Unitary massless representations of conformal superalgebras

Volume 166B, n u m b e r 1

PHYSICS LETTERS

2 January 1986

UNITARY M A S S L E S S R E P R E S E N T A T I O N S OF C O N F O R M A L S U P E R A L G E B R A S B MOREL Cahforma Institute of Technology, Pasadena, CA 91125, USA

A SCIARRINO Dlparttmento dl Ftslca, Unwerslta dl Napoh, 1-80125 Naples, Italy and INFN, Sezlone dl Napoh, 1-80125 Naples, Italy

and P SORBA LAPP, F-74019 Annecy-le-Vteux, France Received 25 September 1985

Massless unitary irreducible representations of the conformal superalgebras SU(2, 2 / N ) are shown to be atypical representauons The existence of such representations appears restricted to N ~<4 if the spin condition s ~< 2 is ~mposed on the states

Lots of efforts are devoted these days m the attempt of acinevmg the unlficaUon of fundamental forces wa a supergravlty theory In such a framework an unavoidable tool appears to be the theory of superalgebras and thmr representations. Smaple superalgebras come almost naturally m the same way the Pomcar6 group can be deformed into the (ant1-) de Sitter group, extended super-Pomcar6 algebras can be obtained by contraction of the orthosymplectlc superalgebras OSp(N/4), whale the conformal algebra generahzes darectly to the extended conformal superalgebras SU(2,2/N) [ 1 ] If the classification by Kac [2] of sample superalgebras as well as their fimte dtmensaonal representaUons has constituted a fundamental step m thxs field, the &fferent contributions wluch have been brought these last years are also of first interest for a complete understandmg and a practical use of these mathematical properties. Due to the existence of fermlomc roots, there exists fimte-dxmensaonal representations of sample superalgebras winch are Irreducible on the quohent space V/V 0, where V is the linear space of representatmn and V 0 an mvarlant subspace In the following, we shall write con&tlons to ldentafy V 0 winch has to be consadered as decoupled from the representatmn Such representatmns are called by Kac [2] "atypical" Tins property can be extended to umtary (mfimte-damensaonal) representatxons We have recently shown that the umtary representatmns of OSp(N/ 4) relevant for supergravaty, m wluch a "shortenmg" of the multaplets occur [3], correspond actually to atyplcal representatmns [4] As we shall see m the followmg, massless umtary representations of SU(2,2[N) are also atyplcal representatmns and tins condxtxon can give rise to the restriction of the spin content ("shortening" of the representatmn) In thxs note, we construct the umtary xrreduclble representatmns (UIR) of SU(2,2/N) startmg from those of the conformal group (SU(2,2) and of the internal symmetry group SU(N) We are mterested m representataons winch must contain only posatlve-energy "massless" representatmns of the conformal group SU(2,2) [5], that is UIRs winch, when restricted to the Pomcar~ group, remain irreducible and massless. Then, we wall restrict our consaderataons to representatmns revolving states wath spin 0 < s ~< 2 0370-2693/86/$ 03 50 © Elsevier Science Pubhshers B V (North-Holland Physacs Pubhshmg DlVaSaon)

69

Volume 166B, number 1

PHYSICS LETTERS

2 January 1986

The (non-) existence of massless umtary representations of SU(2,2/N) has already been subject to controversies m ref. [6], the authors pointed out an error m the commutation relations used m ref [7] where a no-go theorem for conformal supersymmetry was proposed. In a recent paper [8], the extstence of an umque umtary representaUon o f N = 8 conformal superalgebra revolving fields with spin s ~< 2 is clatmed As wdl be shown, tlus supermultlplet appears m our constructaon with negatwe-norm states Let us stress that the authors of ref [8] used a method completely different from ours. Indeed they consadered an oscillator-hke approach to describe the umtary representataons of "non-compact" supergroups In the case of SU(2,2/8) their technics lead us to express the corresponding UIRs m terms of "compact" supergroups SU(2/p) × SU(2/8 - p), 0 ~

(+-(e,-el),z,] =1,

,4,+(pl-Pm),l,m=l,

,8)

(1)

related to SU(2,2) X SU(8) X U(1), and its fermlomc part A 1= {+(e, -- p/), t = l,

,4,1=1,

,8),

(2)

which builds a (4",8) • (4,8) of SU(2,2) × SO(8) The Cartan subalgebra can be obtained by (anti-) commuting the roots [Eez_ ez÷l ,Ee~+l - e~] =hz,

[Epl_Pl+1 ,Ep/+l_Pl]

~Ee,_pl,Eth_e4)=h4,

t= 1,2,3,

= hl+4, l = 1,

,7

(3)

In partacular the compact SU(2) X SU(2) X U(1) subalgebra of SU(2,2) is generated by (E± (ca - e4)' h 3) ~ H with

H=~(2h 2 +h 1 +h3),

(E±(el-e2), h l )

~

(4)

while the fermiomc generators read (Weyl representataon)

Qal=Ept_ea,

S&/=Ep/_,r,+2 ,

~&l=gea+2_t~l , Sal=Eea_pl ,

(5)

with a (and &) = 1,2 and l = 1, . , 8 The commutation relataon between the Cartan part and the (non-zero) roots are obtained from the Cartan matrix /2 -1 0

(a)=

-1 2 -1

0 -1 2 -1

-1 0 -1

1 2 -1

-1 2

\

(6)

,

-1 -1

2/

that is

[hm ,E ±(et-el+l) ] : +amtE±(ei_et+l) [hm' E±(e4-pl)] = ±am4E±(e4-pl)' 70

m--l,

,11,

l= 1 , 2 , 3 , (7)

Volume 166B, number 1

PHYSICS LETTERS

[h m,E±(pk_pk+l) ] = +-am,k+4E±(pk_Pk+l),

k = 1,

2 January 1986

,7

(7 cont'd)

Note finally that the generator associated to the second U(1) commuting vath SU(2,2) X SU(8) is k =2(h 1 +2h 2 + 3 h 3 + 4 h 4 ) - ( h l l

+ 2 h 1 0 + 3 h 9+

+7h7)

(8)

Denotmg E 0 the elgenvalue o f H m an umtary Irreducible representataon of SU(2,2), massless representations, which In the PoIncar~ lunlt are massless and irreducible, are characterized by the relataon [5] E 0=1+11,

12 = 0

or

E0=1+12,

11 = 0 '

(9)

where (11,12) associated with the SU(2) X SU(2) part, labels also a Finite-dimensional irreducible representation of the Lorentz group SL(2,C), itself appeanng m the Cartan decomposition as SU(2) X SU(2). In order to budd up a positive energy UIR of SU(2,2/8), we will - as already done for UIR of OSp(4/N) start by defining an SU(2,2/8) starting weight A with the properties (1) A IS a SU(2) × SU(2) X SU(8) tughest weight, (n) A IS associated to the lowest-energy elgenvalue E ~ m It will therefore be annflaflated by any fermiomc generator decreasing the E 0 value, 1 e

Ee4_plA=Epl_elA=O

(l=1,

,8),

(10)

wluch lmphes, because of (I), Eea _pl A = E l_e2A = 0

(11)

In the same way the tughest weight of a f'mite-dunensmnal SU(4/8) representation is ann~halated by any root butlt from a linear combination with positive integer coefficients of the sunple roots e l - e 2, e 2 - e 3 , e 3 - e 4, e4 P l , P 1--P2, , P7-Ps, the starting weight can be seen as anmhalated by any root built from the system e 1 - e 2, e3-e4, e 4 - - P l , P I--P2, , PT-P8, P S - e l We note that in ttus last set two ferrmomc roots are present Then, Kac -Dynkm Indices for an UIR of SU(2 ~2/8) (a 1, a2, a3, a4, as, • , a 11), are de fined by

hzA =aiA

(12)

The action of the other fermIomc generators (m antlsymmetnc eombmataons when more than one) on A will allow to obtaan the other SU(2,2) X SU(8) X U(1) representalaons in this SU(22/8) UIR Denoting now the state A as

A = (E 0,I 1 = al/2, m 1,12 = a3/2, m 2, a 4 , as,

(13)

, all),

with --11 ~
, a l 0 , a l l + 1),

Ee2_psA= t82(EO + 1/2,11 + 1/2,11 - 1/2,12,12,a4,as,

, a l 0 , a l l + 1)

+ t'~(E 0 + 1/2,11 - 1/2,11 - 1/2,12,12,a4,as,.

(14)

, a l 0 , a l l + 1),

and E p l _ e 4 A = t l ( E 0 + 1/2,11,11,12 + 1/2,12 + 1/2,a4,a 5 + 1,a 6,

,all),

Epl_e3A= t~(E 0 + 1/2,11,11,12 + 1/2,12 - 1/2,a 4 + 1,a 5 + 1,a6, + i~(E 0 + 1/2,11,11 '12 - 1/2,12 - 1/2,a4 + 1,a 5 + 1,a6,

, a l l ).

,all ) (15)

In order to compute the coefficients t~, we have now to make precise our defimUon of unitanty for the fernuomc 71

Volume 166B, number 1

PHYSICS LETTERS

2 January 1986

generators [10], we mean here (16)

(Ee,-pl)+ = E l - e, ' and we reqmre It~l 2/> 0 Use of the anU-commutatlon relations 4

4

(E

{ E e , _ p x , E l _ e , } = l ~ h1 '

et-p l

,E

l

}=~h-F2h3+m

pl-el

i=t

]

, l~>2,

=

(17)

leads to [t~[ 2 = 4 1 + It~12=a4,

+a 4 - a 5 -

-

It~l 2 + l [ l l 2 = a 3 + a 4 = [ t l [

2+a3,

(18)

some coefficients being related by thetr SU(2) X SU(2) properUes, namely It/[ 2 = 1t~12/(211 + 1),

It/I 2 = 1t1412/(212+ 1)

(19)

With the conmdered choice of A, that is SU(2) X SU(2) X SU(8) lughest wexght and satlsfymg eqs (10), (11), we wdl get atypical representaUons of the first level (1 e by action of only one ferrmomc generator) if E_ (ea -ps)E~I -o8 A = 0, or

(20)

(and)

E_(ol _e4)Eol _e4A = 0

(21)

The subspace V 0 with the weight Eel-Ps A = A' or (and)Epl_ e4 A = A" has then to be conmdered as decoupled from the representation Actually, eq. (20) lmphes a l+a 2-

+a 4 - a 5 -

-all=0,

(22)

whtle eq (21) gwes a 4 = 0,

(23)

wtuch means, because of the umtanty condmon [eq (16)] that the states A' (respecUvely A") are of zero norm In order to decouple the subspace assocmted with A' or (and) A" we must at the zth level of A (1 e by actaon of antlsymmetnc combinations of t ferrmomc generators) throw away the weights appearing at the (7 - 1)th level of A' or (and) A" To perform exphclfly such a procedure, we remark that the pomtlve fermlonlc rootsE%_ol (a = 1,2, l = 1, ,8) increase E 0 by 1[2, the elgenvalue K of the generator k [see eq (8)] by 1 and transform under SU(2) X SU(2) X SU(8) as (2, 1,8), whtle the negatave fenmomc rootsEp/_ea.2 increase E 0 by 1/2, decrease K by 1, and transform as (1,2, 8) Denoting A = (E 0, I1,12, [X],K),

(24)

where [X] = [X1 , , X7] is the Young tableau charaetenzmg the SU(8) representaUon, we get at the first level the weights (where the E 0 label is omitted) (11 X 1/2,12, [Xl X [171,K+ 1) • (I1'12 X 1/2, [X] X [ I ] , K -

1),

(25)

from wluch we have to decouple the weights A' or (and) A". The second level is obtained by applying to A the anUsymmetnc product of two fermlomc generators transforming as follows (1,0,[12 ],~K=2), (0, 0, [2], AK = --2), 72

(0,0,[27 ] , A K = 2 ) , (1/2, 1/2, [1]

X

(0,1,[12 ] , A K = - - 2 ) ,

[17], AK

=

0),

(26)

Volume 166B, number 1

PHYSICS LETTERS

2 January 1986

and then subtracting the weights which appear m the first level of states built from A' or (and) A" and so on for the next levels Thus these rules allow to build up the atypical UIRs of SU(2,2/8) just knowing how to perform the Kronecker product of SU(8) Young tableaux Now, we will show the masslessness unphes atypacahty at the first level. Let us first consider the case11 4: 0,• 2 = 0 , E 0 =11 + 1 which lmphes [eq (4)] a 2 = 1 In order to keep masslessness, we have to impose [eq (15)] a 4 = 0 which is the second atyplcallty con&tmn [eq (23)]. Now from eq (14) we have also to require [t8212 = 0, which leads, using relations gwen in eqs (18), (19), either to a 4 = a 5 = = a l l = 0 or to a 1 = 211 = 0. The corresponding representations are therefore labelled as (a I =11/2, a 2 = 1, a 3 = 34 = = a l 1 = 0) and the starting weight chosen as (27)

A = (E 0 =I 1 + 1,11,rn =11 , 0, [0] , K = 411 + 4) Applying our procedure we find at the i th level (z = 1, 1

1

(E0 +-it,ll +-~t,O, [18-~]

,8) the states

(28)

,411 + 4 + 0

All these states are massless, but, as we can see, the spin can increase up tOll + 4. Let us remark that in this case an atypical condition at second level Los _elEps _e2Ee2_psEel _ p s A = 0 that is a 2 + a 3 + a 4 - a 5 - all - 1 = 0 has come to play It is this second condition which insures the masslessness and restricts the number of levels to 8 The case11 = 0,12 :/: 0 , E 0 =12 + 1 can easily be ehmmated Indeed, masslessness would demand [eq (15)] [~1 [2 = 0 which le ads, using again relations (18), (19) 12 = 0. So, we are led to study the casell =12 = 0 Then states such that X = Ep 1 - e4Eei - Ps A

(29)

would be massive The decouphng of this state implies the corresponding norm to be zero, 1 e

[X[2=a4(al +

+a 4-a 5-

-a11)=0.

(30)

It follows that either eq (22) or eq. (23) has to be satisfied Actually, both atyplcahty con&tlons at first level have to hold if we demand only massless states with s p i n / ~ 2 Indeed condition (23) alone will imply states with spin up tOl = 4, while condltmn (22) without a 4 = 0 does not allow to throw away massive state showing up at second level since the second-level atyplcahty condition Ee4 _olEea_olEol _ eaEol _ e 4 A = 0 leads to (a 3 + a 4 + 1)a 4 = 0 Therefore, let us consider representations of SU(2,2/8) such that

a1=33=a4=0,

32=1=a5+

+all.

(31)

Then the starting weight Awfll belong to an antlsymmetrlc SU(8) representatmn We know already that atyplcalaty conditions insures us that at tth level (t ~< 8) the remmnmg states are massless and of spin t/2 In order to have only states of spin <~ 2, it will be sufficient to require that all the states that might appear at tth level, t > 4, have to be thrown away as appearing at 0 - 1)th level of A ' = (3/2, 1/2,0, [X+ 1]),

A" = ( 3 / 2 , 0 , 1/2, [~'1 + 1,X2'

" ,~k7])

(32)

(where the K label has been omitted), that is [X] X [1'] C [X1 + 1 , X 2 ,

,XT] X [ 1 ' - 1 1 ,

[X] × [t 8 - , ] C [ X + l ] X [ 1 8 - ' + 1 1 .

(33)

The starting SU(8) representatmn is then determined [X] = [14] = 70,

(34)

and the corresponding SU(2 ~2/8) representation reads (a l = 0 , a 2 = 1 , a 3 = a 4 =

=a 7=0,a 8=1,a9=

=all=0),

(35) 73

Volume 166B, number 1

PHYSICS LETTERS

2 January 1986

and decomposes as follows (E 0 = 1,11 = 0 , 1 2 = 0, [14 ] = 7 0 , K = 0 )

= (3/2, 1/2,0, [13 ] = 56, 1) • (3/2,0, 1/2, [1 s ] = ~ , - 1 ) (2, 1 , 0 , [12] = 2 8 , 2 ) e ( 2 , 0 , l, [16 ] = 2 , - 2 ) ( 5 / 2 , 3 / 2 , 0 , [1] = 8 , 3 ) ~ ( 5 / 2 , 0 , 3 / 2 , [17 ] = 8 , - 3 ) ( 3 , 2 , 0 , [01 = 1,4) ~ ( 3 , 0 , 2, [181 = T, --4).

(36)

We recoginze the supermultlplet of ref [8], but remark that certain states are of negatwe norm as an example, the state ~ = Eos - e , A IS such that IxI'l2 = (AEe¢ _osEps _ e , A ) = (Alh 4 - h 5 - h 6 - h 7 - h81A) = - 1 .

(37)

It is stra,ghtforward to ganerahze the above results for N < 8 and to deduce that negative norm states will be present If the starting SU(N) representataon is not the trixqal one However, as exphcitly shown for N = 8, choosing the starting weight In the trivial representation o f SU(N) will allow to construct (positive-norm) states of spin/1 , . ,11 +N/2 So m order to have no spin greater than 2 and no pathology, we have to limit to N < 4 In thts last case, we obtain the umque SU(2,2/4) UIR (a l = 0 , a 2 = l , a 3 =a 4 =

=a 7 = 0 )

(38)

wtuch develops as ( r 0 = 1,11 = 0 , 1 2 = 0 ,

[0l = I , K = 4) • (3/2, 1/2,0, [13] = 4 , 5 )

(2, 1,0, [12] = 6 , 6 ) • ( 5 / 2 , 3 / 2 , 0 , [1] = 4 , 7 ) • ( 3 , 2 , 0 , [0] = 1 , 8 )

(39)

As a conclusion, let us first emphasize that the techniques developed for the construction of fimte-dtmenslonal representations of simple superalgebras can be extended to the unitary representataons o f SU(2,2/N). We have shown that the condition of masslessness for the states at the SU(2,2) level leads naturally to atypicahty Restrictlng to states with spin s < 2 in a posmve-deFinxte Hflbert space, acceptable supermultlplets occur only for N < 4 We are indebted to G Girard1 and R Grimm for interesting dtscusslons One o f us (A S ) thanks the LAPP theory group for its warm hospltahty This work has been pamally supported m the framework of the French-Itahan Scientific Cooperation, and we are grateful to D. Gabay for help and goodwill

References [1] s Ferrara, M Kaku, P K Townsend and P Van Nleuwenhmzen, Nucl Plays B129 (1977) 125, E Bergshoeff, M de Roo and B de Wit, Nucl Phys B192 (1982) 173 121 V Kae, Lecture Notes m Mathematms, Vol 676 (Springer, Berlin, 1978) p 597 Ial D Freedman and H Nmolaa,Nucl Phys B239 (1984) 342 [41 B Morel, A Semrrmo and P Sorba, Nuel Phys, to be pubhshed [51 G Mack, Commun Math Phys 55 (1977)1 [61 M Flato and C Fronsdal, Lett Math Phys 8 (1984) 159 [7] L Castell and W Heldememh, Plays Rev D25 (1982) 1745, D26 (1982) 1485 [8] M Gllnaydm and N Marcus, Caltech prepnnt CALT-68-1208 [9] J.P HurmandB Morel, J Math. Phys 24(1983)157 [10] M Scheunert, W Nahrn and V Rlttenberg, J Math Phys 18 (1977) 146 74