Octonionic M-theory and D=11 generalized conformal and superconformal algebras

Following [1] we further apply the octonionic structure to supersymmetric D=11 $M$-theory. We consider the octonionic $2^{n+1} \times 2^{n+1}$ Dirac matrices describing the sequence of Clifford algebras with signatures ($9+n,n$) ($n=0,1,2, ...$) and derive the identities following from the octonionic multiplication table. The case $n=1$ ($4\times 4$ octonion-valued matrices) is used for the description of the D=11 octonionic $M$ superalgebra with 52 real bosonic charges; the $n=2$ case ($8 \times 8$ octonion-valued matrices) for the D=11 conformal $M$ algebra with 232 real bosonic charges. The octonionic structure is described explicitly for $n=1$ by the relations between the 528 Abelian O(10,1) tensorial charges $Z_\mu Z_{\mu\nu}, Z_{\mu \gt... \mu_5}$ of the $M$-superalgebra. For $n=2$ we obtain 2080 real non-Abelian bosonic tensorial charges $Z_{\mu\nu}, Z_{\mu_1 \mu_2 \mu_3}, Z_{\mu_1 ... \mu_6}$ which, suitably constrained describe the generalized D=11 octonionic conformal algebra. Further, we consider the supersymmetric extension of this octonionic conformal algebra which can be described as D=11 octonionic superconformal algebra with a total number of 64 real fermionic and 239 real bosonic generators.

Using the first relation in (1.3) it has been proposed [13] that the standard D = 10 Poincaré superalgebra can be described by a pair of octonionic supercharges (Q 1 , Q 2 ), with the following basic relations where Z ab = Z + ba and P µ = (P 0 , P 1 , . . . P 9 ) describe D = 10 momentum generators. Recently we proposed also to impose on the M-superalgebra generators (see formulae (1.1) and (1.2)) the octonionic structure [1]. The relations (1.1-1.2) we replaced by octonionic M-superalgebra (r, s = 1, 2, 3, 4) where four real octonion-valued supercharges Q r replace 32 real supercharges Q A and in place of the 528 real Abelian charges present in the rhs of (1.1-1.2) we get only 52 independent real Abelian supercharges described by the real components ( As a consequence, the octonionic M-algebra can be fully described by 11 four-momenta P µ and only 41 generators Z µν describing together the Abelian contractions of all the generators in the coset O(10,2) G 2 ; another way of providing the bosonic generators of (1.6) is to introduce suitably constrained five-tensor charges Z [µ 1 ...µ 5 ] .
In this paper we continue the considerations presented in [1]. In Sect. 2 we shall consider more in detail the properties of the octonionic M-superalgebra (1.6) and present it as a member of generalized octonionic supersymmetry algebras in D = (9 + n, n) (n = 0, 1, 2, . . .). For n = 0 we obtain the generalized supersymmetry algebra for D = 9 Euclidean theory. If n = 1 we get the octonionic M-superalgebra considered in [1], and the case n = 2 provides the extension of M-superalgebra to D = 13 with signature (11,2) and D = 14 with signature (11,3), considered by Bars [14,15] 2 . It appears however that the relation (1.6) if r, s, = 1, 2, . . . , 2 n , can also be used for the supersymmetrization of the octonionic algebras U α (2 n |O), which describe for n = 1 the octonionic D = (9, 1) algebra with ten curved translations, for n = 2 the octonionic conformal algebra (see (1.3)) and for n = 3 the D = 11 octonionic generalized conformal algebra U α (8|O) with 232 real charges. We shall argue that U α (8|O) can be obtained from the generalized D = 11 conformal algebra Sp(64|R) [16,17] by imposing constraints describing the octonionic structure. It is known [17] that the Sp(64|R) generators can be described by O(11, 2) two-tensors, three-tensors and six-tensors. We shall show that the octonionic conformal M-algebra U α (8|O) will be completely described only in terms of the two-tensor generators from the coset supplemented by a suitably restricted set of three-tensor generators. It appears that U α (8|O) supplemented with S 7 ≃ U(1|O) generators describe the bosonic sector of the supersymmetric extension U α (8; 1|O) of the octonionic conformal M-algebra.
The conformal algebras in D = 10, since fundamental D=10 octonionic conformal spinors have four components, belong to the framework of conformal Jordan algebras [9][10][11]18,12]). Beyond D = 10 and in particular for the eleven dimensional conformal M-algebra U α (8; O), we are outside of the framework of conformal algebras associated with Jordan algebras. The construction of the conformal algebra can be however linked with the group of invariance of the metric for "doubled Lorentz spinors". This procedure we propose to apply to octonionic conformal spinors in dimensions D = (9 + n, n), providing D = 11, 13 etc. We consider in Sect. 3 the generalized octonionic conformal algebras and superalgebras in the form of octonionic-valued (super-)matrix realizations which admit closed algebraic relations with the (super-)matricial (anti-)commutator structure. For D=11, considering 8 × 8 octonionic matrices U α (8|O) ≃ Sp(8|O), one can show that it contains the generators parametrizing the coset O(11,2) G 2 (64 real generators), but with additional 168 real generators not closing to any subset of Lie algebra generators. Such algebra is also not of Malcev type [20]; it is an interesting task to elucidate its algebraic characterization.
Further, in Sect. 4, we provide the table listing the number of independent n-fold antisymmetric products of octonionic Dirac matrices in odd dimensions D = 7 + 2k (k = 0, 1, 2, 3) and we interprete them as the relation between the p-brane degrees of freedom in corresponding supersymmetric D-dimensional theory with the most general set of central charges. Let us observe that (0,7) spinors are real eight-dimensional and C = C T 3 . Introducing seven 8 × 8 real Γ i matrices, i = 1, . . . , 7, If we use 8 real supercharges one obtains the D=7 generalized Euclidean superalgebra which unfortunately does not contain the 7-momentum sector which should be linear in Γ i . One obtains the octonionic C O (0, 7) Clifford algebra with generators Γ i (7) satisfying the relation (2.1) by assuming where t i are seven octonionic units with the multiplication table (1.4). The Hermitean N = 1 octonionic superalgebra generated by the supercharge describes the octonionic N = 8 supersymmetric mechanics, with the real generator Z playing the role of the Hamiltonian [18].
One can introduce the following 2 × 2 matrix realizations of the octonionic-valued Clifford algebras: where C = Γ 0 . The generalized octonionic D=9 Poincaré algebra takes the form (r, s = 1, 2; µ = 0, 1 . . . 8) We see that the generator Z in D=9 is the central charge. Due however to the first relation (1.3) the superalgebra (2.7) can be promoted to D= (1,9) superPoincaré algebra [10], where now the D = 9 central charge Z is the component P 9 of the ten-dimensional momenta. One can also assume that the generators on the rhs of (2.7) are nonAbelian and form the algebra U α (2|O) containing only ten curved (1.9) translations, which is an octonionic counterpart of the D = 10 AdS algebra.
We see that in the D = 11 supersymmetric theory with basic superalgebra (2.13) the two-brane and five-brane tensorial central charges are strongly constrained, and the theory can be described completely -either by one-brane and two-brane sectors ((P µ , Z µν O ) in generalized momentum space or, using the dual picture, by (X µ , X µν O ) in generalized space-time, -or by the constrained five-brane sector ( c2) The D = 10 octonionic conformal superalgebra U α (4|O) The form (1.6) of the octonionic superalgebra is also obtained for the generalized octonionic D = 10 conformal superalgebras. For that purpose one can write the basic relation in O(10, 2) -covariant form (r, s = 1, . . . , 4; µ 1 , µ 2 = 0, 1, . . . 11) where (µ 1 , µ 2 = 0, 1, . . . 10) and M 12µ the curved D = 11 AdS translations. In particular if we introduce the rescaling of the generators M 12µ = R · P µ by performing the limit R → ∞ one can obtain from the relations (2.15) the D=11 octonionic M-algebra, given by (1.6).
We add here that by doubling the realization of C O (10, 1) given in footnote 4) one obtains the realizations of the Clifford algebras C O (2, 9) and C O (2, 10).

D=11 Octonionic Generalized (Super)conformal Transformations as Automorphisms
It is known that the conformal algebra can be introduced as the algebra of transformations leaving invariant the inner product of fundamental conformal spinors called also twistors. We shall apply this method to derivation of octonionic conformal algebra from octonionic spinors with inner product. In D = (10, 1) such inner product is given by ψ † Cη, where ψ, η are eightdimensional octonionic conformal O(11, 2) spinors described by pairs of octonionic O(10, 1) Lorentz spinors and the matrix C given by the product of the two space-like Clifford's Gamma matrices Γ (13) 0 , Γ 12 (see (2.17)) is real-valued and totally antisymmetric. Therefore, the conformal transformations are realized by the octonion-valued 8-dimensional matrices M leaving C invariant, i.e. satisfying This allows identifying the octonionic conformal transformations with the octonionic unitary-symplectic transformations U α (8|O). The most general octonionic-valued matrix leaving invariant Ω can be expressed as follows where the 4 × 4 octonionic matrices B, C are hermitian It is easily seen that the total number of independent components in (3.2) is precisely 232, as we expected from the previous considerations. It should be noticed that the set of octonionic matrices M of (3.2) type forms a closed algebraic structure under the usual matrix commutation. Indeed one gets [M, M] ⊂ M, endowing the structure of U α (8|O) to M. As recalled in the introduction, U α (2n; O) for n > 3 is no longer a conformal algebra associated with a Jordan-algebra (see e.g [18]), nevertheless it admits the Lie-algebraic commutation relations which, however, do not satisfy the Jacobi identities.
In the procedure of supersymmetric extension to the superconformal algebra we have to accommodate the components of 8 octonionic spinors of (11, 2) into a supermatrix enlarging U α (8|O). This can be achieved as follows. The two 4-column octonionic spinors α and β can be accommodated into a supermatrix of the form Under anticommutation, the lower bosonic diagonal block reduces to U α (8|O). On the other hand, extra real seven generators, associated to the 1-dimensional antihermitian matrix A i.e. described by seven imaginary octonions, are obtained in the upper bosonic diagonal block. Therefore, the generic bosonic element is of the form with A, B and C satisfying (3.5) and (3.3).

Octonionic Structure and p-Superbranes
We have shown that the 52 independent components of the Hermitian-octonionic Z ab matrix can be represented either as the 11 + 41 bosonic generators entering or as the 52 bosonic generators entering The reason for that lies in the fact that, unlike in the real case, the sectors individuated by (4.1) and (4.2) are not independent. This is a consequence of the multiplication table of the octonions. Indeed, when we multiply antisymmetric products of k octonionic-valued Gamma matrices, a certain number of them are redundant. For k = 2, due to the G 2 automorphisms, 14 such products have to be erased. In the general case [22] a table can be produced, which we write down (see Table 1) for 7 ≤ D ≤ 13 odd-dimensional spacetime corresponding to octonionic realizations of Clifford algebras considered in Sect. 2. The Table 1 was constructed from the D = 7 results (which can be easily computed), by taking into account that out of the D Gamma matrices, 7 of them are octonion-valued, while the remaining D − 7 are purely real. The columns in Table 1 are labeled by k, the number of antisymmetrized Gamma matrices.  Table 1: Number of independent octonionic tensorial charges with underlined octonionic-Hermitean matrices. The signatures entering the tables are respectively given by (0, 7), (9, 0)or (1,8), (10,1)or (2,9), (11,2)or (3,10).
The Table 1 is valid for octonionic generalized Poincaré superalgebras, with Abelian generators, as well as for their nonAbelian counterparts U α (k, O) (k = 2 for D = 9, k = 4 for D = 11, k = 8 for D = 13) describing octonionic D-dimensional AdS superbranes. The octonionic equivalence of different sectors, via generalized Poincaré or generalized AdS supersymmetry algebras interpreted as branes sectors, can be symbolically expressed in different odd space-time dimensions according to the Table 2. Table 2. The relation between octonionic super-p-branes M p .
In D = 11 dimensions the relation between M1 + M2 and M5 can be made explicit as follows. The 11 vectorial indices µ are split into the 4 real indices, labeled by a, b, c, . . . and the 7 octonionic indices labeled by i, j, k, . . .. We get: which shows the equivalence of the two sectors, as far as the octonionic content and tensorial properties are concerned. Please notice that the correct total number of 52 independent components is recovered 52 = 2 × 7 + 28 + 6 + 4.
It would be very interesting to find a dynamical realization of presented above octonionic super-p-branes framework. Similarly one can reproduce the count of independent degrees of freedom for octonionic M2, M3, M6 in D = 13.

Outlook
The octonions are the basic ingredients of many exceptional structures in mathematics. It is very well known, that the octonions provide the algebraic and geometric framework for the exceptional Lie algebras. Indeed, G 2 is the automorphism group of the octonions, while F 4 is the automorphism group of the 3 × 3 octonionic-valued hermitian matrices realizing the exceptional J 3 (O) Jordan algebra. F 4 and the remaining exceptional Lie algebras (E 6 , E 7 , E 8 ) are recovered from the so-called "magic square Tit's construction" which associates a Lie algebra to any given pair of division algebras, if one of these algebras coincide with the octonionic algebra [12]. We would like also to point out here that exceptional Lie algebras have numerous applications in elementary particle physics (see e.g. [21]) We have applied the octonionic structure to the description of a new version of M-theory. The main outcome of our considerations, which is symbolically represented in Table 2, implies that in such a framework the different brane sectors are no longer independent. We would like also to point out (see formula (2.14)) that octonionic structure imposes in extended space-times [23,24,19] additional constraints on central charge tensor coordinates, without restricting however D=11 spacetime.
Our considerations here are purely algebraic -the step which would be desirable is to provide some corresponding geometrical notions. It should be pointed out, however, that for nonassociative algebras the distinction between algebraic and geometric considerations rather disappears.