An octonionic formulation of the M-theory algebra

We give an octonionic formulation of the N = 1 supersymmetry algebra in D = 11, including all brane charges. We write this in terms of a novel outer product, which takes a pair of elements of the division algebra A and returns a real linear operator on A. More generally, with this product comes the power to rewrite any linear operation on R^n (n = 1,2,4,8) in terms of multiplication in the n-dimensional division algebra A. Finally, we consider the reinterpretation of the D = 11 supersymmetry algebra as an octonionic algebra in D = 4 and the truncation to division subalgebras.


INTRODUCTION
A recurring theme in the study of supersymmetry and string theory is the connection to the four division algebras: the real numbers R, the complex numbers C, the quaternions H and the octonions O. See, for example, [1][2][3][4][5][6][7][8]. The octonions are of particular interest in this context since they may be used to describe representations of the Lorentz group in spacetime dimensions D = 10, 11, where string and M-theory live. Furthermore, the octonions provide a natural explanation [9,10] for the appearance of exceptional groups as the U-dualities of supergravities [11] and M-theory [12,13].
A D = 11 spinor with 32 components may be packaged as a 4-component octonionic column vector [5,14]. This has prompted the question of how to write the algebra of D = 11 supergravity (or 'M-algebra') using octonionic supercharges Q. This was explored in [14] where the problem was highlighted that the apparently natural choice of octonionic matrices could not provide enough degrees of freedom to account for all of M-theory's brane charges. Another fundamental question that arises when writing the {Q, Q} algebra in this way is whether or not the usual anti-commutator is really the appropriate object to study, given that the fermionic supercharges are written over a non-commutative and non-associative algebra O.
In the present paper we tackle this problem by introducing a novel outer product, which takes a pair of elements belonging to a division algebra A and returns a real linear operator on A, expressed using multiplication in A. This product enables one to rewrite any expression involving n × n matrices and n-dimensional vectors in terms of multiplication in the n-dimensional division algebra A. We solve the problem of the octonionic M-algebra using this product, which allows a derivation of the correct {Q, Q} bracket. In the final section we consider "Cayley-Dickson halving" the octonionic M-algebra, which corresponds to its reinterpretation as the maximal supergravity algebra in D = 7, 5, 4. For example, the M-algebra may be considered to be an oc-tonionic rewriting of the D = 4, N = 8 supersymmetry algebra; from this perspective the D = 4, N = 1 algebra comes from a truncation O → R.

THE DIVISION ALGEBRAS
A normed division algebra is an algebra A equipped with a positive-definite norm satisfying the condition ||xy|| = ||x|| ||y||. (1) Remarkably, there are only four such algebras: R, C, H and O, with dimensions n = 1, 2, 4 and 8, respectively. A division algebra element x ∈ A is written as the linear combination of n basis elements with real coefficients: x = x a e a , with x a ∈ R and a = 0, · · · , (n − 1). One basis element e 0 = 1 is real; the other (n − 1) e i are imaginary: where i = 1, · · · , (n − 1). In analogy with the complex case, we define a conjugation operation indicated by *, which changes the sign of the imaginary basis elements: The multiplication rule for the basis elements of a division algebra is given by: where we define the structure constants 1 Γ a bc = δ a0 δ bc + δ b0 δ ac − δ ab δ c0 + C abc , Γ a bc = δ a0 δ bc − δ b0 δ ac + δ ab δ c0 − C abc ⇒ Γ a bc =Γ a cb . The tensor C abc is totally antisymmetric with C 0ab = 0, so it is identically zero for A = R, C. For the quaternions C ijk is simply the permutation symbol ε ijk , while for the octonions the non-zero C ijk are specified by the set of oriented lines of the Fano plane, see [15].
One of the most important properties of the division algebras is that they provide a representation of the SO(n) Clifford algebra. This is reflected in the structure constants, which satisfy In other words, we have the interpretation that multiplying a divison algebra element ψ by the basis element e a has the effect of multiplying ψ's components by the gamma matrixΓ a : This property is essential for many of the applications of division algebras to physics, including that of this paper.
A natural inner product [15] on A is given by: This is just the canonical inner product on R n .

A NEW OUTER PRODUCT
It is interesting to see what other linear operations on R n look like when written in terms of the divisionalgebraic multiplication rule. This was explored in [16], but we take a different approach here. Consider the following general problem. Given some linear operator on R n expressed as an n × n matrix M ab , we would like to find an operatorM on the division algebra A such thatM has the effect of multiplying the components of x = x a e a ∈ A by M ab : An explicit form for this operator can be found using the inner product above. First we rewrite Now it is clear that the operator where a dot represents a slot for an octonion, has matrix elements This suggests that we write the outer product for division algebra elements using their multiplication rule, defining: With the new product comes the power to rewrite any expression involving n × n matrices and n-dimensional vectors in terms of multiplication in the n-dimensional division algebra A.
It is useful to note various equivalent ways of writing the outer product above: Due to the alternativity of the division algebras we also have and similarly for the other four possibilities above.

OCTONIONIC SPINORS IN D = 11
In D = 11 the Majorana spinor may be written as a 32component real column vector. However, if we consider R 32 as the tensor product R 4 ⊗ R 8 ∼ = R 4 ⊗ O then we can write this as a 4-component octonionic column vector A natural set of generators {γ M } = {γ 0 , γ a+1 , γ 9 , γ 10 }, M = 0, 1, . . . , 10 for the 4 × 4 octonionic Clifford algebra is then given by with a = 0, 1, . . . , 7. These matrices satisfy and the infinitesimal Lorentz transformation of the spinor λ is where ω M N = −ω N M . In general, the action of the rank r Clifford algebra element on λ can be written The positioning of the brackets in the above expression follows from repeated application of (7); nonassociativity matters only for the imaginary gamma matrices γ i+1 , which provide a representation of the SO (7) Clifford algebra. If we define an operatorγ M , whose action is left-multiplication by γ M , then we can think of the rank r Clifford algebra element as the operator where the operatorsγ M must be composed aŝ This ensures that the action ofγ [M1M2...Mr] on a spinor is given by (20), as required.

THE OCTONIONIC M-ALGEBRA
The anti-commutator of two supercharges in the D = 11 supergravity theory is conventionally written as the 'M-algebra' [17,18] and if we think of αa as a composite spinor indexᾱ = 1, . . . , 32, then the set of {γ Mᾱβ } generates the usual real Clifford algebra as in (23). For the charge conjugation matrix, we define the 4 × 4 real matrix (which is numerically equal to γ 0 but with a different index structure) The octonionic matrix elements of this are then trivially which can be identified with the 32 × 32 matrix: Armed with these tools, the right-hand side can then be written over O simply by replacingᾱ → α and putting hats on the gammas: With the identificationᾱ = αa we can also write the left-hand side of (23) in terms of the composite indices: Now, the expression (29) is an octonionic operator with matrix elements as on the right-hand side of (23), so on the left we require an octonionic operator with matrix elements given by (30). The required operator is obtained simply by contracting (30) with the outer product e a × e b defined in (13): The octonionic formulation of the M-algebra is then Using the first two versions of the outer product given in (14), we could write the left-hand side as The first two terms look similar to the more intuitive anticommutator {Q α , Q * β }, explored in [14], but to reproduce the full M-algebra we require all four terms above.

RELATION TO LOWER DIMENSIONS
It is interesting to consider the octonionic version of the supersymmetry algebra after an 11 = 4 + 7 split: Seven of the Clifford algebra generators γ i+1 are imaginary, while the other four are real. This suggests that we split the dimensions as follows: In D = 4 we regard the D = 11 octonionic spinor Q αa e a as eight 4-component Majorana spinors Q αa , which we may leave packaged as an 'internal' octonion. This transforms as the spinor 8 of SO (7). The D = 4 interpretation of the octonionic gamma matrices is as follows:γ whereê i denotes the operator whose action is leftmultiplication by e i and γ * (otherwise known as γ 5 ) is the highest rank Clifford element: The matrix C αβ is just the charge conjugation matrix in D = 4. We do not split the M, N indices of equation (33) into µ and i parts here, as the expression of the right-hand side itself is not particularly illuminating. The result is a copy of the N = 8 supersymmetry algebra written over the octonions. The interesting point is that the D = 11 supersymmetry algebra can be reinterpreted as an octonionic D = 4 algebra.
More generally, the spinor and associated gamma matrices defined in (16) and (17) correspond to those of D = 4, 5, 7 if we replace O with R, C, H, respectivelysee Table I. This means that in this framework the minimal supersymmetry algebra in these dimensions is written over R, C, H, while doubling the amount of supersymmetry corresponds to Cayley-Dickson doubling the division algebra. This process terminates when we reach maximal supersymmetry, i.e. when the Cayley-Dickson process takes us to O, the largest normed division algebra.    The above discussion serves to emphasise the correspondence between the octonions and maximal supersymmetry in various dimensions. Rather than thinking of the M-theory algebra as an eleven-dimensional real algebra, it may be fruitful to think of it as a four-dimensional octonionic one, as in Table II.
The work of LB is supported by an Imperial College Junior Research Fellowship. The work of MJD is supported by the STFC under rolling grant ST/G000743/1.