Abstract
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.
Highlights
JHEP11(2014)022 supersymmetry algebra; from this perspective the D = 4, N = 1 algebra comes from a truncation O → R
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
Summary
A normed division algebra is an algebra A equipped with a positive-definite norm satisfying the condition. 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. We have the interpretation that multiplying a divison algebra element ψ by the basis element ea has the effect of multiplying ψ’s components by the gamma matrix Γa: eaψ = eaebψb = Γabcecψb = ecΓacbψb. This property is essential for many of the applications of division algebras to physics, including that of this paper.
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