SU(2) deformations of the minimal unitary representation of OSp(8*|2N) as massless 6D conformal supermultiplets
aa r X i v : . [ h e p - t h ] A ug Preprint typeset in JHEP style - HYPER VERSION SU (2) deformations of the minimal unitaryrepresentation of OSp (8 ∗ | N ) as massless D conformalsupermultiplets Sudarshan Fernando ∗ and Murat G¨unaydin † Physical Sciences DepartmentKutztown UniversityKutztown, PA 19530, USA Center for Fundamental TheoryInstitute for Gravitation and the CosmosPhysics DepartmentPennsylvania State UniversityUniversity Park, PA 16802, USA
Abstract:
Minimal unitary representation of SO ∗ (8) ≃ SO (6 ,
2) realized over the Hilbertspace of functions of five variables and its deformations labeled by the spin t of an SU (2) subgroup correspond to massless conformal fields in six dimensions as was shown in arXiv:1005.3580 . In this paper we study the minimal unitary supermultiplet of OSp (8 ∗ | N )with the even subgroup SO ∗ (8) × U Sp (2 N ) and its deformations using quasiconformal meth-ods. We show that the minimal unitary supermultiplet of OSp (8 ∗ | N ) admits deformationslabeled uniquely by the spin t of an SU (2) subgroup of the little group SO (4) of lightlikevectors in six dimensions. We construct the deformed minimal unitary representations andshow that they correspond to massless 6 D conformal supermultiplets. The minimal unitarysupermultiplet of OSp (8 ∗ |
4) is the massless supermultiplet of (2 ,
0) conformal field theorythat is believed to be dual to M-theory on
AdS × S . We study its deformations in furtherdetail and show that they are isomorphic to the doubleton supermultiplets constructed byusing twistorial oscillators. Keywords:
AdS/CFT, Minimal Unitary Representations, Conformal Group. ∗ [email protected] † [email protected] ontents
1. Introduction 12. Deformations of the Minimal Unitary Representation of SO ∗ (8) SO ∗ (8) with respect to the subgroup SO ∗ (4) × SU (2) × SO (1 ,
1) 62.2 The noncompact 3-grading of SO ∗ (8) D with respect to the subgroup SU ∗ (4) × SO (1 ,
1) 112.3 The compact 3-grading of SO ∗ (8) D with respect to the subgroup SU (4) × U (1) 132.4 Distinguished SU (1 , K subgroup of SO ∗ (8) D generated by the isotonic (sin-gular) oscillators 142.5 SU (2) T × SU (2) A × U (1) J × U (1) H basis of the deformed minimal unitaryrepresentations of SO ∗ (8) 16
3. Deformations of the Minimal Unitary Representation of
OSp (8 ∗ | N ) OSp (8 ∗ | N ) D OSp (8 ∗ | N ) as D MasslessConformal Supermultiplets 266. Deformed Minimal Unitary Supermultiplets of
OSp (8 ∗ | U Sp (2 N ) in termsof Fermionic Oscillators 34
1. Introduction
Motivated by the problem of constructing the relevant unitary representations of noncompactU-duality groups of extended supergravity theories, a general oscillator method was developedin [1–3]. The method, as formulated in [2], generalized and unified the special constructionsthat had previously appeared in the physics literature. The general oscillator constructionwas later extended to noncompact supergroups in [4] using bosonic as well as fermionic oscil-lators. In the generalized oscillator constructions of [2] and [4], one realizes the generators ofnoncompact groups or supergroups as bilinears of an arbitrary number P C (“colors”) of setsof oscillators transforming in a definite representation (typically fundamental) of their max-imal compact subgroups or subsupergroups. For symplectic groups Sp (2 n, R ), the minimum– 1 –alue of P C is one, and the resulting unitary representations are the singleton representations,which are referred to as metaplectic representations in the mathematics literature. Symplec-tic groups Sp (2 n, R ) admit only two singleton irreducible representations (irreps). In general,the minimum allowed value of P C is two, and the resulting unitary representations of suchnoncompact groups were later called doubleton representations. For example, for the groups SU ( n, m ) and SO ∗ (2 n ), with maximal compact subgroups SU ( m ) × SU ( n ) × U (1) and U ( n ),respectively, one finds that P Cmin = 2. When P Cmin = 2, the noncompact group or supergroupadmits an infinite number of doubleton irreps. The positive energy singleton or doubletonirreps of noncompact groups or supergroups do not belong to the discrete series represen-tations. However, by tensoring them, one obtains positive energy unitary representationsthat, in general, belong to the holomorphic discrete series representations of the respectivenoncompact group or supergroup.The Kaluza-Klein spectrum of IIB supergravity spontaneously compactified over theproduct space
AdS × S of 5 D anti-de Sitter space AdS with the five sphere S was firstobtained via the oscillator method by simple tensoring of the CPT self-conjugate doubletonsupermultiplet of N = 8 AdS superalgebra P SU (2 , |
4) repeatedly with itself and restrictingto the CPT self-conjugate short supermultiplets of
P SU (2 , |
4) [5]. The CPT self-conjugatedoubleton supermultiplet of the symmetry superalgebra
P SU (2 , |
4) of
AdS × S solutionof IIB supergravity does not have a Poincar´e limit in five dimensions and decouples fromthe Kaluza-Klein spectrum as gauge modes. This led the authors of [5] to propose that thefield theory of CPT self-conjugate doubleton supermultiplet of P SU (2 , |
4) must live on theboundary of
AdS , which can be identified with 4 D Minkowski space on which SO (4 ,
2) actsas a conformal group, and the unique candidate for this theory is the four dimensional N = 4super Yang-Mills theory that was known to be conformally invariant.The spectra of spontaneous compactifications of eleven dimensional supergravity over AdS × S and AdS × S , that had been obtained by other methods previously, were fittedinto supermultiplets of the symmetry superalgebras OSp (8 | , R ) and OSp (8 ∗ |
4) constructedby oscillator methods in [6] and [7], respectively. Furthermore, the entire Kaluza-Klein spectraof eleven dimensional supergravity over these two spaces were obtained by tensoring thesingleton and scalar doubleton supermultiplets of
OSp (8 | , R ) and OSp (8 ∗ | OSp (8 | , R ) and scalar doubleton supermultipletof OSp (8 ∗ |
4) do not have a Poincar´e limit in four and seven dimensions, respectively, anddecouple from the respective spectra as gauge modes. Again it was proposed that the fieldtheories of the singleton and scalar doubleton supermutiplets live on the boundaries of
AdS and AdS as superconformally invariant theories [6, 7].These results have become an integral part of the work on AdS/CFT dualities in M-/superstring theory since the famous paper of Maldacena [8] and subsequent works of Witten[9] and of Gubser et al. [10].Noncompact groups entered physics as spectrum generating symmetry groups during the1960s. The work of physicists on spectrum generating symmetry groups motivated Josephto introduce the concept of minimal unitary representations of Lie groups in [11]. These are– 2 –efined as unitary representations of corresponding noncompact groups over Hilbert spacesof functions of smallest possible (minimal) number of variables. Joseph gave the minimalrealizations of the complex forms of classical Lie algebras and of the exceptional Lie algebra g in a Cartan-Weyl basis. The minimal unitary representation of the split exceptionalgroup E was first identified within Langland’s classification by Vogan [12]. Later, Kostantstudied the minimal unitary representation of SO (4 ,
4) and its relation to triality in [13]. Ageneral study of minimal unitary representations of simply laced groups was given by Kazhdanand Savin [14] and by Brylinski and Kostant [15, 16]. Pioline, Kazhdan and Waldron [17]reformulated the minimal unitary representations of simply laced groups given in [14] anddetermined the spherical vectors for the simply laced exceptional groups. The minimal unitaryrepresentations of quaternionic real forms of exceptional Lie groups were studied by Grossand Wallach in [18] and those of SO ( p, q ) in [19–22]. The relation of minimal representationsof SO ( p, q ) to conformal geometry was studied rather recently in [23].Over the last decade, there has been a great deal of progress made towards the goalof constructing physically relevant unitary representations of U-duality groups of extendedsupergravity theories. This was partly motivated by the proposals that certain extensions ofU-duality groups act as spectrum generating symmetry groups of extremal black hole solu-tions in these theories. For example, the classification of the orbits of extremal black holesolutions in N = 8 supergravity and N = 2 Maxwell-Einstein supergravity theories withsymmetric scalar manifolds led to the proposal that four dimensional U-duality groups actas spectrum generating conformal symmetry groups of corresponding five dimensional super-gravity theories [24–29]. Extension of this proposal to corresponding spectrum generatingsymmetry groups of extremal black hole solutions of four dimensional supergravity theo-ries with symmetric scalar manifolds led to the discovery of novel geometric quasiconformalrealizations of three dimensional U-duality groups [25]. Quasiconformal extensions of fourdimensional U-duality groups were then proposed as spectrum generating symmetry groupsof the corresponding supergravity theories [25–29]. A concrete framework for the implemen-tation of the proposal that three dimensional U-duality groups act as spectrum generatingquasiconformal groups was given in [30–32]. This framework was based on the equivalenceof equations of attractor flows of spherically symmetric stationary BPS black holes of fourdimensional supergravity theories and the geodesic equations of a fiducial particle moving inthe target space of three dimensional supergravity theories obtained by reduction of the 4 D theories on a timelike circle [33].Quasiconformal realization of three dimensional U-duality group E of maximal super-gravity in three dimensions is the first known geometric realization of any real form of E [25].As a quasiconformal group the action of E leaves invariant a generalized light-cone withrespect to a quartic distance function in 57 dimensions. Quasiconformal realizations existfor various real forms of all noncompact groups as well as for their complex forms [25, 34].Furthermore, the quantization of geometric quasiconformal action of a noncompact groupleads directly to its minimal unitary representation. This was first shown explicitly for the– 3 –aximally split exceptional group E with the maximal compact subgroup SO (16) [35] andfor the three dimensional U-duality group E − of the exceptional supergravity theory [36]in [37]. The minimal unitary representations of U-duality groups F , E , E − , E − and SO ( d + 2 ,
4) of 3
D N = 2 Maxwell-Einstein supergravity theories with symmetric scalarmanifolds were studied in [34, 37]. A unified formulation of the minimal unitary representa-tions of certain noncompact real forms of groups of type A , G , D , F , E , E , E and C n was given in [38]. The minimal unitary representations of Sp (2 n, R ) are simply the singletonrepresentations. In [38], minimal unitary representations of noncompact groups SU ( m, n ), SO ( m, n ), SO ∗ (2 n ) and SL ( m, R ) obtained by quantization of their quasiconformal real-izations were also given explicitly. Furthermore, this unified approach was generalized todefine and construct the corresponding minimal representations of non-compact supergroups G whose even subgroups are of the form H × SL (2 , R ) with H compact.In mathematics literature, the term minimal unitary representation, in general, refersto a unique representation of a noncompact group. Symplectic groups Sp (2 N, R ) admit twosingleton irreps whose quadratic Casimirs take on the same value. They are both minimalunitary representations, though in some of the mathematics literature only the scalar single-ton is referred to as the minrep. Similarly the supergroups OSp ( M | N, R ) with the evensubgroup SO ( M ) × Sp (2 N, R ) admit two inequivalent singleton supermultiplets [6, 39, 40].For noncompact groups or supergroups that admit only doubleton irreps, this raises thequestion as to whether any of the doubleton unitary representations can be identified withthe minimal unitary representation, and if so, how the infinite set of doubletons are relatedto the minrep. This question was addressed for 5 D anti-de Sitter or 4 D conformal group SU (2 ,
2) and corresponding supergroups SU (2 , | N ) in our earlier work [41]. We showedthat the minimal unitary representation of the group SU (2 ,
2) obtained by quantization of itsquasiconformal realization coincides with the scalar doubleton representation correspondingto a massless scalar field in four dimensions. Furthermore the minrep of SU (2 ,
2) admits aone-parameter ( ζ ) family of deformations, and for a positive (negative) integer value of thedeformation parameter ζ , one obtains a positive energy unitary irreducible representationof SU (2 ,
2) corresponding to a massless conformal field in four dimensions transforming in (cid:16) , ζ (cid:17) (cid:16)(cid:16) − ζ , (cid:17)(cid:17) representation of the Lorentz subgroup, SL (2 , C ). We showed that theserepresentations are simply the doubletons of SU (2 ,
2) that describe massless conformal fieldsin four dimensions [42, 43]. They were referred to as ladder (or most degenerate discreteseries) unitary representations in some of the earlier literature on conformal group and it wasshown by Mack and Todorov that they remain irreducible under restriction to the Poincar´esubgroup [44]. Therefore the deformation parameter ζ can be identified with twice the helic-ity h of the corresponding massless representation of the Poincar´e group. We also extendedthese results to the minimal unitary representations of supergroups SU (2 , | N ) with the evensubgroup SU (2 , × U ( N ) and their deformations. The minimal unitary supermultiplet of SU (2 , | N ) coincides with the CPT self-conjugate (scalar) doubleton supermultiplet, and for P SU (2 , |
4) it is simply the four dimensional N = 4 Yang-Mills supermultiplet. We showed– 4 –hat there exists a one-parameter family of deformations of the minimal unitary supermulti-plet of SU (2 , | N ), and each integer value of the deformation parameter ζ leads to a uniqueunitary supermultiplet of SU (2 , | N ). The minimal unitary supermultiplet of SU (2 , | N )and its deformations coincide with the unitary doubleton supermultiplets that were con-structed and studied using the oscillator method earlier [5, 42, 43]. These results extend tothe minreps of SU ( m, n ) and of SU ( m, n | N ) and their deformations in a straightforwardmanner.More recently we gave a detailed study of the minimal unitary representation (minrep) of SO (6 , ≃ SO ∗ (8) over an Hilbert space of functions of five variables, obtained by quantizingits quasiconformal realization, and its deformations, and we constructed the minimal unitarysupermultiplet of OSp (8 ∗ | N ) [45]. We showed that there exists a family of “deformations”of the minrep of SO ∗ (8) labeled by the spin t of an SU (2) T subgroup of the little group SO (4) of lightlike vectors. These deformed minreps labeled by t are positive energy unitaryirreducible representations of SO ∗ (8) that describe massless conformal fields in six dimensions.The SU (2) T spin t is the six dimensional analog of U (1) deformations of the minrep of4 D conformal group SU (2 ,
2) labeled by helicity. The minimal unitary representation of
OSp (8 ∗ | N ) describes a massless six dimensional conformal supermultiplet. In particular, theminimal unitary supermultiplet of OSp (8 ∗ |
4) is the massless supermultiplet of (2 ,
0) conformalfield theory that is believed to be dual to M-theory on
AdS × S . It is simply the scalardoubleton supermultiplet of OSp (8 ∗ |
4) first constructed in [7].The oscillator construction of the positive energy unitary supermultiplets of
OSp (8 ∗ | N )was first given in [7]. The unitary supermultiplets of OSp (8 ∗ | N ) and their applicationsto AdS /CF T dualities were further studied in [46, 47], where it was shown that the dou-bleton supermultiplets correspond to massless conformal supermultiplets in six dimensions.Construction of positive energy unitary supermultiplets of OSp (8 ∗ | N ) using harmonic super-space methods as well as their applications to AdS /CF T dualities were studied in [48, 49].A classification of positive energy unitary supermultiplets of 6 D superconformal algebrasusing Cartan-Kac formalism was given in [50, 51]. The oscillator construction of positiveenergy unitary representations of general supergroups OSp (2 M ∗ | N ) with even subgroups SO ∗ (2 M ) × U Sp (2 N ) was given much earlier in [52].In this paper we extend the results of [45] and show that the minimal unitary supermul-tiplet of OSp (8 ∗ | N ) admits deformations labeled uniquely again by the spin t of an SU (2) T subgroup and construct all such deformed minimal unitary supermultiplets. In section 2,we review our results on the minimal unitary representation of SO ∗ (8) and its deformationsrealized over the Hilbert space of functions of five variables. In particular we give a “particlebasis” for these unitary representations over the tensor product of Fock space of four bosonicoscillators with the state space of a conformal (singular) oscillator. Their transformations un-der a distinguished SO (4) × U (1) × U (1) subgroup are also given. In section 3, we present thedeformations of the minimal unitary representation of OSp (8 ∗ | N ) labeled by the spin t of an SU (2) T subgroup. Section 4 presents the compact 3-graded decomposition of the Lie super-algebra of OSp (8 ∗ | N ) with respect to the subsuperalgebra of U (4 | N ). In section 5, we give– 5 –he general deformed minimal unitary representations of OSp (8 ∗ | N ) as 6 D massless confor-mal supermultiplets. In section 6, we study the deformed minimal unitary supermultiplets of OSp (8 ∗ |
4) which is the symmetry superalgebra of eleven dimensional supergravity compacti-fied over
AdS × S . We show that the minimal unitary supermultiplet of OSp (8 ∗ |
4) and itsdeformations are precisely the doubleton supermultiplets that were constructed and studiedusing the twistorial oscillator construction [7, 46, 47]. Appendix A reviews the construction ofrelevant representations of
U Sp (2 N ) using “supersymmetry fermions.”
2. Deformations of the Minimal Unitary Representation of SO ∗ (8) In our previous work [45], we gave a detailed study of the minimal unitary representation of
AdS or Conf group SO ∗ (8) ≃ SO (6 ,
2) obtained by the quantization of its quasiconformalrealization [38]. This minrep coincides with the scalar doubleton representation of SO ∗ (8),which corresponds to a massless conformal scalar field in six dimensions [7, 46, 47]. Thereare infinitely many other doubleton representations of SO ∗ (8), corresponding to 6 D masslessconformal fields of higher spin [7, 46, 47]. In the oscillator approach [1, 2], all the doubletonrepresentations can be constructed over the Fock space of two pairs of twistorial oscillatorstransforming in the spinor representation of SO ∗ (8) [7,46]. In [45], we obtained all these higherspin doubleton representations from the minimal unitary representation via a “deformation”in a manner similar to what happens in the case of 4 D conformal group SU (2 ,
2) [41].In this section, we shall review how one deforms the minimal unitary representation of SO ∗ (8) so as to obtain infinitely many irreducible unitary representations that are isomorphicto the irreducible doubleton representations of SO ∗ (8). SO ∗ (8) with respect to the subgroup SO ∗ (4) × SU (2) × SO (1 , so ∗ (8) has a 5-grading with respect to its subalgebra g (0) = so ∗ (4) ⊕ su (2) ⊕ so (1 ,
1) [38]: so ∗ (8) = g ( − ⊕ g ( − ⊕ [ so ∗ (4) ⊕ su (2) ⊕ ∆] ⊕ g (+1) ⊕ g (+2) (2.1)such that h ∆ , g ( m ) i = m g ( m ) (2.2)where ∆ is the SO (1 ,
1) generator. In this decomposition, the subspaces g ( ± are one-dimensional, and the subspaces g ( ± transform in the ( , ) dimensional representation of SO ∗ (4) × SU (2). Since so ∗ (4) = su (1 , ⊕ su (2), the grade zero subalgebra can be written as g (0) = su (1 , N ⊕ su (2) A ⊕ su (2) T ⊕ so (1 ,
1) (2.3) We use the standard convention of denoting the groups with capital letters and the corresponding Liealgebras with small case letters. – 6 –here we denoted the su (1 ,
1) and su (2) subalgebras of so ∗ (4) as su (1 , N and su (2) A , re-spectively, and the su (2) that commutes with so ∗ (4) in equation (2.1) as su (2) T . In theundeformed case, this su (2) was denoted as su (2) S in [45]. In the deformation of the mini-mal unitary representation of SO ∗ (8), the subalgebra su (2) S gets extended to the diagonalsubalgebra su (2) T ⊂ su (2) S ⊕ su (2) G (2.4)where the generators of su (2) S are realized as bilinears of bosonic oscillators and those of of su (2) G are realized in terms of fermionic oscillators. To realize the minrep of SO ∗ (8) and its deformations, one first introduces bosonic annihi-lation operators a m , b m and their hermitian conjugates a m = ( a m ) † , b m = ( b m ) † ( m, n, · · · =1 ,
2) that satisfy the commutation relations:[ a m , a n ] = [ b m , b n ] = δ nm [ a m , a n ] = [ a m , b n ] = [ b m , b n ] = 0 (2.5)and a single “central-charge coordinate” x and its conjugate momentum p such that[ x , p ] = i . (2.6)Now the generators of su (2) S are realized as follows: S + = a m b m S − = ( S + ) † = a m b m S = 12 ( N a − N b ) (2.7)where N a = a m a m and N b = b m b m are the respective number operators. They satisfy:[ S + , S − ] = 2 S [ S , S ± ] = ± S ± (2.8)The quadratic Casimir of su (2) S is C [ su (2) S ] = S = S + 12 ( S + S − + S − S + )= 12 ( N a + N b ) (cid:20)
12 ( N a + N b ) + 1 (cid:21) − a [ m b n ] a [ m b n ] (2.9)where square bracketing a [ m b n ] = ( a m b n − a n b m ) represents antisymmetrization of weightone.To realize SU (2) G , we introduce an arbitrary number P pairs of fermionic annihilation op-erators ξ x and χ x and their hermitian conjugates ξ x = ( ξ x ) † and χ x = ( χ x ) † ( x = 1 , , . . . , P )that satisfy the usual anti-commutation relations: { ξ x , ξ y } = { χ x , χ y } = δ xy { ξ x , ξ y } = { ξ x , χ y } = { χ x , χ y } = 0 (2.10)The generators of SU (2) G are given by the following bilinears of these fermionic oscillators: G + = ξ x χ x G − = χ x ξ x G = 12 ( N ξ − N χ ) (2.11) We should note that in our previous paper [45], we added a “ ◦ ” above all deformed generators to distinguishthem from the undeformed generators. In this paper, we drop those circles for the sake of simplicity. – 7 –here N ξ = ξ x ξ x and N χ = χ x χ x are the respective number operators. They satisfy thecommutation relations: [ G + , G − ] = 2 G [ G , G ± ] = ± G (2.12)Then the generators of su (2) T are simply: T + = S + + G + = a m b m + ξ x χ x T − = S − + G − = b m a m + χ x ξ x T = S + G = 12 ( N a − N b + N ξ − N χ ) (2.13)The su (2) G components realized in terms of fermions represent the deformations of the minrep.The quadratic Casimir of the subalgebra su (2) T is C [ su (2) T ] = T = T + 12 ( T + T − + T − T + ) . (2.14)The generators of su (2) A and su (1 , N , which we denote as A ± , and N ± , , respectively,are realized purely in terms of bosonic oscillators: A + = a a + b b A − = ( A + ) † = a a + b b A = 12 (cid:0) a a − a a + b b − b b (cid:1) N + = a b − a b N − = ( N + ) † = a b − a b N = 12 ( N a + N b ) + 1 (2.15)and they do not get modified by ξ - and χ -type fermionic oscillators under deformation. Theysatisfy the commutation relations:[ A + , A − ] = 2 A [ A , A ± ] = ± A ± [ N − , N + ] = 2 N [ N , N ± ] = ± N ± (2.16)The quadratic Casimirs of these subalgebras C [ su (2) A ] = A = A + 12 ( A + A − + A − A + ) C [ su (1 , N ] = N = N −
12 ( N + N − + N − N + ) (2.17)coincide and are equal to that of su (2) S in the minrep: S = A = N (2.18)The generator ∆ that defines the 5-grading is realized in terms of the “central chargecoordinate” x and its conjugate momentum p as∆ = 12 ( xp + px ) . (2.19) We should note that SU (2) G , as defined in equation (2.11), commutes with the USp (2 P ) group generatedby the bilinears ξ ( x χ y ) , ( ξ x ξ y − χ y χ x ) and ξ ( x χ y ) . – 8 –he generator in grade − K − = 12 x (2.20)and the eight generators in grade − U m = x a m V m = x b m U m = x a m V m = x b m (2.21)Together with K − , they form an Heisenberg algebra:[ U m , U n ] = [ V m , V n ] = 2 δ nm K − [ U m , U n ] = [ U m , V n ] = [ V m , V n ] = 0 (2.22)The single generator in grade +2 subspace is realized as follows: K + = 12 p + 14 x (cid:18) T + 32 (cid:19) (2.23)The generators ∆, K ± form a distinguished su (1 ,
1) subalgebra, that we denote as su (1 , K :[ K − , K + ] = i ∆ [∆ , K ± ] = ± i K ± (2.24)Its quadratic Casimir operator turns out to be equal to that of su (2) T : C [ su (1 , K ] = K = 12 ( K + K − + K − K + ) −
14 ∆ = T (2.25)The generators in grade +1 subspace can be obtained by taking the commutators of theform (cid:2) g ( − , g (+2) (cid:3) : e U m = i [ U m , K + ] e U m = (cid:16) e U m (cid:17) † = i [ U m , K + ] e V m = i [ V m , K + ] e V m = (cid:16) e V m (cid:17) † = i [ V m , K + ] (2.26)Explicitly they are given by: e U m = − p a m + 2 ix (cid:20)(cid:18) T + 34 (cid:19) a m + T − b m (cid:21)e U m = − p a m − ix (cid:20)(cid:18) T − (cid:19) a m + T + b m (cid:21)e V m = − p b m − ix (cid:20)(cid:18) T − (cid:19) b m − T + a m (cid:21)e V m = − p b m + 2 ix (cid:20)(cid:18) T + 34 (cid:19) b m − T − a m (cid:21) (2.27)– 9 –hey form an Heisenberg algebra with K + as its “central charge”: h e U m , e U n i = h e V m , e V n i = 2 δ nm K + h e U m , e U n i = h e U m , e V n i = h e V m , e V n i = 0 (2.28)The commutators (cid:2) g ( − , g (+1) (cid:3) close into grade − h e U m , K − i = i U m h e V m , K − i = i V m h e U m , K − i = i U m h e V m , K − i = i V m (2.29)The non-vanishing commutators of the form (cid:2) g ( − , g (+1) (cid:3) are: h U m , e U n i = − δ nm ∆ − i δ nm N − i δ nm T − i A nm h V m , e V n i = − δ nm ∆ − i δ nm N + 2 i δ nm T − i A nm h U m , e V n i = − i δ nm T − h V m , e U n i = − i δ nm T + h U m , e V n i = − i ǫ mn N − h V m , e U n i = +2 i ǫ mn N − (2.30)where ǫ mn is the Levi-Civita tensor ( ǫ = +1) and we have labeled the generators of su (2) A as A mn : A = − A = A A = A + A = (cid:0) A (cid:1) † = A − (2.31)With the generators defined above, the 5-graded decomposition of the deformed minimalunitary realization, which we denote as so ∗ (8) D , takes the form: so ∗ (8) D = g ( − D ⊕ g ( − D ⊕ [ so ∗ (4) ⊕ su (2) T ⊕ ∆] ⊕ g (+1) D ⊕ g (+2) D = ⊕ ( , ) ⊕ [ su (2) A ⊕ su (1 , N ⊕ su (2) T ⊕ so (1 , ∆ ] ⊕ ( , ) ⊕ = K − ⊕ [ U m , U m , V m , V m ] ⊕ [ A ± , ⊕ N ± , ⊕ T ± , ⊕ ∆ ] ⊕ h e U m , e U m , e V m , e V m i ⊕ K + (2.32)The quadratic Casimir of so ∗ (8) D is given by C [ so ∗ (8) D ] = C [ su (2) T ] + C [ su (2) A ] + C [ su (1 , N ] + C [ su (1 , K ] − i F (cid:16) U, e U , V, e V (cid:17) (2.33)where F (cid:16) U, e U , V, e V (cid:17) = (cid:16) U m e U m + V m e V m + e U m U m + e V m V m (cid:17) − (cid:16) U m e U m + V m e V m + e U m U m + e V m V m (cid:17) (2.34)– 10 –nd reduces to C [ so ∗ (8) D ] = 2 G − G is the quadratic Casimir of su (2) G . Thus the quadratic Casimir of the deformedminrep of SO ∗ (8) depends only on the quadratic Casimir of SU (2) G constructed out of ξ -and χ -type fermionic oscillators used to deform the minrep. SO ∗ (8) D with respect to the subgroup SU ∗ (4) × SO (1 , SO ∗ (8) D has a noncompact 3-gradingdetermined by the dilatation generator D : so ∗ (8) D = N − D ⊕ N D ⊕ N + D (2.36)where N D = su ∗ (4) ⊕ so (1 , su ∗ (4) ≃ so (5 ,
1) corresponding to the six dimensionalLorentz group. The so (1 ,
1) dilatation generator is given by D = 12 [∆ − i ( N + − N − )] . (2.37)The generators that belong to N ± D and N D subspaces are as follows: N − D = K − ⊕ (cid:20) N −
12 ( N + + N − ) (cid:21) ⊕ (cid:0) U − V (cid:1) ⊕ (cid:0) U + V (cid:1) ⊕ (cid:0) V + U (cid:1) ⊕ (cid:0) V − U (cid:1) N D = D ⊕
12 [∆ + i ( N + − N − )] ⊕ T ± , ⊕ A ± , ⊕ (cid:0) U + V (cid:1) ⊕ (cid:0) U − V (cid:1) ⊕ (cid:0) V − U (cid:1) ⊕ (cid:0) V + U (cid:1) ⊕ (cid:16) e U − e V (cid:17) ⊕ (cid:16) e U + e V (cid:17) ⊕ (cid:16) e V + e U (cid:17) ⊕ (cid:16) e V − e U (cid:17) N + D = K + ⊕ (cid:20) N + 12 ( N + + N − ) (cid:21) ⊕ (cid:16) e U + e V (cid:17) ⊕ (cid:16) e U − e V (cid:17) ⊕ (cid:16) e V − e U (cid:17) ⊕ (cid:16) e V + e U (cid:17) (2.38)In terms of the above operators, the Lorentz group generators M µν ( µ, ν, · · · = 0 , , , . . . , N D are given by: M = 14 h(cid:0) U + V (cid:1) + (cid:0) V + U (cid:1) + i (cid:16) e U − e V (cid:17) + i (cid:16) e V − e U (cid:17)i M = i h(cid:0) U + V (cid:1) − (cid:0) V + U (cid:1) + i (cid:16) e U − e V (cid:17) − i (cid:16) e V − e U (cid:17)i M = i h(cid:0) U − V (cid:1) + (cid:0) V − U (cid:1) + i (cid:16) e U + e V (cid:17) + i (cid:16) e V + e U (cid:17)i M = − h(cid:0) U − V (cid:1) − (cid:0) V − U (cid:1) + i (cid:16) e U + e V (cid:17) − i (cid:16) e V + e U (cid:17)i (2.39a)– 11 – = 14 h(cid:0) U + V (cid:1) + (cid:0) V + U (cid:1) − i (cid:16) e U − e V (cid:17) − i (cid:16) e V − e U (cid:17)i M = i h(cid:0) U + V (cid:1) − (cid:0) V + U (cid:1) − i (cid:16) e U − e V (cid:17) + i (cid:16) e V − e U (cid:17)i M = i h(cid:0) U − V (cid:1) + (cid:0) V − U (cid:1) − i (cid:16) e U + e V (cid:17) − i (cid:16) e V + e U (cid:17)i M = − h(cid:0) U − V (cid:1) − (cid:0) V − U (cid:1) − i (cid:16) e U + e V (cid:17) + i (cid:16) e V + e U (cid:17)i (2.39b) M = T + A M = i T + − T − − A + + A − ) M = −
12 ( T + + T − − A + − A − ) M = 12 ( T + + T − + A + + A − ) M = i T + − T − + A + − A − ) M = T − A (2.39c) M = 12 [∆ + i ( N + − N − )] (2.39d)They satisfy the commutation relations[ M µν , M ρτ ] = i ( η νρ M µτ − η µρ M ντ − η ντ M µρ + η µτ M νρ ) (2.40)where η µν = diag( − , + , + , + , + , +). The six generators of grade +1 space are the momenta P µ that generate translations, and the six generators of grade − K µ ( µ = 0 , , , . . . , P = K + + (cid:20) N + 12 ( N + + N − ) (cid:21) P = − h(cid:16) e U + e V (cid:17) + (cid:16) e V + e U (cid:17)i P = − i h(cid:16) e U + e V (cid:17) − (cid:16) e V + e U (cid:17)i P = − i h(cid:16) e U − e V (cid:17) + (cid:16) e V − e U (cid:17)i P = 12 h(cid:16) e U − e V (cid:17) − (cid:16) e V − e U (cid:17)i P = K + − (cid:20) N + 12 ( N + + N − ) (cid:21) K = (cid:20) N −
12 ( N + + N − ) (cid:21) + K − K = i (cid:2)(cid:0) U − V (cid:1) + (cid:0) V − U (cid:1)(cid:3) K = − (cid:2)(cid:0) U − V (cid:1) − (cid:0) V − U (cid:1)(cid:3) K = − (cid:2)(cid:0) U + V (cid:1) + (cid:0) V + U (cid:1)(cid:3) K = − i (cid:2)(cid:0) U + V (cid:1) − (cid:0) V + U (cid:1)(cid:3) K = (cid:20) N −
12 ( N + + N − ) (cid:21) − K − (2.41)They satisfy the commutation relations:[ D , P µ ] = + i P µ [ D , K µ ] = − i K µ [ D , M µν ] = [ P µ , P ν ] = [ K µ , K ν ] = 0[ P µ , M νρ ] = i ( η µν P ρ − η µρ P ν )[ K µ , M νρ ] = i ( η µν K ρ − η µρ K ν )[ P µ , K ν ] = 2 i ( η µν D + M µν ) (2.42)– 12 –nterestingly, none of the special conformal transformations K µ ( µ = 0 , , , . . . ,
5) receivesany contributions from ξ - or χ -type fermionic oscillators used to deform the minrep of SO ∗ (8).Furthermore, the six dimensional Poincar´e mass operator vanishes identically: M = η µν P µ P ν = 0 (2.43)for the deformed minimal unitary realization of SO ∗ (8) given above. Hence each deformedirreducible minrep corresponds to a massless conformal field in six dimensions. SO ∗ (8) D with respect to the subgroup SU (4) × U (1)The Lie algebra of so ∗ (8) D can be given a compact 3-grading so ∗ (8) D = C − D ⊕ C D ⊕ C + D (2.44)with respect to its maximal compact subalgebra C D = su (4) ⊕ u (1), determined by the u (1)generator: H = N + 12 ( K + + K − ) (2.45)This u (1) generator plays the role of the AdS energy or the conformal Hamiltonian when SO ∗ (8) ≃ SO (6 ,
2) is taken as the seven dimensional
AdS group or the six dimensionalconformal group, respectively. In terms of the time components of momenta and specialconformal generators defined in the noncompact 3-graded basis (equation (2.41)), we have H = 12 ( K + P ) . (2.46)The grade − Y m = 12 (cid:16) U m − i e U m (cid:17) = 12 ( x + i p ) a m + 1 x (cid:20)(cid:18) T + 34 (cid:19) a m + T − b m (cid:21) Z m = 12 (cid:16) V m − i e V m (cid:17) = 12 ( x + i p ) b m − x (cid:20)(cid:18) T − (cid:19) b m − T + a m (cid:21) N − = a b − a b B − = i i ( K + − K − )] = 14 ( x + i p ) − x (cid:18) T + 316 (cid:19) (2.47)and the grade +1 operators are given by their hermitian conjugates: Y m = 12 (cid:16) U m + i e U m (cid:17) = 12 ( x − i p ) a m + 1 x (cid:20)(cid:18) T − (cid:19) a m + T + b m (cid:21) Z m = 12 (cid:16) V m + i e V m (cid:17) = 12 ( x − i p ) b m − x (cid:20)(cid:18) T + 34 (cid:19) b m − T − a m (cid:21) N + = a b − a b B + = − i − i ( K + − K − )] = 14 ( x − i p ) − x (cid:18) T + 316 (cid:19) (2.48)– 13 –he su (4) subalgebra has the maximal subalgebra su (4) ⊃ su (2) T ⊕ su (2) A ⊕ u (1) J where the u (1) J generator is given by: J = N −
12 ( K + + K − ) = 12 ( K − P ) (2.49)The generators T ± , and A ± , of su (2) T and su (2) A were given in equations (2.13) and (2.15)and the generators belonging to the coset SU (4) / [ SU (2) T × SU (2) A × U (1) J ]are as follows: C m = 12 (cid:16) U m + i e U m (cid:17) = 12 ( x − i p ) a m − x (cid:20)(cid:18) T + 34 (cid:19) a m + T − b m (cid:21) C m = 12 (cid:16) U m − i e U m (cid:17) = 12 ( x + i p ) a m − x (cid:20)(cid:18) T − (cid:19) a m + T + b m (cid:21) C m = 12 (cid:16) V m + i e V m (cid:17) = 12 ( x − i p ) b m + 1 x (cid:20)(cid:18) T − (cid:19) b m − T + a m (cid:21) C m = 12 (cid:16) V m − i e V m (cid:17) = 12 ( x + i p ) b m + 1 x (cid:20)(cid:18) T + 34 (cid:19) b m − T − a m (cid:21) (2.50)The u (1) generator H that defines the compact 3-grading is an operator whose spectrumis bounded from below. Hence the minrep of SO ∗ (8) and its deformations are all positiveenergy unitary (lowest weight) representations. The unitary lowest weight representations of SO ∗ (8) D are uniquely labeled by a lowest energy K-type, that transforms irreducibly underthe SU (4) subgroup, with the lowest energy eigenvalue with respect to the U (1) generator H , and are annihilated by all the grade − C − D . Since SU (2) T × SU (2) A × U (1) J is a maximal subgroup of SU (4), one can label these lowest energy K-types by the SU (2) T × SU (2) A × U (1) J quantum numbers of their highest weight vectors as irreps of SU (4). SU (1 , K subgroup of SO ∗ (8) D generated by the isotonic (sin-gular) oscillators Note that the u (1) generator H , given in equation (2.45), that determines the compact 3-grading of so ∗ (8) D can be written as H = H a + H b + H ⊙ (2.51)where H a = 12 ( N a + 2) H b = 12 ( N b + 2) (2.52)– 14 –re simply the Hamiltonians of standard bosonic oscillators of a - and b -type. On the otherhand, H ⊙ = 12 ( K + + K − ) = 14 (cid:0) x + p (cid:1) + 1 x (cid:18) T + 316 (cid:19) = 14 (cid:18) x − ∂ ∂x (cid:19) + 1 x (cid:18) T + 316 (cid:19) (2.53)is the Hamiltonian of a singular harmonic oscillator with a singular potential function V D ( x ) = G D x where G D = 2 T + 38 . (2.54) H ⊙ also arises as the Hamiltonian of conformal quantum mechanics [53] with G D playingthe role of the coupling constant [35]. In some literature it is also referred to as the isotonicoscillator [54, 55].Together with the generators B ± belonging to C ± D subspaces of so ∗ (8) D (equations (2.47)and (2.48)): B − = i i ( K + − K − )] = 14 ( x + ip ) − x (cid:18) T + 316 (cid:19) = 14 (cid:18) x + ∂∂x (cid:19) − x (cid:18) T + 316 (cid:19) B + = − i − i ( K + − K − )] = 14 ( x − ip ) − x (cid:18) T + 316 (cid:19) = 14 (cid:18) x − ∂∂x (cid:19) − x (cid:18) T + 316 (cid:19) (2.55) H ⊙ generates the distinguished su (1 , K subalgebra: [ B − , B + ] = 2 H ⊙ [ H ⊙ , B ± ] = ± B ± (2.56)For a given eigenvalue t ( t + 1) of the quadratic Casimir T of su (2) T , the wave functionscorresponding to the lowest energy eigenvalue of this singular harmonic oscillator Hamilto-nian will be superpositions of functions of the form ψ ( α t )0 ( x ) Λ ( t, m t ), where Λ ( t, m t ) is aneigenstate of T and T , independent of x : T Λ ( t, m t ) = t ( t + 1) Λ ( t, m t ) T Λ ( t, m t ) = m t Λ ( t, m t ) (2.57)and ψ ( α t )0 ( x ) is a function of x that satisfies B − ψ ( α t )0 ( x ) Λ ( t, m t ) = 0 (2.58)whose solution is given by [56] ψ ( α t )0 ( x ) = C x α t e − x / (2.59) This is the su (1 ,
1) subalgebra generated by the longest root vector. – 15 –here C is a normalization constant and α t = 12 + p t ( t + 1) = 2 t + 32 . (2.60)The normalizability of the state imposes the constraint α t ≥ . (2.61)A state of the form ψ ( α t =2 t +3 / ( x ) Λ ( t, m t ) is an eigenstate of H ⊙ with eigenvalue ( t + 1): H ⊙ ψ (2 t +3 / ( x ) Λ ( t, m t ) = ( t + 1) ψ (2 t +3 / ( x ) Λ ( t, m t ) (2.62)which is the lowest energy eigenvalue in the deformed case with the deformation parameter t . Higher energy eigenstates of H ⊙ can be obtained from ψ (2 t +3 / ( x ) Λ ( t, m t ) by acting onit repeatedly with the raising generator B + : ψ (2 t +3 / n ( x ) Λ ( t, m t ) = C n ( B + ) n ψ (2 t +3 / ( x ) Λ ( t, m t ) (2.63)where C n are normalization constants. They correspond to energy eigenvalues n + t + 1: H ⊙ ψ (2 t +3 / n ( x ) Λ ( t, m t ) = ( n + t + 1) ψ (2 t +3 / n ( x ) Λ ( t, m t ) (2.64)We shall denote the corresponding states as (cid:12)(cid:12)(cid:12) ψ (2 t +3 / n ( x ) ; Λ ( t, m t ) E = (cid:12)(cid:12)(cid:12) ψ (2 t +3 / n ( x ) E ⊗ | Λ ( t, m t ) i and refer to them as the particle basis of the state space of the (isotonic) singular oscillator.The (2 t + 1) states belonging to the subspace corresponding to an irrep of SU (2) T labeledby spin t will all have the same eigenvalue of H ⊙ . SU (2) T × SU (2) A × U (1) J × U (1) H basis of the deformed minimal unitary repre-sentations of SO ∗ (8)The fermionic Fock vacuum | i F is chosen such that: ξ x | i F = χ x | i F = 0 x = 1 , , . . . , P (2.65)A “particle basis” of states in the fermionic Fock space is provided by the action of creationoperators ξ x and χ y on the Fock vacuum | i F . A state of the form χ [ x χ x χ x . . . χ x P ] | i F has a definite eigenvalue − P of G and is annihilated by the lowering operator G − . Byrepeatedly acting on this state with the raising operator G + , one can obtain P other statesof the form: ξ [ x χ x χ x . . . χ x P ] | i F ⊕ ξ [ x ξ x χ x . . . χ x P ] | i F ⊕ . . . . . . ⊕ ξ [ x ξ x ξ x . . . ξ x P ] | i F Note that square bracketing of fermionic indices implies complete anti-symmetrization of weight one. – 16 –e shall denote these P + 1 states as (cid:12)(cid:12)(cid:12)(cid:12) P , m P (cid:29) where m P = − P , − P , . . . , + P . (2.66)They transform irreducibly under su (2) G in the spin P representation. We shall denote thebosonic Fock vacuum annihilated by all bosonic oscillators a m , b m ( m = 1 ,
2) as | i B : a m | i B = b m | i B = 0 (2.67)and the tensor product of fermionic and bosonic vacua simply as | i .The tensor products of the states of the form ( a m ) n a,m | i B , ( b m ) n b,m | i B , ξ x | i F and χ x | i F , where n a,m and n b,m are non-negative integers, form the “particle basis” of statesin the full Fock space. As the “particle basis” of the Hilbert space of the deformed minimalunitary representation of SO ∗ (8), we shall take the following tensor products of the abovestates with the state space of the singular (isotonic) oscillator: (cid:0) a (cid:1) n a, (cid:0) a (cid:1) n a, (cid:0) b (cid:1) n b, (cid:0) b (cid:1) n b, | i B ⊗ ξ [ x . . . ξ x k χ x k +1 . . . χ x P ] | i F ⊗ (cid:12)(cid:12)(cid:12) ψ ( α t ) n E where square brackets imply full anti-symmetrization with weight one. We denote them as (cid:0) a (cid:1) n a, (cid:0) a (cid:1) n a, (cid:0) b (cid:1) n b, (cid:0) b (cid:1) n b, ξ [ x . . . ξ x k χ x k +1 . . . χ x P ] (cid:12)(cid:12)(cid:12) ψ ( α t ) n E or simply as (cid:12)(cid:12)(cid:12)(cid:12) ψ ( α t ) n ; n a, , n a, , n b, , n b, ; P , k − P (cid:29) (2.68)where k = 0 , , . . . , P . For a fixed N = n a, + n a, + n b, + n b, , these states transform in the (cid:0) N + P , N (cid:1) representation under the SU (2) T × SU (2) A subgroup. They are, in general, noteigenstates of J . The ( P + 1) states of the form (cid:12)(cid:12)(cid:12)(cid:12) ψ ( P + )0 ; 0 , , , P , k − P (cid:29) ( k = 0 , , . . . , P )that transform in the (cid:0) P , (cid:1) representation of SU (2) T × SU (2) A are, however, all eigenstatesof J with eigenvalue J = − P . These P + 1 states are annihilated by grade − C − D of so ∗ (8) D (given in equation (2.47)). The action of the coset generators SU (4) / [ SU (2) T × SU (2) A × U (1) J ]given in equation (2.50) on the above states leads to a set of states transforming in anirreducible representation of SU (4) with Dynkin labels (2 t, ,
0) = ( P, , (cid:12)(cid:12)(cid:12) Ω ( P + ) ( P, , E . They are all eigenstates of H ( AdS energy), with the lowesteigenvalue of E = t + 2 = P + 2, and are annihilated by all grade − SO ∗ (8), labeled by the SU (2) G spin g = t = P . The resulting unitary– 17 –rreducible representations correspond to deformations of the minimal unitary representation.With respect to the SU (2) T × SU (2) A × U (1) J subgroup of SU (4), the states (cid:12)(cid:12)(cid:12) Ω ( P + ) ( P, , E has the following decomposition: (cid:12)(cid:12)(cid:12) Ω ( P + ) ( P, , E = (cid:18) P , (cid:19) − P ⊕ (cid:18) P − , (cid:19) − P +1 ⊕ (cid:18) P − , (cid:19) − P +2 ⊕ . . . · · · ⊕ (cid:18) , P (cid:19) + P (2.69)where we have labeled the irreps of SU (2) T × SU (2) A × U (1) J as ( t, a ) J . All the other statesof the “particle basis” of the deformed minrep can be obtained from (cid:12)(cid:12)(cid:12) Ω ( P + ) ( P, , E byrepeatedly acting on them with the six operators in C + D subspace of SO ∗ (8) D . These sixoperators Y m , Z m , N + and B + in C + D transform under SU (2) T × SU (2) A × U (1) J as follows:6 = (1 / , / ⊕ (0 , +1 ⊕ (0 , − . The generators (cid:0) Y , Z (cid:1) and (cid:0) Y , Z (cid:1) form two doublets under SU (2) T , and the generators (cid:0) Y , Y (cid:1) and (cid:0) Z , Z (cid:1) form two doublets under SU (2) A . N + and B + are both singlets under SU (2) T and SU (2) A . The generators Y m and Z m have zero J -charge, while the generators N + and B + have J -charges +1 and −
1, respectively.In Table 1, we give the SU (4) × U (1) H decomposition of the deformed minreps of SO ∗ (8)uniquely determined by the ( P + 1) states (cid:12)(cid:12)(cid:12)(cid:12) ψ ( P + )0 ; 0 , , , P , k − P (cid:29) k = 0 , , . . . , P . (2.70)and labeled by the SU (2) G spin g = t = P . Table 1: In this table, we give the SU (4) × U (1) H decomposition ofthe deformed minimal unitary representation of SO ∗ (8), defined bythe “lowest weight state” (cid:12)(cid:12)(cid:12) ψ ( P + )0 ; 0 , , , P , − P E for any non-negative P . These are massless representations of SO ∗ (8), consid-ered as the 6 D conformal group. As massless 6 D conformal fields,their SU ∗ (4) transformations coincide with SU (4) transformationsof the lowest energy K-type (cid:12)(cid:12)(cid:12) Ω ( P + ) E and whose eigenvalue E of H is the negative of the conformal dimension ℓ . First column givesthe states, second column gives the energy eigenvalues, and thirdcolumn gives the SU (4) Dynkin labels.States E SU (4)Dynkin (cid:12)(cid:12)(cid:12) Ω ( P + ) ( P, , E P + 2 ( P, , – 18 – able 1: (continued)State E = − ℓ SU (4)Dynkin C + D (cid:12)(cid:12)(cid:12) Ω ( P + ) ( P, , E P + 3 ( P, , (cid:0) C + D (cid:1) (cid:12)(cid:12)(cid:12) Ω ( P + ) ( P, , E P + 4 ( P, , (cid:0) C + D (cid:1) n (cid:12)(cid:12)(cid:12) Ω ( P + ) ( P, , E P + n + 2 ( P, n,
The states with SU (4) Dynkin labels ( P, n,
0) and
AdS energy P + n + 2 decompose intothe following SU (2) T × SU (2) A × U (1) J irreps labeled as ( t, a ) J :( P, n,
0) = (cid:18) P , (cid:19) − P − nP ≥ ,n ≥ ⊕ (cid:18) P − , (cid:19) − P − n +1 P ≥ ,n ≥ ⊕ (cid:18) P , (cid:19) − P − n +1 P ≥ ,n ≥ ⊕ (cid:18) P − , (cid:19) − P − n +2 P ≥ ,n ≥ ⊕ (cid:18) P , (cid:19) − P − n +2 P ≥ ,n ≥ ⊕ (cid:18) P , (cid:19) − P − n +2 P ≥ ,n ≥ ⊕ (cid:18) P , (cid:19) − P − n +2 P ≥ ,n ≥ ... ⊕ (cid:18) , P (cid:19) + P + nP ≥ ,n ≥ (2.71)with the subscripts denoting the allowed values of P and n .From the above table, it is clear that the deformed minrep of SO ∗ (8) with deformationparameter t is, in fact, the doubleton representation of SO ∗ (8) whose lowest energy K-typehas the SU (4) Young tableau | Ω i = | . . . | {z } t = P i – 19 –reviously constructed by the oscillator method in [7, 46]. We should stress again that in theoscillator construction of [7, 46], one realizes the generators of SO ∗ (8) as bilinears of two setsof twistorial bosonic oscillators transforming in the spinor representation of SO ∗ (8), and theFock space of these oscillators decomposes into the direct sum of infinitely many doubletonirreps of SO ∗ (8). In the quasiconformal approach, each deformation labeled by SU (2) G spin( g = t ) leads to a unique unitary irrep as explained above.
3. Deformations of the Minimal Unitary Representation of
OSp (8 ∗ | N ) In our previous work [45], we constructed the “undeformed” minimal unitary supermultipletof osp (8 ∗ | N ), with particular emphasis on that of osp (8 ∗ | osp (8 ∗ |
4) is simply the supermultiplet of the six dimensional (2,0) conformal fieldtheory that is believed to be dual to the M-theory on
AdS × S .In this section, we will extend the deformed minimal unitary representations of SO ∗ (8),constructed in section 2, to deformed minimal unitary supermultiplets of OSp (8 ∗ | N ). Forthis purpose, in addition to the ξ - and χ -type fermionic oscillators introduced for deformingthe minrep of SO ∗ (8), we introduce a new set of fermionic oscillators required by supersym-metry. More specifically to realize the compact Lie algebra usp (2 N ), we introduce N copiesof α - and β -type fermionic oscillators (with indices r = 1 , , . . . , N ), as reviewed in AppendixA (see equation (A.1)). We shall refer to the fermions ξ and χ as deformation fermions andthe fermions α and β as supersymmetry fermions .Recall that we labeled the su (2) subalgebra that commutes with so ∗ (4) subalgebra in the5-graded decomposition of the minimal unitary realization of so ∗ (8) as su (2) S such that g (0) = su (2) A ⊕ su (1 , N ⊕ su (2) S ⊕ so (1 , . Under deformation of the minimal unitary realization of so ∗ (8), the subalgebra su (2) S getscontributions from deformation fermions ξ and χ and goes over to su (2) T , which is thediagonal subalgebra of su (2) S and su (2) G (as defined in equation (2.11)). Now in extending thedeformed minimal unitary realization of so ∗ (8) to the deformed minimal unitary realization of osp (8 ∗ | N ), it gets further contributions from supersymmetry fermions α and β . In particularthe subalgebra su (2) T gets extended to su (2) T , which is the diagonal subalgebra of su (2) T and su (2) F , which involves only supersymmetry fermions α and β (as given in equation (A.12)).The generators of su (2) T turn out to be given by: T + = S + + F + + G + = a m b m + α r β r + ξ x χ x T − = S − + F − + G − = b m a m + β r α r + χ x ξ x T = S + F + G = 12 ( N a − N b + N α − N β + N ξ − N χ ) (3.1)The quadratic Casimir of su (2) T is C [ su (2) T ] = T = T T + 12 ( T + T − + T − T + ) . (3.2)– 20 –ow osp (8 ∗ | N ) D has a 5-graded decomposition osp (8 ∗ | N ) D = g ( − D ⊕ g ( − D ⊕ g (0) D ⊕ g (+1) D ⊕ g (+2) D (3.3)with respect to the subsuperalgebra g (0) D = osp (4 ∗ | N ) ⊕ su (2) T ⊕ so (1 , ∆ (3.4)such that grade ± osp (4 ∗ | N ) belonging to grade zero subspace do notget any contributions from the deformation fermions ξ and χ . Its generators have the samerealization as in the undeformed case. The generators of so ∗ (4) = su (2) A ⊕ su (1 , N , thoseof usp (2 N ), and the 8 N supersymmetry generators are given by: A + = a a + b b A − = ( A + ) † = a a + b b A = 12 (cid:0) a a − a a + b b − b b (cid:1) N + = a b − a b N − = ( N + ) † = a b − a b N = 12 ( N a + N b ) + 1 (3.5) S rs = α r β s + α s β r M rs = α r α s − β s β r S rs = β r α s + β s α r = ( S rs ) † (3.6)Π mr = a m β r − b m α r Σ rm = a m α r + b m β r Π mr = (Π mr ) † = a m β r − b m α r Σ mr = (Σ rm ) † = a m α r + b m β r (3.7)In the undeformed minimal unitary realization, the quadratic Casimir of the subsuper-algebra osp (4 ∗ | N ) turns out to be equal to the quadratic Casimir of the diagonal su (2)subalgebra of su (2) S and su (2) F , that commutes with so ∗ (4), modulo an additive constantthat depends on N [45]. However, in the deformed realization, no such simple relation holdsbetween the quadratic Casimir of osp (4 ∗ | N ) and that of su (2) T . In the deformed minimalunitary realization, the quadratic Casimir of the subsuperalgebra osp (4 ∗ | N ) is given by C [ osp (4 ∗ | N )] = C [ su (2) T ] + C [ su (2) G ] − (cid:20) T G + 12 ( T + G − + T − G + ) (cid:21) − N ( N − . (3.8)The single bosonic generator in grade − N supersymmetry generators in grade − osp (8 ∗ | N ) D are also unchanged: K − = K − = 12 x (3.9) U m = U m = x a m V m = V m = x b m U m = U m = x a m V m = V m = x b m (3.10)– 21 – r = Q r = x α r S r = S m = x β r Q r = Q r = x α m S r = S r = x β m (3.11)They form a super Heisenberg algebra by (anti-)commuting into K − . However, since su (2) S has now been extended to su (2) T , the grade +2 generator now depends on T and is givenby K + = 12 p + 14 x (cid:18) T + 32 (cid:19) . (3.12)Therefore the generators in grade +1 subspace get modified, since they are obtained from thecommutators of the form h g ( − D , g (+2) D i : e U m = i [ U m , K + ] e U m = (cid:16) e U m (cid:17) † = i [ U m , K + ] e V m = i [ V m , K + ] e V m = (cid:16) e V m (cid:17) † = i [ V m , K + ] (3.13) e Q r = i [ Q r , K + ] e Q r = (cid:16) e Q r (cid:17) † = i [ Q r , K + ] e S r = i [ S r , K + ] e S r = (cid:16) e S r (cid:17) † = i [ S r , K + ] (3.14)The explicit form of these 8 bosonic generators and 4 N supersymmetry generators of grade+1 subspace are as follows: e U m = − p a m + 2 ix (cid:20)(cid:18) T + 34 (cid:19) a m + T − b m (cid:21)e U m = − p a m − ix (cid:20)(cid:18) T − (cid:19) a m + T + b m (cid:21)e V m = − p b m − ix (cid:20)(cid:18) T − (cid:19) b m − T + a m (cid:21)e V m = − p b m + 2 ix (cid:20)(cid:18) T + 34 (cid:19) b m − T − a m (cid:21) (3.15) e Q r = − p α r + 2 ix (cid:20)(cid:18) T + 34 (cid:19) α r + T − β r (cid:21)e Q r = − p α r − ix (cid:20)(cid:18) T − (cid:19) α r + T + β r (cid:21)e S r = − p β r − ix (cid:20)(cid:18) T − (cid:19) β r − T + α r (cid:21)e S r = − p β r + 2 ix (cid:20)(cid:18) T + 34 (cid:19) β r − T − α r (cid:21) (3.16)These grade +1 generators (anti-)commute into the grade +2 generator and form a superHeisenberg algebra. – 22 –he anticommutators between the supersymmetry generators in g ( − D and g (+1) D givenabove close into the bosonic generators in g (0) D : n Q r , e Q s o = 0 n Q r , e Q s o = − δ sr ∆ − i δ sr T + 2 i M sr n S r , e S s o = 0 n S r , e S s o = − δ sr ∆ + 2 i δ sr T + 2 i M sr n Q r , e S s o = − i S rs n Q r , e S s o = − i δ sr T − n S r , e Q s o = +2 i S rs n S r , e Q s o = − i δ sr T + (3.17)The commutators between the bosonic (even) and fermionic (odd) generators of g ( − D and g (+1) D subspaces close into the fermionic (odd) generators of g (0) D : h U m , e Q r i = 0 h U m , e Q r i = − i Σ rm h U m , e S r i = − i Π mr h U m , e S r i = 0 h V m , e Q r i = +2 i Π mr h V m , e Q r i = 0 h V m , e S r i = 0 h V m , e S r i = − i Σ rm (3.18) h Q r , e U m i = 0 h Q r , e U m i = − i Σ rm h S r , e U m i = − i Π mr h S r , e U m i = 0 h Q r , e V m i = +2 i Π mr h Q r , e V m i = 0 h S r , e V m i = 0 h S r , e V m i = − i Σ rm (3.19)Thus the 5-grading of the Lie superalgebra osp (8 ∗ | N ) D , defined by the generator ∆,takes the form: osp (8 ∗ | D = g ( − D ⊕ g ( − D ⊕ [ osp (4 ∗ | N ) ⊕ su (2) T ⊕ so (1 , ∆ ] ⊕ g (+1) D ⊕ g (+2) D = K − ⊕ [ U m , U m , V m , V m , Q r , Q r , S r , S r ] ⊕ (cid:2) A ± , , N ± , , S rs , M rs , S rs , Π mr , Π mr , Σ rm , Σ mr , T ± , , ∆ (cid:3) ⊕ h e U m , e U m , e V m , e V m , e Q r , e Q r , e S r , e S r i ⊕ K + (3.20)The quadratic Casimir of osp (8 ∗ | N ) D is given by C [ osp (8 ∗ | N ) D ] = C [ osp (4 ∗ | N )] − C [ su (2) T ] + C [ su (1 , K ] − i F ( U , V ) + i F ( Q , S ) (3.21)– 23 –here F ( U , V ) = (cid:16) U m e U m + V m e V m + e U m U m + e V m V m (cid:17) − (cid:16) U m e U m + V m e V m + e U m U m + e V m V m (cid:17) F ( Q , S ) = (cid:16) Q r e Q r + S r e S r − e Q r Q r − e S r S r (cid:17) + (cid:16) Q r e Q r + S r e S r − e Q r Q r − e S r S r (cid:17) (3.22)and reduces to C [ osp (8 ∗ | N ) D ] = G − (cid:0) N − N + 32 (cid:1) (3.23)where G is the quadratic Casimir of su (2) G . Hence each deformed irreducible minimalunitary supermultiplet of osp (8 ∗ | N ) can be labeled by the eigenvalues g ( g + 1) of G as weshall explicitly show later.
4. The Compact 3-Grading of
OSp (8 ∗ | N ) D The Lie superalgebra osp (8 ∗ | N ) D can be given a 3-graded decomposition with respect to itscompact subsuperalgebra u (4 | N ) = su (4 | N ) ⊕ u (1): osp (8 ∗ | N ) D = C − D ⊕ C D ⊕ C + D (4.1)where C − D = 12 (cid:16) U m − i e U m (cid:17) ⊕ (cid:16) V m − i e V m (cid:17) ⊕ N − ⊕ i i ( K + − K − )] ⊕ S rs ⊕ (cid:16) Q r − i e Q r (cid:17) ⊕ (cid:16) S r − i e S r (cid:17) ⊕ Π mr C D = (cid:20) T ± , ⊕ A ± , ⊕ (cid:20) N −
12 ( K + + K − ) (cid:21) ⊕ (cid:16) U m + i e U m (cid:17) ⊕ (cid:16) U m − i e U m (cid:17) ⊕ (cid:16) V m + i e V m (cid:17) ⊕ (cid:16) V m − i e V m (cid:17) ⊕ M rs ⊕ (cid:20)
12 ( K + + K − ) + 2 N M (cid:21)(cid:21) ⊕ H ⊕ (cid:16) Q r + i e Q r (cid:17) ⊕ (cid:16) Q r − i e Q r (cid:17) ⊕ (cid:16) S r + i e S r (cid:17) ⊕ (cid:16) S r − i e S r (cid:17) ⊕ Σ rm ⊕ Σ mr C + D = 12 (cid:16) U m + i e U m (cid:17) ⊕ (cid:16) V m + i e V m (cid:17) ⊕ N + ⊕ − i − i ( K + − K − )] ⊕ S rs ⊕ (cid:16) Q r + i e Q r (cid:17) ⊕ (cid:16) S r + i e S r (cid:17) ⊕ Π mr (4.2)The u (1) generator H that defines the compact 3-grading of osp (8 ∗ | N ) D is given by H = 12 ( K + + K − ) + N + M = 14 (cid:0) x + p (cid:1) + 1 x (cid:18) T + 316 (cid:19) + 12 ( N a + N b + N α + N β ) + 2 − N osp (8 ∗ | N ).In the supersymmetric extension of the deformed minrep, the u (1) generator that corre-sponds to the AdS energy and determines a 3-grading of so ∗ (8) D is given by H = 12 ( K + + K − ) + N = 14 (cid:0) x + p (cid:1) + 1 x (cid:18) T + 316 (cid:19) + 12 ( N a + N b ) + 1= H ⊙ + H a + H b (4.4)where H ⊙ is the Hamiltonian of the singular oscillator: H ⊙ = 12 ( K + + K − ) = 14 (cid:0) x + p (cid:1) + 1 x (cid:18) T + 316 (cid:19) (4.5)and H a and H b are the Hamiltonians corresponding to a - and b -type bosonic oscillators,respectively: H a = 12 ( N a + 1) H b = 12 ( N b + 1)We shall label the bosonic operators that belong to the subspace C − D of the deformed osp (8 ∗ | N ) D in the compact 3-grading as follows: Y m = 12 (cid:16) U m − i e U m (cid:17) = 12 ( x + i p ) a m + 1 x (cid:20)(cid:18) T + 34 (cid:19) a m + T − b m (cid:21) Z m = 12 (cid:16) V m − i e V m (cid:17) = 12 ( x + i p ) b m − x (cid:20)(cid:18) T − (cid:19) b m − T + a m (cid:21) N − = a b − a b B − = i i ( K + − K − )] = 14 ( x + i p ) − x (cid:18) T + 316 (cid:19) S rs = α r β s + α s β r (4.6)and the 4 N supersymmetry generators in C − D subspace as: Q r = 12 (cid:16) Q r − i e Q r (cid:17) = 12 ( x + i p ) α r + 1 x (cid:20)(cid:18) T + 34 (cid:19) α r + T − β r (cid:21) S r = 12 (cid:16) S r − i e S r (cid:17) = 12 ( x + i p ) β r − x (cid:20)(cid:18) T − (cid:19) β r − T + α r (cid:21) Π mr = a m β r − b m α r (4.7)The generators that belong to C + D subspace are the Hermitian conjugates of those in C − D .– 25 –hen the bosonic operators in C + D are: Y m = 12 (cid:16) U m + i e U m (cid:17) = 12 ( x − i p ) a m + 1 x (cid:20)(cid:18) T − (cid:19) a m + T + b m (cid:21) Z m = 12 (cid:16) V m + i e V m (cid:17) = 12 ( x − i p ) b m − x (cid:20)(cid:18) T + 34 (cid:19) b m − T − a m (cid:21) N + = a b − a b B + = − i − i ( K + − K − )] = 14 ( x − i p ) − x (cid:18) T + 316 (cid:19) S rs = α r β s + α s β r (4.8)and the 4 N supersymmetry generators in C + D subspace are: Q r = 12 (cid:16) Q r + i e Q r (cid:17) = 12 ( x − i p ) α r + 1 x (cid:20)(cid:18) T − (cid:19) α r + T + β r (cid:21) S r = 12 (cid:16) S r + i e S r (cid:17) = 12 ( x − i p ) β r − x (cid:20)(cid:18) T + 34 (cid:19) β r − T − α r (cid:21) Π mr = a m β r − b m α r (4.9)Once again, we have the important relation Y Z − Y Z = N + B + (4.10)
5. Deformed Minimal Unitary Representations of
OSp (8 ∗ | N ) as D MasslessConformal Supermultiplets
Since the quadratic Casimir of
OSp (8 ∗ | N ) D depends only on the quadratic Casimir of SU (2) G constructed out of deformation fermions ξ and χ (see equation (3.23)), just as thequadratic Casimir of SO ∗ (8) D depends only on the quadratic Casimir of SU (2) G (see equa-tion (2.35)), one expects to obtain an irreducible unitary supermultiplet of OSp (8 ∗ | N ) foreach spin g labeling the irreps of SU (2) G . Let us show that this indeed is the case. For each SU (2) T spin t = g = P = 0 there is a multiplet of states that are annihilated by all theoperators in grade − C − D and transforms irreducibly under the subsuperalgebra u (4 | N ), which is the grade zero subspace C D . Let us call this supermultiplet of states the“lowest energy K-type” of OSp (8 ∗ | N ) D . To obtain this lowest energy K-type for a given t = g , consider the tensor product of states of the form (cid:12)(cid:12)(cid:12)(cid:12) ψ ( α t ) n ; n a, , n a, , n b, , n b, ; P , k − P (cid:29) constructed earlier, with the states created by the supersymmetry fermions α [ r . . . α r nα β r nα +1 . . . β r nα + nβ ] | i F – 26 –here | i F is the fermionic Fock vacuum annihilated by all the fermionic annihilation opera-tors ξ x , χ x , α r and β r ( x = 1 , . . . , P and r = 1 , . . . , N ), and denote them as (cid:12)(cid:12)(cid:12)(cid:12) ψ ( α t ) n ; n a, , n a, , n b, , n b, ; n α , n β ; P , k − P (cid:29) . (5.1)The following ( P + 1) states of the form (cid:12)(cid:12)(cid:12)(cid:12) ψ ( α t )0 ; 0 , , , , P , k − P (cid:29) ( k = 0 , , . . . , P ) (5.2)transform in the (cid:0) P , (cid:1) representation of SU (2) T × SU (2) A with a definite eigenvalue J = − P with respect to the U (1) J generator J = N − ( K + + K − ), and are annihilated by the sixbosonic operators and N ( N + 1) / − C − D of osp (8 ∗ | N ) D (given in equations (4.6) and (4.7)) for the choice α t = 2 t + 32 = P + 32 . (5.3)These states uniquely define a positive energy unitary supermultiplet of OSp (8 ∗ | N ), labeledby the SU (2) G spin g = t = P , which corresponds to a deformation of the minimal unitarysupermultiplet. By acting on the states in equation (5.2) with the coset generators SU (4 | N ) / [ SU (2) T × SU (2) A × U (1) J ]one obtains a set of states transforming in an irreducible representation of SU (4 | N ) with theYoung supertableau (cid:0)(cid:0) . . . (cid:0) | {z } t . We denote these states as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω ( P + ) , (cid:0)(cid:0) . . . (cid:0) | {z } P =2 t + For a given value of the deformation parameter t (= g ), they have the lowest “total energy” H = t + 1 = g + 1 = P + 1 and are all annihilated by grade − OSp (8 ∗ | N ). By repeatedly acting on this set of states (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω ( P + ) , (cid:0)(cid:0) . . . (cid:0) | {z } P =2 t + with the operators in grade +1 subspace C + D , one obtains an infiniteset of states that form a basis of a unitary irreducible representation of OSp (8 ∗ | N ). Thisinfinite set of states can be decomposed into a finite number of irreducible representations ofthe even subgroup SO ∗ (8) × U Sp (2 N ), with each irrep of SO ∗ (8) corresponding to a masslessconformal field in six dimensions.In Table 2, we present the general deformed minimal unitary supermultiplet of osp (8 ∗ | N )corresponding to the deformation parameter t (= g ), obtained by starting from the lowestenergy K-type (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω ( P + ) , (cid:0)(cid:0) . . . (cid:0) | {z } P =2 t =2 g + . – 27 – able 2: The general deformed minimal unitary supermultiplet of osp (8 ∗ | N ) corresponding to the deformation parameter t = g = P/
2. First column gives the
AdS energy H which is equal to neg-ative conformal dimension ℓ of the corresponding conformal field.Last column gives the SU (4) Dynkin labels of the lowest energyK-type of SO ∗ (8), which coincides with the Dynkin labels of thecorresponding conformal field under the Lorentz group SU ∗ (4), andthe Dynkin labels with respect to U Sp (2 N ). The decompositionof SU (4) irreps with respect to SU (2) T × SU (2) A × U (1) J is de-noted by ( t , a ) J and is listed in the third column. Second columngives the eigenvalues of “total energy” H . Note that for P < N thestates with negative entries in their Dynkin labels do not occur. H = − ℓ H ( t , a ) J SU ∗ (4) Dynkin = SU (4) Dynkin
U Sp (2 N ) Dynkin t + 2 t + 2 − N ( t , − t ⊕ (cid:0) t − , (cid:1) − t +1 (2 t , , SU (4) ⊕ · · · ⊕ (0 , t ) t (0 , . . . , | {z } ( N − , USp (2 N ) t + t + 3 − N (cid:0) t + , (cid:1) − t − ⊕ (cid:0) t , (cid:1) − t + (2 t + 1 , , SU (4) ⊕ · · · ⊕ (cid:0) , t + (cid:1) t + (0 , . . . , | {z } ( N − , , USp (2 N ) ... ... ... ... t + 2 + N t + 2 + N (cid:0) t + N , (cid:1) − t − N (2 t + N, , SU (4) ⊕ (cid:0) t + N − , (cid:1) − t − N − (0 , . . . , | {z } N ) USp (2 N ) ⊕ · · · ⊕ (cid:0) , t + N (cid:1) t + N t + t + 2 − N (cid:0) t − , (cid:1) − t + (2 t − , , SU (4) ⊕ (cid:0) t − , (cid:1) − t + (0 , . . . , | {z } ( N − , , USp (2 N ) ⊕ · · · ⊕ (cid:0) , t − (cid:1) t − – 28 – able 2: (continued) H = − ℓ H ( t , a ) J SU ∗ (4) Dynkin = SU (4) Dynkin
U Sp (2 N ) Dynkin t + 2 t + 3 − N ( t , − t ⊕ (cid:0) t − , (cid:1) − t +1 (2 t , , SU (4) ⊕ · · · ⊕ (0 , t ) t (0 , . . . , | {z } ( N − , , , USp (2 N ) ... ... ... ... t + 1 + N t + 1 + N (cid:0) t + N − , (cid:1) − t − N − (2 t − N, , SU (4) ⊕ (cid:0) t + N − , (cid:1) − t − N − (0 , . . . , | {z } N ) USp (2 N ) ⊕ · · · ⊕ (cid:0) , t + N − (cid:1) t + N − ... ... ... ...... ... ... ... t + 2 − n t + 2 − N (cid:0) t − n , (cid:1) − t + n (2 t − n, , SU (4) ⊕ (cid:0) t − n +12 , (cid:1) − t + n +22 ( 0 , . . . , | {z } ( N − n − , , , . . . , USp (2 N ) ⊕ · · · ⊕ (cid:0) , t − n (cid:1) t − n t + − n t + 3 − N (cid:0) t − n − , (cid:1) − t + n − (2 t − n + 1 , , SU (4) ⊕ (cid:0) t − n , (cid:1) − t + n +12 ( 0 , . . . , | {z } ( N − n − , , , . . . , USp (2 N ) ⊕ · · · ⊕ (cid:0) , t − n − (cid:1) t − n − ... ... ... ... – 29 – able 2: (continued) H = − ℓ H ( t , a ) J SU ∗ (4) Dynkin = SU (4) Dynkin
U Sp (2 N ) Dynkin t + 2 − n + N t + 2 − n + N (cid:0) t − n + N , (cid:1) − t + n − N (2 t − n + N, , SU (4) ⊕ (cid:0) t − n + N − , (cid:1) − t + n − N − (0 , . . . , | {z } N ) USp (2 N ) ⊕ · · · ⊕ (cid:0) , t − n + N (cid:1) t − n + N ... ... ... ...... ... ... ... t + 2 − N t + 2 − N (cid:0) t − N , (cid:1) − t + N (2 t − N, , SU (4) ⊕ (cid:0) t − N +12 , (cid:1) − t + N +22 (0 , . . . , | {z } N ) USp (2 N ) ⊕ · · · ⊕ (cid:0) , t − N (cid:1) t − N
6. Deformed Minimal Unitary Supermultiplets of
OSp (8 ∗ | Due to its importance as the symmetry superalgebra of the S compactification of the elevendimensional supergravity, we shall discus the results for the case N = 2, i.e OSp (8 ∗ | r, s, . . . of α - and β -type (supersymmetry) fermionic oscillators takethe values 1,2 in this case.Recall that the undeformed minimal unitary supermultiplet is obtained by taking thedeformation parameter t = 0, which we present in Table 3.– 30 – able 3: The minimal unitary supermultiplet of osp (8 ∗ |
4) definedby the lowest weight vector (cid:12)(cid:12) Ω (3 / , (cid:11) , which corresponds to thedeformation parameter t = 0. The decomposition of SU (4) irrepswith respect to SU (2) T × SU (2) A × U (1) J is denoted by ( t , a ) J . H is the AdS energy (negative conformal dimension), and H is thetotal energy. The Dynkin labels of the lowest energy SU (4) rep-resentations of SO ∗ (8) coincide with the Dynkin labels of the cor-responding massless 6 D conformal fields under the Lorentz group SU ∗ (4). U Sp (4) Dynkin labels of these fields are also given. H = − ℓ H ( t , a ) J SU (4) = SU ∗ (4) U Sp (4)Dynkin Dynkin2 1 (0 , (0,0,0) (0,1) (cid:0) , (cid:1) − ⊕ (cid:0) , (cid:1) + (1,0,0) (1,0)3 3 (1 , − ⊕ (cid:0) , (cid:1) ⊕ (0 , +1 (2,0,0) (0,0) The simplest deformed case is when the deformation parameter t = g = . This makes α t = . In this case, we must choose only one pair of deformation fermions ξ and χ (i.e. P = 1). Then we act on states χ (cid:12)(cid:12)(cid:12) ψ (5 / E ⊕ ξ (cid:12)(cid:12)(cid:12) ψ (5 / E (6.1)with the coset generators SU (4 | N ) / [ SU (2) T × SU (2) A × U (1) J ] of grade zero subspace toobtain a set of lowest energy states (cid:12)(cid:12) Ω (5 / , (cid:0) (cid:11) transforming in an irreducible representationof SU (4 |
2) with the Young supertableau (cid:0) . Repeatedly acting on these states with thesupersymmetry generators in grade +1 subspace C + D , one obtains the supermultiplet given inTable 4.This supermultiplet coincides with the doubleton supermultiplet given in [46,47] with thelowest energy K-type whose supertableu with respect to SU (4 |
2) is | (cid:0) i .– 31 – able 4: The deformed minimal unitary supermultiplet of osp (8 ∗ | t = g = 1 /
2. The de-composition of SU (4) irreps with respect to SU (2) T × SU (2) A × U (1) J is denoted by ( t , a ) J . H is the AdS energy (negative con-formal dimension), and H is the total energy. The SU (4) Dynkinlabels and the U Sp (4) Dynkin labels are also given. The Dynkinlabels of the lowest energy SU (4) representations of SO ∗ (8) co-incide with the Dynkin labels of the corresponding massless 6 D conformal fields under the Lorentz group SU ∗ (4). H = − ℓ H ( t , a ) J SU (4) U Sp (4)Dynkin Dynkin
52 32 (cid:0) , (cid:1) − ⊕ (cid:0) , (cid:1) + (1,0,0) (0,1)3 (1 , − ⊕ (cid:0) , (cid:1) (0 , +1 (2,0,0) (1,0)
72 72 (cid:0) , (cid:1) − ⊕ (cid:0) , (cid:1) − ⊕ (cid:0) , (cid:1) + ⊕ (cid:0) , (cid:1) + (3,0,0) (0,0)2 (0 , (0,0,0) (1,0)
52 52 (cid:0) , (cid:1) − ⊕ (cid:0) , (cid:1) + (1,0,0) (0,0) All higher spin doubleton supermultiplets can be obtained similarly as deformations ofthe minimal unitary supermutiplet by choosing the deformation parameter t = g = P/ t = 1 / , , / , . . . . The resulting general deformed supermultiplet(higher spin doubleton) is give in Table 5.This supermultiplet matches exactly the corresponding doubleton supermultiplet withlowest energy K-type | (cid:0)(cid:0) . . . (cid:0) | {z } t =2 g = P i given in [46, 47].– 32 – able 5: The general deformed minimal unitary supermultipletof osp (8 ∗ |
4) corresponding to the deformation parameter t = g = P/
2. The decomposition of SU (4) irreps with respect to SU (2) T × SU (2) A × U (1) J is denoted by ( t , a ) J . H is the AdS energy (negative conformal dimension), and H is the total energy.The SU (4) Dynkin labels and the U Sp (4) Dynkin labels are alsogiven. The Dynkin labels of the lowest energy SU (4) representa-tions of SO ∗ (8) coincide with the Dynkin labels of the correspond-ing massless 6 D conformal fields under the Lorentz group SU ∗ (4). H = − ℓ H ( t , a ) J SU (4) U Sp (4)Dynkin Dynkin t + 2 t + 1 ( t , − t ⊕ (cid:0) t − , (cid:1) − t +1 ⊕ . . . . . . (2 t , ,
0) (0,1) . . . · · · ⊕ (0 , t ) + t t + t + 2 (cid:0) t + , (cid:1) − t − ⊕ (cid:0) t , (cid:1) − t + ⊕ . . . . . . (2 t + 1 , ,
0) (1,0) . . . · · · ⊕ (cid:0) , t + (cid:1) t + t + 3 t + 3 ( t + 1 , − t − ⊕ (cid:0) t + , (cid:1) − t − ⊕ . . . . . . (2 t + 2 , ,
0) (0,0) . . . · · · ⊕ (0 , t + 1) t +1 t + t + 1 (cid:0) t − , (cid:1) − t + ⊕ (cid:0) t − , (cid:1) − t + ⊕ . . . . . . (2 t − , ,
0) (1,0) . . . · · · ⊕ (cid:0) , t − (cid:1) t − t + 2 t + 2 ( t , − t ⊕ (cid:0) t − , (cid:1) − t +1 ⊕ . . . . . . (2 t , ,
0) (0,0) . . . · · · ⊕ (0 , t ) + t t + 1 t + 1 ( t − , − t +1 ⊕ (cid:0) t − , (cid:1) − t +2 ⊕ . . . . . . (2 t − , ,
0) (0,0) . . . · · · ⊕ (0 , t − t − Acknowledgements:
We would like to thank Oleksandr Pavlyk for many stimulatingdiscussions and his generous help with Mathematica. S.F. would like to thank the Center forFundamental Theory of the Institute for Gravitation and the Cosmos at Pennsylvania StateUniversity, where part of this work was done, for their warm hospitality.This work was supported in part by the National Science Foundation under grants numbered– 33 –HY-0555605 and PHY-0855356. Any opinions, findings and conclusions or recommendationsexpressed in this material are those of the authors and do not necessarily reflect the views ofthe National Science Foundation.
AppendixA. Construction of Finite-Dimensional Representations of
U Sp (2 N ) in termsof Fermionic Oscillators To realize the generators of the compact Lie algebra usp (2 N ), we define two new sets of N fermionic oscillators α r , β r and their hermitian conjugates α r = ( α r ) † , β r = ( β r ) † ( r =1 , , . . . , N ), such that they satisfy the usual anti-commutation relations: { α r , α s } = { β r , β s } = δ sr { α r , α s } = { α r , β s } = { β r , β s } = 0 (A.1)The Lie algebra usp (2 N ) has a 3-graded decomposition with respect to its subalgebra u ( N ) as follows: usp (2 N ) = g ( − ⊕ g (0) ⊕ g (+1) = S rs ⊕ M rs ⊕ S rs (A.2)where the generators M rs form the u ( N ) subalgebra. They can be realized as bilinears of thefermionic oscillators: S rs = α r β s + α s β r M rs = α r α s − β s β r S rs = β r α s + β s α r = ( S rs ) † . (A.3)The usp (2 N ) generators satisfy the following commutation relations: (cid:2) S rs , S tu (cid:3) = − δ ts M ur − δ tr M us − δ us M tr − δ ur M ts [ M rs , S tu ] = − δ ru S st − δ rt S su (cid:2) M rs , S tu (cid:3) = δ us S rt + δ ts S ru (cid:2) M rs , M tu (cid:3) = δ ts M ru − δ ru M ts (A.4)The quadratic Casimir of usp (2 N ) is given by C [ usp (2 N )] = M rs M sr + 12 ( S rs S rs + S rs S rs )= N ( N + 2) − ( N α + N β ) [( N α + N β ) + 2] − α ( r β s ) α ( r β s ) (A.5) Note that realizing the generators of
USp (2 N ) as bilinears of a single pair of fermionic oscillators leads toa finite set of irreps, which are the compact analogs of ”doubleton” irreps. To construct more general irrepsof USp (2 N ) one needs to take an arbitrary number (color) of pairs of these oscillators and sum over the colorindex. See [52] for a general treatment. – 34 –here “( rs )” represents symmetrization of weight one, α ( r β s ) = ( α r β s + α s β r ).We choose the Fock vacuum of these fermionic oscillators such that α r | i F = β r | i F = 0 . (A.6)To generate an irrep of U Sp (2 N ) in this Fock space in a U ( N ) basis, one chooses a set ofstates | Ω i , transforming irreducibly under U ( N ) and is annihilated by all grade − S rs , and act on it with grade +1 generators S rs [52].The possible sets of states | Ω i , that transform irreducibly under U ( N ) and are annihilatedby S rs , are of the form α r α r . . . α r m | i F (A.7)or of the equivalent form β r β r . . . β r m | i F . (A.8)where m ≤ N . They lead to irreps of U Sp (2 N ) with Dynkin labels [52]( 0 , . . . , | {z } ( N − m − , , , . . . , | {z } ( m ) ) . (A.9)In addition, we have the following states α [ r β s ] | i F = 12 ( α r β s − α s β r ) | i F (A.10)that are annihilated by all grade − S tu . They lead to the irrep of U Sp (2 N ) withDynkin labels ( 0 , . . . , | {z } ( N − , , , . (A.11)Note that in the special case of usp (4), the states α r α s | i F , β r β s | i F and α [ r β s ] | i F all leadto the trivial representation.Also note that the following bilinears of these α - and β -type fermionic oscillators: F + = α r β r F − = β r α r F = 12 ( N α − N β ) (A.12)where N α = α r α r and N β = β r β r are the respective number operators, generate a usp (2) F ≃ su (2) F algebra [ F + , F − ] = 2 F [ F , F ± ] = ± F ± (A.13)that commutes with the usp (2 N ) algebra defined above. Nonetheless, the equivalent irrepsof U Sp (2 N ) constructed from the states | Ω i involving only α -type excitations or β -typeexcitations can form non-trivial representations of this U Sp (2) F .For example, the two irreps labeled by (1 ,
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