙ ) ) ˙ γ α † ˙ ˙ α β βα = ab ab ) , but the possibility is certainly =(¯ ˙ α ˙ σ σ β β β 11 ˙ † y α y y , ˙ ˙ ˙ )= β β α +( +(¯ =¯ = α =( δ 3 = AdS ) ˙ α σ α ˙ β ˙ α ˙ a α β β α α ¯ y δ ¯ , y αβ δ δ σ − ( 2 ) , ab ab 3 2 − η η 51 = σ , ,y ,σ = = = , − αβ 2 ˙ β α , † ˙ β β ¯ ) , y y ˙ 1 ˙ ββ ˙ α α ,σ α β ˙ ) α ) ) σ ˙ 1 αβ α y +++) and work with two-component Weyl spinors b b β a ) − σ σ σ − a (¯ (¯ ( ,σ , ˙ ˙ = =( = σ α α α ˙ α α α ˙ ˙ α α α ) ) ) = 11 supergravity theory in a profound way. The first ¯ ¯ y a y y a a =( σ := (1 := (1 σ σ D ( ( (¯ ˙ ˙ = diag( αβ β we define the van der Waerden symbols ( = 4 in a certain limit, or a new kind of ) ˙ α αβ 3 a ) ) , ab (9) group as the little group classifying the massless degrees of 2 a a D σ η , 1 σ σ ( (¯ σ SO =(¯ † ˙ β α ) a 1) we take σ , ( (3 The van der Waerden symbols obey the completeness relations SO the Pauli matrices tantalizing and we expect that ittheory would in be the highly M-theory relevant to framework.representations the of In massless this higher spin context, it is interesting to note that the with the hermicity properties where the charge conjugation matrix A. Spinor conventions For We thank M. Duff, J.grateful Maldacena to and E. P.K. Witten Townsendus for for to useful his add discussions. comments several We on clarifying are the remarks. first version of this paper which led Acknowledgments AdS supergravity in freedom of an eleveninteresting dimensional results supergravity that hint has at been the studied possibility recently of higher [44] spin with massless fields. modifies the well known scenario is in lineconnection with [8, the 9, previous 10,struction studies 11, of on 12, the 13]. the singleton/doubleton supermembrane-supersingleton The representations latter of scenario a is motivated candidate by a recent con- supergroup in [43], which turnif out an to action, be or ofsentations rather equations in unusual the of kind. ten motion, It dimensional can is boundary be not of known written yet down to describe these repre- JHEP11(1998)016 = 2 ) 5 (B.1) (B.2) (A.8) (A.7) (A.9) (A.6) (A.5) γ .These ˙ α ˙ β can be used ) ˙ , ab . α ] ¯ σ ψ , b , | ] C. ˙ β , ˙ c β 5) and work with ˙ ˙ ˙ T , = α α β ¯ η y , ) 0 ) (such that ( =(¯ δ ˙ † ) ˙ ij α 3 β ) β ab ¯ 3 ± y , θ ˙ ¯ y ( β β α α α i 2 1 2 σ γ 2 ˙ ˙ ) α α δ (¯ 0 ψ 2 , + d ) ¯ i y [ i (Ψ 1 γ | − ˙ = = ba θ β a 1 , , − [ ˙ ± ˙ ˙ α α c σ γ β ij ] σ ` , ¯ ˙ (¯ 0 y =1. b α = = ˙ a, =0 = η α − ˙ + iγ β i 0 ˙ ˙ β α ˙ β β γ = =5 ) 0123 α α = † +4( ¯ = y ) ) ) 4) cd α 5 5 A A a ) | ˙ A, B [ − β β σ α y γ γ ± +2¯ ˙ (¯ σ α ( δ (8), supersymmetry, Lorentz trans- ˙ ( ,M β i (8 ,T ( ) )( = ˙ for for α θ β i i E ab a ¯ − a α y θ ] abcd β SO ˙ σ α y α d α ¯ 1 2 y y 0123 η ) β , as well as a purely imaginary Majorana c 1 2 1 2 5 iα shs +2 [ α := (Ψ +2 b γ ) 52 ! η ˙ β a 5 β = = ˙ +++ (+) α and (¯ β ) and Γ-matrices y γ γ α ˙ α ( ( ) β , α ± ˙ − αi ) +2 i ( i α d α +2 y a βα 0 σ Ψ αβ ij ( , -matrices and ) σ ( ) θ αβ γ ab ` abcd ! ij = ab and its hermitian conjugate ( σ ˙ abcd λ ˙ σ β + 1) αβ β i α ˙ ( α ) , 0 = diag( α , i i ψ a ) i ,P 0( =( 1 ,Q (3 σ 4 A ! = = = β i (¯ AB ˙ αβ θ y α 0 ˙ η β β (Γ − α α α ¯ αβ SO αβ ψ γ γ ) y y ) ψ ) )
subalgebra of 1 2 1 2 ):= ± c are the parameters for ba cd cd σ ˙ σ -symbol is defined by
= = σ β σ ( ( = = ( ( 4) α γ Y,θ β | γ ( = − α αβ α αi α 2 ,a ) αβ ) ) ) abcd (8 C P a Q ± ab = αβ ( α M ab 2) we take 1 2 γ 1) Weyl spinor σ , ( σ , ( Ψ ,` ( αβ (3 : iα (3 ) OSp ) are symmetric , − ab α ( SO ij a σ SO λ γ For An four-component, Majorana spinors Ψ quantities obey the decomposition rules 1). In the Dirac representation we choose where In the expansion (2.19), we find the quadratic, homogeneous polynomial B. The where ( where to represent a real Majorana spinor Ψ spinor Ψ formations and translations respectively. The corresponding generators where the tensorial and they have the duality properties JHEP11(1998)016 , ˙ δ , α (B.8) (B.7) (B.6) (B.5) (B.3) 4) en- . P | ] 3 (B.4) 2) bases m (8 βγ , i ) i (3 m , − a ,..., . + , OSp 2) subalgebra 1 ω +2 + 3 terms ˙ , SO , δ 4 , , ˙ ˙ k α βi β 4 ˙ il β α (3 α Q P ) T α AB a =0 P jk σ αγ αγ ,..., SO M ?Q ( 4) does not need to 4) relations (B.3) in ij 1 | | 1 2 δ iδ i i with a pair of bosonic , AB (8 (8 ··· − i ω = = = = E 4) and the i 0 ? 4) 1 | =0 2 | | ? ? ? ? = 1 ] ] ] i } ˙ [ (8 ˙ α (8 OSp δ ,a,b shs γi = kl γ α m ˙ ˙ E βj β 2 4) is consider tensor products α | − ˙ ,T α ,Q OSp ¯ shs ˙ +3 y ,Q (8 M ,,P β ij ˙ k α β α E 4 T αi α αβ y α [ P ) ˙ ,ω α ,A,B [ Q 4) can be identified with the subspace a 4) M α | shs | ab σ { [ ?Q ω ( (8 ω 1 2 ˙ (8 β m E ˙ 2 α E +2 − ) = , 53 ˙ β ˙ , ab shs , 4 δ +2 , ¯ ˙ y k a β σ ˙ 4 α shs αi (¯ α αj , + 3 terms ¯ y M 1 4 ˙ Q m ,M β Q ij 2 ˙ ˙ α β αγ − T = AD ˙ βγ ik ω α ˙ +1 β i M ˙ αβ k i iδ α M +¯ 4 ˙ β + 3 terms + α β ˙ + 2) connection one-form given in the two − BC α + y , ) βδ αγ iη α βi αi ab (3 ?M y αβ − Q M σ M Q ,ω (¯ ˙ δ αβ M ˙ 1 4 = = jk β αγ αγ SO ··· ω ab ij ? n + ? ω ] i i i i δ iδ obtained by acting on a ground state 4). Moreover, a representation of = 8 anti de Sitter superalgebra i | 2 1 ··· 4 αβ αβ α 1 Φ = = = = = i (8 1 ) CD N θ M α ? ? ? ? ? E . One may notice that ( ab ] ] ] ] n 4 } ˙ − σ αβ a δ Ω= ,M M γi ( 4) enveloping algebra which is spanned by odd, fully (anti-)symmetrized ) γ γδ αk ω | βj shs +2 1 4 =4, k ab 4 = (8 AB σ ,P ,Q α ,Q = := ,M ( ,Q D ˙ β ··· 1 4 M a ij α 1 αβ αi [ αβ αβ α T ω P OSp = [ y Q M ω [ M [ Hence, considered as a vector space, If we let Ω be the { [ ab M yields where represent the relation (B.8),representation as (C.7). is the case, for instance , with the tensor product then we find the following relation between the components gauge fields: of the Fock space C. Oscillator realization of One way to obtain unitary representations of obey the and hermitian conjugates. The following change of basis for the of the functions. However, when considered as algebras, veloping algebra differ from each other. Actually, the by combination with (B.8) doelements not in suffice to determine the commutator of two general JHEP11(1998)016 ? 4) 4) | (see (8 (C.2) (C.6) (C.3) (C.5) (C.1) (C.4) (C.7) E remain , ,..., e, o ] o s shs 8 ,sincethe Φ = =1 Φ . 0 in ⊗ ) A ) , ( A P 1 2 ) and by the ordinary 2 λ, λ , i ˆ A θ e i ˆ Φ v ψ (1 | Φ arepresentationof + D ˆ P [ 1 ( − actually splits into two ⊕ not 4 A ] ,where 2 , Φ c with an even (odd) total ˆ i⊗ θ 4) and, using the notation 3 8 | , ( , u i S,A 1 2 ] | 2 2 0 ⊗ 0 (8 | , , λ E = uP 0) Φ , given in (2.3) is given by A 4) acts reducible on 1) =1 =1 1 2 | ˆ ⊗ shs ψ ( − A λ (8 , , D Φ ) E . ˆ of G. o ˆ θ +( ˆ ; Φ F =[ i , ¯ F ˆ y shs v =0 ) o ⊕ 2 y, i ˆ e y (ˆ )= 54 i 0 ,A,B ⊗| Φ | ,p,q F ) + A i = pq AB ˙ ˆ = 1 d u δ δ ψ ,Φ ¯ ˆ y | Φ F?G ] ˆ ( F c ( ˆ = = 1 2 P 8 ]= is fully (anti)symmetrized, or Weyl ordered. The i } † q = ⊗ 0 ˆ † B ˆ a | F ) 2) and fermionic creation operators 2 is defined by of the element ˆ ψ p , 1 2 4) subalgebra (B.3) of , ˆ a | ˆ a , p , Φ Φ ) are made up by acting on a (8 o A ):=( (1 given in (2.7)–(2.8) is then represented in [ˆ =1 ⊗ ˆ i Φ ψ D ( v p A [ , Φ { ( e ) OSp ˙ are given by 2 ⊕ † p Φ ¯ 4) are even polynomials, and as a result ] ˆ y a i | i s i⊗| ˆ θ 8 u 4), namely (8 . The tensor product form a reducible, unitary representation of in the exponents denote Grassmann parities (the vacuum is taken + | | E ( ⊗ 1 A (8 ˆ P y and P (ˆ E 0) ˙ ) are the two 8-dimensional subspaces of the 16-dimensional Fock space of shs 1 2 α , c ¯ 1 2 ˆ y ( shs and (8 = , α 4), containing the invariant subspaces [ s 1 D | u y algebra ˆ a (8 =[ ? Notice that the operator The representation on The resulting unitary representation of E e Φ where ˆ where 8 irreducible under the elements of product of elements in operator product: introduced in (2.29), they are labeled as follows: creation operators ˆ the fermionic oscillators, obtained bycreation acting operators with on an the even vacuum. (odd) The number representation of of fermionic the element where the states in on the tensor product obeying [36, 37] number of fermionic and bosonic oscillators. The two spaces UIR’s of where to be even).the Eq. (C.7) impliesshs that the tensor product is JHEP11(1998)016 (C.8) (C.9) (C.10) , , is derived S ] e ⊕··· ⊕··· ) of the tensor Φ , , , , ) A ) ) ) 6) 6) , , ⊗ , , , , , ) ) e (7 (7 . Φ , Z, Y Z, Y Z, Y Z, Y D D ( ( ( ( i i Z, Y Z, Y ⊕··· u F u ( F F ( F ⊕ ⊕··· ⊕··· ⊕ ) # # # # F F 5 2 ˙ α 4) 5) 5) 4) α α α , # # , , , , ¯ y ∂ 7 2 ∂ ∂ ∂ ˙ i⊗| α α i⊗| , ( ∂y ∂ ∂y ∂y v ¯ y (5 (6 (6 (5 v ∂ ∂ 2) content i i i i | , , | , ∂ ∂y D has been taken into account in D D D D i i − A + S + − − + uv c uv (3 ⊕ +56 ⊕ ⊕ ⊕ ⊕ ˙ + + α α α α 1) ) 1) v (8) content of [ ¯ z ∂ ∂ ∂ ∂ ˙ 3 2 α α SO − 2) 3) 3) 2) − +35 ∂z ∂z +35 ∂ ∂z , ¯ ( z ∂ , , , , ∂ i i i i 5 2 S ∂ =8 A ∂z SO ( (3 (4 (4 (3 i − − − − i +( s " " " " " " × D i− 8 D D D D i v v 55 +28 +28 ⊕ ⊗ 2) ⊕ ⊕ ⊕ ⊕ )+ )+ )+ )+ )+ )+ ) and anti-symmetrization ( , c A ) S 1 2 S 0) 1) 1) 0) (3 , i⊗| , , , , i⊗| 3 2 u =8 =1 =1 u Z, Y Z, Y Z, Y Z, Y Z, Y Z, Y ( (1 (2 (2 (2 | | ( ( ( ( ( ( SO c c s 8 8 D D D D D 8 F F F F F F = = ˙ ˙ α α α α α α ⊗ ⊗ ⊗ A y z y z y z = = = = = S i i c s s S S A A 8 8 8 i = = i i i i i v S,A v ) ) )=¯ )=¯ )= )= α α i 1 2 0) 2 1 0) ) , , , , 1 2 1 2 1 2 ?z ?y , ( (1 ( (1 i⊗| ) i⊗| ) Z, Y Z, Y Z, Y Z, Y ( ( ( ( (1 u u D D D D | | D h h ⊗ ⊗ ⊗ ⊗ ?F ?F ?F ?F Z, Y Z, Y ) ( ( ) ⊗ ˙ ˙ α α α α 1 2 0) 2 1 0) ¯ ¯ z F z y y F , , , , 0) 1 2 1 2 , (8) content ( (1 ( (1 1 2 ( D D D D h h h h SO D h The result in table 1 for the the last equation. D. Symplectic differentiation and integrationUsing formula (4.11), one finds the following contraction rules: product according to the rule (C.5)), obtained by symmetrization ( using the following decomposition rules for the where the odd Grassmann parity of the states in 8 and the JHEP11(1998)016 ? tz and (D.4) (D.3) (D.6) (D.5) (D.8) (D.1) (D.2) (D.7) by (1984) -space tz z Z by − z and ¯ , , B 138 i i z B , ) ) ˙ β α ¯ ¯ z z and ¯ α ∂ , . ∂z h z ) , ) ˙ α tz, t tz, t i ¯ z ( ( ) d ˙ ˙ ¯ α α A? z Z, Y Z, Y α α Phys. Lett. α ( ( ∧ h h ∂ ˙ α F F α α tz, t ∂z z z ( # # one replaces ˙ dz α ˙ ˙ α αβ α )thenwehave − g ¯ ¯ y ¯ y ∂ y ∂ ˙ + α ) )+¯ i , . ∂ ∂ z ¯ ¯ z z 2 ˙ ˜ h i ) i α 2 g ¯ before zt z − ˙ , edited by A. Salam and E. Sezgin, ¯ z,y, − − α ( )+¯ d ) ¯ tz, t tz, t z ˙ ˙ h ¯ α 1 2 α z ) an arbitrary function. In proving ( ( d ¯ ¯ z ¯ z z α ∂ ∂ ˜ h h tz, t ∂ ∂ + ∂ ˙ ( + α α i z, i ∂z tz, t ¯ z z ( h α B " " ( B. A?B α 2 h h g k ˙ ( α β z α 56 g ¯ # z ∂ := dz α )+ ˙ )+ ∂ 1 2 α dt t dt t dz z ¯ z ∂ B 1 1 )= h 0 0 ∂ = = ˙ Z Z tz α Z, Y Z, Y A? such that if one would replace dt ( g h ( ( z ˙ α 1 h − − z 0 ¯ z F F ∂ = ) ) +¯ )and ˙ d ˙ Z α α ∂ dt ¯ ¯ z z ¯ y )= ˙ y z t α β ˙ α products then (D.7) is no longer valid. As a matter of ˙ α ∂ z, z, ∂f and ¯ f ( ( =¯ =¯ ∂z ˙ ? α 0) + i z k k ¯ ˙ α ˙ z,y, ¯ z , α α 2 ˙ α α z ¯ ¯ z y d ¯ z − " ∂ ∂ (0 ? ? ∂z ∂ f + tz, t , Trieste preprint IC-83-220, Nov 1983, in ) ) ( α A )= f α )= )= )= Z, Y ¯ ¯ ¯ Z, Y z z z := ( ( dz The spectrum of the eleven dimensional supergravity compactified on the Supergravities in Diverse Dimensions ( z, z, z, F F A ) is an arbitrary function. The linear differential equations in ( ( ( A?B ∂ ˙ ( α α f f f d dt Z, Y t ( (0) is an arbitrary constant and as indicated in (D.4)–(D.6). This is so because the contractions alter the f F ¯ z t round seven sphere 57, and in World Scientific, 1988. Therefore, to apply them correctly to (5.27)–(5.29), one must expand the before one expands the [1] E. Sezgin, ¯ z encountered in the perturbativeanti expansion de of Sitter the vacuum, higher have the spin solutions equations [28] around the where these formula one makes use of where of the type References functional dependence on products in the rightand hand sides of (5.27)–(5.29) fact, if we let t JHEP11(1998)016 , , , , 7 S 180 Lett. × , (1988) B 143 4 5 S ,Vol. J. Math. Phys. , (1988) 229. Phys. Lett. 4), | B 203 N ( supersingleton field theory OSp =8 Compactifications of the eleven- Supersymmetric supermembrane (88) 483. Singletons, higher spin massless Class. and Quant. Grav. (88) 237. N , (1989) 71. Lecture Notes in Physics Phys. Lett. , Spectrum generating algebras in Kaluza- B 305 Supermembranes and eleven dimensional ,in B 205 B 224 (87) 69. A quantum consistent supermembrane theory 57 A stable supermembrane vacuum with a discrete (1984) 401. glu, 1983. 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