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Supersymmetrized D = 11 de-sitter-like algebra from octonions
Volume 145B, number 1,2
PHYSICS LETTERS
13 September 1984
SUPERSYMMETRIZED D = 11 DE-SITTER-LIKE ALGEBRA FROM OCTONIONS
Z. HASIEWICZ and J. LUKIERSKI Institute of Theoretical Physics, University o f Wroclaw, ul. Cybulskiego 36, 50-205 Wroclaw, Poland Received 8 May 1984
We consider S 7 as a soft Lie group manifold. We introduce a graded de-Sitter-like algebra in D=I 1 as described by the octonionic extension of OSp (1 ; 4), with bosonic sector parametrized as (SO(10, 2)/G2) × S 7. We provide constructive remarks concerning the meaning of the octonionic extensions of Lie algebras and Lie superalgebras.
1. Recent studies in dimensional reductions of D = 1 1 simple supergravity to D = 4 theory (see e.g., ref. [ 1 ] ) brought special features o f the seven-sphere S 7 into light, namely (i) Only S 7 shares with any Lie group the property that it is parallelizable, i.e. S 7 can be globally described by a pure torsion Tab c [Ea,Eb] (x) = --TabC(X)Ea(X),
a, b = 1 ..... 7 ,
(1)
where E a denote the linear basis o f parallelizing vector fields. We see therefore that S 7 is an example o f a soft Lie group manifold [ 2 - 4 ] . (ri) S 7 can be described by unit octonions [ 5 - 7 ] $7: = (x E O ; x = x 0 + eaXa;.gx = 1} ,
(2a)
eaeb = --Sab +lab Cec ,
(2b)
= ~o(x) + (x, ,~, adx) = co(x) + (x, a, adx), and it satisfies the equation dco(x) = ~d~ 1 - A dx -- -5' co(x) ^ co(x) + ~(co(x)2, x, co(x)) .
with left and right translations described by the octonionic multiplication L a ( x ) = ax E S 7 ,
Ra(x ) = xa ,
a E S7 .
(3)
Due to the nonassociativity o f octonionic multiplication S 7 is not a Lie group. The way o f observing this property is to study the octonionic one-forms on S 7 co(x) = £ d x : - d g x = - f f ) ( x ) = coa(x)ea ,
(co(x)x, x, ~o(x)) a = 2q~'bca(X) cob(x) ^ coO(x),
(7)
and eq. (6) takes the form dcoa(x) = T b c a ( x ) c o b ( x ) A COo(x),
(8)
where (9)
The extension o f Cartan-Maurer equations for S 7 has been proposed by Nahm [8] and Gtirsey and Tze [6], but here we would like to point out that S 7 can be described globally by the soft Lie algebra, with point-dependent structure constants. Eq. (8) is equivalent to (1). Indeed, if the E a are dual to coa (coa(Eb) = 6ab), one gets dcoa(Eb, E c) = - c o i ( [ E b , Le ] ) = Tbca
(4)
which is not invariant under the left or right translations (3). Indeed, for example
(L~ ¢o) (x) = La(x)aLa(x ) : ( ~ ) (aax) = x(a(adx)) - x ( , ~ (adx)) + ( ~ ) ( a d x )
(6)
One can write
Tbca(x) = --fbc a + Cbbca(x) . ea = --ea ,
(5 cont'd)
(5)
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
= - [ G , G ] a,
(10)
i.e. Tbc a is the parallelizing torsion. For soft Lie algebras the integrability condition impries Bianchi equations for torsion which are the generalized Jacobi identities. Indeed, in the case o f S 7 the integrability condition d2co a = 0 implies 65
Volume 145B, number 1,2
PHYSICS LETTERS
dTbc a A cob A COc + 2TbcaTd?coc A ¢od A COl= O. (11) Introducing the three-form
E =dco a A coa = Tbcacoa ^ COb A COC ,
(12)
one can show (see also ref. [5,6] ) that from the relation (11) one can derive the Cartan-van S c h o u t e n Englert (CSE) equation
13 September 1984
(super) group as a real (super) manifold with constraints which follow from the definition of the respective octonionic extension. For example the octonionic extension of the unitary group U(n; 0) is the set of eight real n X n matrices (U O, U a) satisfying the condition U + U = 1, where U = U° + ea U a, i.e. u°Tu° + uaTu a = 1 , uaTuO
- u O T u a -- f b c a U b T u
(20) c =O.
where
The infinitesimal generators of U(n; 0) at (U 0 = 1, Ua = 0) are described by the octonion-valued antihermitean matrices
"I~ = (1/3 ! ) e a b c d f g h T g h f c o a A cob A COc A c o d , (14)
H=AO +eaS a,
Let us observe further that Tbca(1 ) = --fbc a. In concrete parametrization of unit octonions
where A 0 is n × n antisymmetric, and S a are symmetric. We calculate the algebra of generators at the identity point U = 1 by taking the commutators of octonionic units from the octonionic multiplication table (2b). The difference between S 7 and other octonionic extensions of Lie (super) groups is that in the latter case one can introduce only locally the anholonomic coordinate system and use in the neighbourhood of the point U = 1 the structure equations in the form (8). Further we shall be interested only in the algebra of generators describing the octonionic extension of (super) groups at a particular point U = 1. From physical reasons it is interesting to consider the candidates for the D = I I de-Sitter groups and its supersymmetrization. We shall choose as such a candidate the octonionic extension Us (4;0) of Sp(4; R), i.e. the set of 8 4 X 4 matrices (U 0, q / a ) satisfying the conditions (C = - C T)
dY, = - * E ,
$7: x = (1 + ~)(1 - ~)-1 ,
(13)
~ = -~ ,
(15)
one gets
coa =(1 + 1~j2)2[(1 - l~12)d~ a + 2~ a ~bd~ b - 2fbca~b~ e] ,
(16)
and
TabC(X) = --lab c + 2(1 + 1~[2) - 2 X [¢adbC(1 --l~i2)~ d -- ¢dfbCfahd~f~ h] ,
(17)
where (Ca, eb, ec) = Cabcaed. Using the relation (see ref. [9], eq. (3.23)) 1_,:" [a6b] fabC~bdfgc -_ ~J[d¢ g] :'
(18)
one can check that the structure function (17) is completely antisymmetric. 2. We have shown that S 7 can be described globally by the soft Lie algebra (1), with the structure functions satisfying eq. (13) with the unitial condition TabC(1) = --lab c. We see therefore that the algebra of octonions can be identified with the algebra (1) at x =1:
[Ea, Eb] (1) = fabCEc(1) .
(19)
Similarly S 1 describes U(1) and S 3 can be identified with U(1; H ) = Sp(1), due to (2a) the real manifold S 7 can be identified with U(1 ; 0). We shall consider the octonionic extensions of the matrix Lie 66
H=-H #.
Q/0TcQ/0 + QLa T c Q ~ a = C ,
(21)
(22)
cff.a T c c ~ O - Qg0Tcc~0 - f b c a c'~b ccff-c = 0 .
The corresponding U~(4; 0) algebra is described by the 4 X 4 octonion-valued matrices 9 ~ = S 0 +ea Aa , 9~C+ ~ g # C = 0 ,
(23)
with 10 + 7 " 6 = 52 generators. We see that this number o f generators does not match with the 66 generators of O(10, 2); so we shall further describe the U~(4; 0) algebra as a D = 11 de-Sitter-like algebra. One can introduce the matrix representation of the D = 11 de-Sitter-like algebra as follows:
Volume 145B, number 1,2 R e ~ :Pu = 7uC,
Muv = o u v C ,
l m ~ :P4+a = (75C)ea ,
M~,4+a = (7574C)ea ' R a = C" e a ,
(24)
where 7u are Majorana, and we use the property that 7u C, ouvC are symmetric, and C, 75 C and 75 7u C antisymmetric. We obtain that (Pu' Muv) form the 0(3, 2) Sp(4) anti-de-Sitter algebra, and besides [Pa+a,Pa+b]
= 2fabCRc
13 September 1984
PHYSICS LETF ER S
,
[Pa+a,Rb]
=
2fabCP4+c '
IRa, Rb] = 2fabCRa ,
de-Sitter-like algebra we propose to consider the octon. ionic extension of OSp(1; 4), which we denote by UaU(4; 1 ; 0). The octonionic 5 × 5 matrices s~ belonging to the Uc~U(4; 1 ; 0) superalgebra do satisfy the equation UaU(4, 1 ; 0):
s~#G=-G~,
G
.
=
(271
One gets
(25a)
s~-\F2(4 X 1) q./(4X 7 )' [M~,4+a, Mv,4+b] = 2171avfabCRc + 28abMla v ,
qgC + CQZ # = 0 , [Mu,4+a, P4+b] = 25abPu,
[Mu,4+a, Rb ] = 2fabCMu,4+b'
(25b)
U= - U ,
F # C = F 2.
(28)
We see that the bosonic sector of UaU(4, 1; 0) is U,~(4; 0) X U(1 ; 0), where Ca(4; 0) X U(1; O)
[Pu,Pa+a] = 2Mu,a+a,
[P~,,Ra]=O, ((SO(IO, 2)/SO(7)) X S 7) X S 7 ,
[Pu, Mv ,4+a ] = 2r/~v Pa+a ,
[M.v, Pa+al = 0 ,
[M.v, Ra] =0 ,
(25c)
(29)
and the respective algebras are (25) and (19). Four octonion-vatued (= 32 real) fermionic generators can be described as the following 5 X 5 matrices
[M~v, Mo,4+a] = 2r/voM~,4+a - 2rluoMv,4+a " Let us write down the 66 generators of SO(10, 2) as (Pu' Pu+a'Muv ' Mu,4+a'Mab)" If we replace the 21 generatorSMab by 7 generators R a describing S7 at the point x = I, we see that 52 generators of Ua(4; 0) are counted in a similar way as the generators of the coset SO(10, 2)/G 2 = (SO(10, 2)/SO(7)) X S 7, what we denote as U~(4; 0) ~ SO(10, 2)/G 2. We would like to mention that similarly one can relate the 0(9, 1) and 0('9) Lie algebras with the octonionic extensions SL(2; 0) and U(2; 0). Calculating in a similar way the algebra at the identity one gets that SL(2; 0) ~ SO(9, 1)/G 2 ,
U(2; 0) ~ SO(9)/G 2 . (26)
3. It is known that the complex and quaternionic extensions of the OSp(1 ; 4) superalgebra describes D = 5 and D = 7 graded de-Sitter algebras [10], with, respectively, the bosonic sectors SO(4, 2) X U(1)and SO(6, 2) X SU(2). In order to describe the D = 1 1 graded
Q~ =
,
ql~
Q~a --
0
(3(I)
e a ql~
0
]'
where a = 1 ..... 7, a -'- 1 ..... 4, and q/~ denotes the Sp(4; R) spinor basis, satisfying the relation c~ c g ~ = C ~ . The bosonic sector is the following Ua(4; 0)
X U(1; 0):
(~-~),
~c+c#~=0,
~=-~
(31)
We denote the generators of Ua(4; O) in accordance with the formulae (24), and the generators L a describing U(1 ; 0) by the imaginary octonion w do satisfy the relation [La, Lb] = 2fabCLc (see (19)). By calculating the anticommutators of the supercharge matrices (30) we obtain the following relations:
(aa, Q3}
=I
2(TuC)a#P
#
1
- ~(ouvC)at~ I
#v
,
(32)
67
PHYSICS LETTERS
Volume 145B, number 1,2
{Qc~a, Q#} = ~ (3"53"uC)a~Mu,4+a -- { (3'5C)c~P4+a
Jacobi identity at U = 1 gives the result
o(o,~a, O~b, Q'ye)
+ ~ CaoRa + 2Ca#L a _I
1
{Qaa, Qpb}- ~6ab [(3"uC)ePPu - 2(cruuC)a#M
lap
1 -- "35ad 6be~ef~defg(C~TQag + CTaQ~g + CaP QTg )'
]
(38)
1 c -- 2lab [(3"57~zC)a(3Mt~,4+c -- ~1 (3'5C)~t3 p 4+e
+ ~Ca~R c - 2ca~re] .
(32 cont'd)
The covariance relations do look as follows:
[Pu' Qa] (3',)~Qt3,
13 September 1984
[Pu'
[Muv, Qa] = (ouv)c, PQ# ,
QJ
= (3",)aOQ~a'
[Mu,, Qc~a] = ( o u u ) J Q ~ ,
i.e. similarly to (35) our superalgebra is closed under triple Jacobi operation. The relation (38) can be understood as describing the generalized CSE equation for UaU(4; 1 ; 0) at the point U = 1, or as characterizing the first term in the Taylor expansion of the structure function of UaU(4; 1 ; 0) around U = 1. In general one can represent the octonionic extension of the Lie n X n matrix (super)algebra G by
Gi -+ ~i = Gi ~ eaFi (a ) = Gi ~ Fai ' [Mu,4+a, Q~] = ( 3 ' 5 % ) ~ Q , ~ a ,
(39)
where the generators G i form the Lie (super) algebra G. The real generators of ~i are L A = (G i, Fai), and they have the following property (i) D(L A , L B, L C) = 0 if at least one generator LA, L B , L C belongs to G (ii) D(Fa/, Fbk, Fcl) 4= 0 but
[Mta,4+a, Qab] = (3"u3"5)a#fabe Q~e ' [P4+a, Qal = (3"s)~C~Qt3a , [P4+a, ac~b] : (3"5)a#fabCaoc ,
fabCD(Faf , F bk, Fcl ) = 0 . [Ra, as] = Qc,a , [La, Qc~] = -Qaa ,
[Ra, Qab] =fabCQac, [La, aab] = -fabCQac •
The trilinear relation (39) characterize therefore the octonionic extensions of Lie (super) algebras. (33)
In order to study the derivation from the Jacobi identity one can introduce the Jacobi function
D ( A , B , C) = ½([[A, B], C] + cycl.) .
(34)
One gets for S 7 the relation
D(ea ' eb ' ec)= ~abcd ed .
(35)
We see from (17) that the RHS of (35) describes the derivative of the structure function at x = 1, and eq. (35) isthe CSE equation at the point x = 1. The trilinear relation, satisfied by (35) is the following
fabCD(ea, e b, ec) = 0 .
(36)
Similarly, one can introduce the graded Jacobi function
o(ac~ ' ap, 07 ) = ½[Q~, {09, a.r} ] + graded cycl., (37) and calculate the derivation from graded Jacobi identities for UaU(4; 1 ; 0). In particular the generalized 68
(40)
4. We would like to add the following remarks: (a) The octonionic extensions of Lie (super) groups (except U(1; 0) -= S 7) due to nonassociativity of octonions are not closed under multiplication, i.e. do not form parallelizable manifolds. Any octonionic extension has, however, two types of parallelizable submanifolds: (i) Describing quaternionic extensions of Lie (supergroups. Due to the property that the octonions (e r, er+ 1, er+3) form three quaterionic algebras, one can select three quaternionic subalgebras (see (39))
L(Ar) = (Gi, Fri, Fr+l,i, Fr+3,i),
r = 1, 2, 3 .
(41)
Three Lie (super) groups obtained by the exponential map of L~) cover the manifold described by the octonionic extension of G. For example UaU(4; 1 ; 0) can be covered by three graded Lie algebras U~U(4; 1 ; H) (see ref. [10]). (ii) Describing embedded seven-spheres S 7, CS 7 • The subalgebras describing S 7 submanifolds are described by Fai (i fixed; a = 1 ..... 7). In particular inside
Volume 145B, number 1,2
PHYSICS LETTERS
U~(4; 0) one can embed six seven-spheres with the generatorsMu,4+a, R4+ a and R a. For fixed/1 we obtain from (25) five subalgebras: (P4+a, Ra) described by (25a) and (M#,4+a, Ra) , which looks as follows: ( +:/l= 1,2,3) [Mu,a+a, Mu,4+b] = ±2fabCRc -:/l
but the concept of their invariance has to be modified. It was shown already by Nahm [8] that seven gauge fields describing linear connection on a generalized principle bundle with an S 7 fibre can be obtained via dimensional reduction o f D = 11 supergravity with the internal sector described by S 7.
0
References
[M•,4+a, Rb] = 2fabCM.,4+c,
[R a, R b] : 2fabeRa .
13 September 1984
(42)
Introducing R+a = R a + P4+a one can check that the 14 parameter parallelizable manifold generated by (Ra, P4+a) is S 7 X S7. Similarly one gets that (Ra, Mu,4+a) for gt = 1,2, 3 are the generators of S 7 × S 7, and for/a = 0 they describe the complexification CS 7 of S 7. (b) It should be mentioned that one can generalize the theory of principal bundles with the structure group G to generalized principal bundles with the group G replaced by a paralMizable manifold, in particular by S 7, which can be called an octonionic line bundle. For a fixed paralMization of the structural manifold it is possible to formulate the theory of connections,
[ 1 ] M. Duff, B.E.W. Nilsson and C.N. Pope, Nucl. Phys. B233 (1984) 433. [2] Y. Ne'eman and T. Regge, Phys. Lett. 74B (1978) 54. [3] R. d'Auria, P. Fr6 and T. Regge, in: Supergravity 81, eds. S. Ferrara and J.C. Taylor, P. 421. [4] M.S. Sohnius, Z. Phys. C18 (1983) 229. [5] T. Dereli, M. Panahinoghaddam, A. Sudberg and R.W. Tucker, Phys. Lett. 126B (1983) 33. [6] F. Gtirsey and C.H. Tze, Phys. Lett. 127B (1983) 191. [7] J. Lukierski and P. Minnaert, Phys. Lett. 129B (1983) 392. [8] W. Nahm, CERN preprint TH-2489 (April 1978), unpublished. [9] R. Diandarer, F. Gtirsey and C.H. Tze, Yale University YTP 81-13 (September 1983). [10] Z. Hasiewicz, J. Lukierski and P. Morawiec, Phys. Lett. 130B (1983) 55.
69
PHYSICS LETTERS
13 September 1984
SUPERSYMMETRIZED D = 11 DE-SITTER-LIKE ALGEBRA FROM OCTONIONS
Z. HASIEWICZ and J. LUKIERSKI Institute of Theoretical Physics, University o f Wroclaw, ul. Cybulskiego 36, 50-205 Wroclaw, Poland Received 8 May 1984
We consider S 7 as a soft Lie group manifold. We introduce a graded de-Sitter-like algebra in D=I 1 as described by the octonionic extension of OSp (1 ; 4), with bosonic sector parametrized as (SO(10, 2)/G2) × S 7. We provide constructive remarks concerning the meaning of the octonionic extensions of Lie algebras and Lie superalgebras.
1. Recent studies in dimensional reductions of D = 1 1 simple supergravity to D = 4 theory (see e.g., ref. [ 1 ] ) brought special features o f the seven-sphere S 7 into light, namely (i) Only S 7 shares with any Lie group the property that it is parallelizable, i.e. S 7 can be globally described by a pure torsion Tab c [Ea,Eb] (x) = --TabC(X)Ea(X),
a, b = 1 ..... 7 ,
(1)
where E a denote the linear basis o f parallelizing vector fields. We see therefore that S 7 is an example o f a soft Lie group manifold [ 2 - 4 ] . (ri) S 7 can be described by unit octonions [ 5 - 7 ] $7: = (x E O ; x = x 0 + eaXa;.gx = 1} ,
(2a)
eaeb = --Sab +lab Cec ,
(2b)
= ~o(x) + (x, ,~, adx) = co(x) + (x, a, adx), and it satisfies the equation dco(x) = ~d~ 1 - A dx -- -5' co(x) ^ co(x) + ~(co(x)2, x, co(x)) .
with left and right translations described by the octonionic multiplication L a ( x ) = ax E S 7 ,
Ra(x ) = xa ,
a E S7 .
(3)
Due to the nonassociativity o f octonionic multiplication S 7 is not a Lie group. The way o f observing this property is to study the octonionic one-forms on S 7 co(x) = £ d x : - d g x = - f f ) ( x ) = coa(x)ea ,
(co(x)x, x, ~o(x)) a = 2q~'bca(X) cob(x) ^ coO(x),
(7)
and eq. (6) takes the form dcoa(x) = T b c a ( x ) c o b ( x ) A COo(x),
(8)
where (9)
The extension o f Cartan-Maurer equations for S 7 has been proposed by Nahm [8] and Gtirsey and Tze [6], but here we would like to point out that S 7 can be described globally by the soft Lie algebra, with point-dependent structure constants. Eq. (8) is equivalent to (1). Indeed, if the E a are dual to coa (coa(Eb) = 6ab), one gets dcoa(Eb, E c) = - c o i ( [ E b , Le ] ) = Tbca
(4)
which is not invariant under the left or right translations (3). Indeed, for example
(L~ ¢o) (x) = La(x)aLa(x ) : ( ~ ) (aax) = x(a(adx)) - x ( , ~ (adx)) + ( ~ ) ( a d x )
(6)
One can write
Tbca(x) = --fbc a + Cbbca(x) . ea = --ea ,
(5 cont'd)
(5)
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
= - [ G , G ] a,
(10)
i.e. Tbc a is the parallelizing torsion. For soft Lie algebras the integrability condition impries Bianchi equations for torsion which are the generalized Jacobi identities. Indeed, in the case o f S 7 the integrability condition d2co a = 0 implies 65
Volume 145B, number 1,2
PHYSICS LETTERS
dTbc a A cob A COc + 2TbcaTd?coc A ¢od A COl= O. (11) Introducing the three-form
E =dco a A coa = Tbcacoa ^ COb A COC ,
(12)
one can show (see also ref. [5,6] ) that from the relation (11) one can derive the Cartan-van S c h o u t e n Englert (CSE) equation
13 September 1984
(super) group as a real (super) manifold with constraints which follow from the definition of the respective octonionic extension. For example the octonionic extension of the unitary group U(n; 0) is the set of eight real n X n matrices (U O, U a) satisfying the condition U + U = 1, where U = U° + ea U a, i.e. u°Tu° + uaTu a = 1 , uaTuO
- u O T u a -- f b c a U b T u
(20) c =O.
where
The infinitesimal generators of U(n; 0) at (U 0 = 1, Ua = 0) are described by the octonion-valued antihermitean matrices
"I~ = (1/3 ! ) e a b c d f g h T g h f c o a A cob A COc A c o d , (14)
H=AO +eaS a,
Let us observe further that Tbca(1 ) = --fbc a. In concrete parametrization of unit octonions
where A 0 is n × n antisymmetric, and S a are symmetric. We calculate the algebra of generators at the identity point U = 1 by taking the commutators of octonionic units from the octonionic multiplication table (2b). The difference between S 7 and other octonionic extensions of Lie (super) groups is that in the latter case one can introduce only locally the anholonomic coordinate system and use in the neighbourhood of the point U = 1 the structure equations in the form (8). Further we shall be interested only in the algebra of generators describing the octonionic extension of (super) groups at a particular point U = 1. From physical reasons it is interesting to consider the candidates for the D = I I de-Sitter groups and its supersymmetrization. We shall choose as such a candidate the octonionic extension Us (4;0) of Sp(4; R), i.e. the set of 8 4 X 4 matrices (U 0, q / a ) satisfying the conditions (C = - C T)
dY, = - * E ,
$7: x = (1 + ~)(1 - ~)-1 ,
(13)
~ = -~ ,
(15)
one gets
coa =(1 + 1~j2)2[(1 - l~12)d~ a + 2~ a ~bd~ b - 2fbca~b~ e] ,
(16)
and
TabC(X) = --lab c + 2(1 + 1~[2) - 2 X [¢adbC(1 --l~i2)~ d -- ¢dfbCfahd~f~ h] ,
(17)
where (Ca, eb, ec) = Cabcaed. Using the relation (see ref. [9], eq. (3.23)) 1_,:" [a6b] fabC~bdfgc -_ ~J[d¢ g] :'
(18)
one can check that the structure function (17) is completely antisymmetric. 2. We have shown that S 7 can be described globally by the soft Lie algebra (1), with the structure functions satisfying eq. (13) with the unitial condition TabC(1) = --lab c. We see therefore that the algebra of octonions can be identified with the algebra (1) at x =1:
[Ea, Eb] (1) = fabCEc(1) .
(19)
Similarly S 1 describes U(1) and S 3 can be identified with U(1; H ) = Sp(1), due to (2a) the real manifold S 7 can be identified with U(1 ; 0). We shall consider the octonionic extensions of the matrix Lie 66
H=-H #.
Q/0TcQ/0 + QLa T c Q ~ a = C ,
(21)
(22)
cff.a T c c ~ O - Qg0Tcc~0 - f b c a c'~b ccff-c = 0 .
The corresponding U~(4; 0) algebra is described by the 4 X 4 octonion-valued matrices 9 ~ = S 0 +ea Aa , 9~C+ ~ g # C = 0 ,
(23)
with 10 + 7 " 6 = 52 generators. We see that this number o f generators does not match with the 66 generators of O(10, 2); so we shall further describe the U~(4; 0) algebra as a D = 11 de-Sitter-like algebra. One can introduce the matrix representation of the D = 11 de-Sitter-like algebra as follows:
Volume 145B, number 1,2 R e ~ :Pu = 7uC,
Muv = o u v C ,
l m ~ :P4+a = (75C)ea ,
M~,4+a = (7574C)ea ' R a = C" e a ,
(24)
where 7u are Majorana, and we use the property that 7u C, ouvC are symmetric, and C, 75 C and 75 7u C antisymmetric. We obtain that (Pu' Muv) form the 0(3, 2) Sp(4) anti-de-Sitter algebra, and besides [Pa+a,Pa+b]
= 2fabCRc
13 September 1984
PHYSICS LETF ER S
,
[Pa+a,Rb]
=
2fabCP4+c '
IRa, Rb] = 2fabCRa ,
de-Sitter-like algebra we propose to consider the octon. ionic extension of OSp(1; 4), which we denote by UaU(4; 1 ; 0). The octonionic 5 × 5 matrices s~ belonging to the Uc~U(4; 1 ; 0) superalgebra do satisfy the equation UaU(4, 1 ; 0):
s~#G=-G~,
G
.
=
(271
One gets
(25a)
s~-\F2(4 X 1) q./(4X 7 )' [M~,4+a, Mv,4+b] = 2171avfabCRc + 28abMla v ,
qgC + CQZ # = 0 , [Mu,4+a, P4+b] = 25abPu,
[Mu,4+a, Rb ] = 2fabCMu,4+b'
(25b)
U= - U ,
F # C = F 2.
(28)
We see that the bosonic sector of UaU(4, 1; 0) is U,~(4; 0) X U(1 ; 0), where Ca(4; 0) X U(1; O)
[Pu,Pa+a] = 2Mu,a+a,
[P~,,Ra]=O, ((SO(IO, 2)/SO(7)) X S 7) X S 7 ,
[Pu, Mv ,4+a ] = 2r/~v Pa+a ,
[M.v, Pa+al = 0 ,
[M.v, Ra] =0 ,
(25c)
(29)
and the respective algebras are (25) and (19). Four octonion-vatued (= 32 real) fermionic generators can be described as the following 5 X 5 matrices
[M~v, Mo,4+a] = 2r/voM~,4+a - 2rluoMv,4+a " Let us write down the 66 generators of SO(10, 2) as (Pu' Pu+a'Muv ' Mu,4+a'Mab)" If we replace the 21 generatorSMab by 7 generators R a describing S7 at the point x = I, we see that 52 generators of Ua(4; 0) are counted in a similar way as the generators of the coset SO(10, 2)/G 2 = (SO(10, 2)/SO(7)) X S 7, what we denote as U~(4; 0) ~ SO(10, 2)/G 2. We would like to mention that similarly one can relate the 0(9, 1) and 0('9) Lie algebras with the octonionic extensions SL(2; 0) and U(2; 0). Calculating in a similar way the algebra at the identity one gets that SL(2; 0) ~ SO(9, 1)/G 2 ,
U(2; 0) ~ SO(9)/G 2 . (26)
3. It is known that the complex and quaternionic extensions of the OSp(1 ; 4) superalgebra describes D = 5 and D = 7 graded de-Sitter algebras [10], with, respectively, the bosonic sectors SO(4, 2) X U(1)and SO(6, 2) X SU(2). In order to describe the D = 1 1 graded
Q~ =
,
ql~
Q~a --
0
(3(I)
e a ql~
0
]'
where a = 1 ..... 7, a -'- 1 ..... 4, and q/~ denotes the Sp(4; R) spinor basis, satisfying the relation c~ c g ~ = C ~ . The bosonic sector is the following Ua(4; 0)
X U(1; 0):
(~-~),
~c+c#~=0,
~=-~
(31)
We denote the generators of Ua(4; O) in accordance with the formulae (24), and the generators L a describing U(1 ; 0) by the imaginary octonion w do satisfy the relation [La, Lb] = 2fabCLc (see (19)). By calculating the anticommutators of the supercharge matrices (30) we obtain the following relations:
(aa, Q3}
=I
2(TuC)a#P
#
1
- ~(ouvC)at~ I
#v
,
(32)
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PHYSICS LETTERS
Volume 145B, number 1,2
{Qc~a, Q#} = ~ (3"53"uC)a~Mu,4+a -- { (3'5C)c~P4+a
Jacobi identity at U = 1 gives the result
o(o,~a, O~b, Q'ye)
+ ~ CaoRa + 2Ca#L a _I
1
{Qaa, Qpb}- ~6ab [(3"uC)ePPu - 2(cruuC)a#M
lap
1 -- "35ad 6be~ef~defg(C~TQag + CTaQ~g + CaP QTg )'
]
(38)
1 c -- 2lab [(3"57~zC)a(3Mt~,4+c -- ~1 (3'5C)~t3 p 4+e
+ ~Ca~R c - 2ca~re] .
(32 cont'd)
The covariance relations do look as follows:
[Pu' Qa] (3',)~Qt3,
13 September 1984
[Pu'
[Muv, Qa] = (ouv)c, PQ# ,
QJ
= (3",)aOQ~a'
[Mu,, Qc~a] = ( o u u ) J Q ~ ,
i.e. similarly to (35) our superalgebra is closed under triple Jacobi operation. The relation (38) can be understood as describing the generalized CSE equation for UaU(4; 1 ; 0) at the point U = 1, or as characterizing the first term in the Taylor expansion of the structure function of UaU(4; 1 ; 0) around U = 1. In general one can represent the octonionic extension of the Lie n X n matrix (super)algebra G by
Gi -+ ~i = Gi ~ eaFi (a ) = Gi ~ Fai ' [Mu,4+a, Q~] = ( 3 ' 5 % ) ~ Q , ~ a ,
(39)
where the generators G i form the Lie (super) algebra G. The real generators of ~i are L A = (G i, Fai), and they have the following property (i) D(L A , L B, L C) = 0 if at least one generator LA, L B , L C belongs to G (ii) D(Fa/, Fbk, Fcl) 4= 0 but
[Mta,4+a, Qab] = (3"u3"5)a#fabe Q~e ' [P4+a, Qal = (3"s)~C~Qt3a , [P4+a, ac~b] : (3"5)a#fabCaoc ,
fabCD(Faf , F bk, Fcl ) = 0 . [Ra, as] = Qc,a , [La, Qc~] = -Qaa ,
[Ra, Qab] =fabCQac, [La, aab] = -fabCQac •
The trilinear relation (39) characterize therefore the octonionic extensions of Lie (super) algebras. (33)
In order to study the derivation from the Jacobi identity one can introduce the Jacobi function
D ( A , B , C) = ½([[A, B], C] + cycl.) .
(34)
One gets for S 7 the relation
D(ea ' eb ' ec)= ~abcd ed .
(35)
We see from (17) that the RHS of (35) describes the derivative of the structure function at x = 1, and eq. (35) isthe CSE equation at the point x = 1. The trilinear relation, satisfied by (35) is the following
fabCD(ea, e b, ec) = 0 .
(36)
Similarly, one can introduce the graded Jacobi function
o(ac~ ' ap, 07 ) = ½[Q~, {09, a.r} ] + graded cycl., (37) and calculate the derivation from graded Jacobi identities for UaU(4; 1 ; 0). In particular the generalized 68
(40)
4. We would like to add the following remarks: (a) The octonionic extensions of Lie (super) groups (except U(1; 0) -= S 7) due to nonassociativity of octonions are not closed under multiplication, i.e. do not form parallelizable manifolds. Any octonionic extension has, however, two types of parallelizable submanifolds: (i) Describing quaternionic extensions of Lie (supergroups. Due to the property that the octonions (e r, er+ 1, er+3) form three quaterionic algebras, one can select three quaternionic subalgebras (see (39))
L(Ar) = (Gi, Fri, Fr+l,i, Fr+3,i),
r = 1, 2, 3 .
(41)
Three Lie (super) groups obtained by the exponential map of L~) cover the manifold described by the octonionic extension of G. For example UaU(4; 1 ; 0) can be covered by three graded Lie algebras U~U(4; 1 ; H) (see ref. [10]). (ii) Describing embedded seven-spheres S 7, CS 7 • The subalgebras describing S 7 submanifolds are described by Fai (i fixed; a = 1 ..... 7). In particular inside
Volume 145B, number 1,2
PHYSICS LETTERS
U~(4; 0) one can embed six seven-spheres with the generatorsMu,4+a, R4+ a and R a. For fixed/1 we obtain from (25) five subalgebras: (P4+a, Ra) described by (25a) and (M#,4+a, Ra) , which looks as follows: ( +:/l= 1,2,3) [Mu,a+a, Mu,4+b] = ±2fabCRc -:/l
but the concept of their invariance has to be modified. It was shown already by Nahm [8] that seven gauge fields describing linear connection on a generalized principle bundle with an S 7 fibre can be obtained via dimensional reduction o f D = 11 supergravity with the internal sector described by S 7.
0
References
[M•,4+a, Rb] = 2fabCM.,4+c,
[R a, R b] : 2fabeRa .
13 September 1984
(42)
Introducing R+a = R a + P4+a one can check that the 14 parameter parallelizable manifold generated by (Ra, P4+a) is S 7 X S7. Similarly one gets that (Ra, Mu,4+a) for gt = 1,2, 3 are the generators of S 7 × S 7, and for/a = 0 they describe the complexification CS 7 of S 7. (b) It should be mentioned that one can generalize the theory of principal bundles with the structure group G to generalized principal bundles with the group G replaced by a paralMizable manifold, in particular by S 7, which can be called an octonionic line bundle. For a fixed paralMization of the structural manifold it is possible to formulate the theory of connections,
[ 1 ] M. Duff, B.E.W. Nilsson and C.N. Pope, Nucl. Phys. B233 (1984) 433. [2] Y. Ne'eman and T. Regge, Phys. Lett. 74B (1978) 54. [3] R. d'Auria, P. Fr6 and T. Regge, in: Supergravity 81, eds. S. Ferrara and J.C. Taylor, P. 421. [4] M.S. Sohnius, Z. Phys. C18 (1983) 229. [5] T. Dereli, M. Panahinoghaddam, A. Sudberg and R.W. Tucker, Phys. Lett. 126B (1983) 33. [6] F. Gtirsey and C.H. Tze, Phys. Lett. 127B (1983) 191. [7] J. Lukierski and P. Minnaert, Phys. Lett. 129B (1983) 392. [8] W. Nahm, CERN preprint TH-2489 (April 1978), unpublished. [9] R. Diandarer, F. Gtirsey and C.H. Tze, Yale University YTP 81-13 (September 1983). [10] Z. Hasiewicz, J. Lukierski and P. Morawiec, Phys. Lett. 130B (1983) 55.
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