We show that the sequence of Jordan algebras M~, M3 z, M34, and M 8, whose elements are in the 3 x 3 Hermitean matrices over ~, C, Q, and 13, respectively, provide an elegant and natural framework in which to describe supersymmetric gauge theories. The four minimal supersymmetric gauge theories are in a one-to-one correspondence with these four Jordan algebras and, hence, with the four division algebras.
Contemporary approaches to unification have supposed supersymmetry to be a fundamental building block of nature. In view of its importance, a deeper understanding of the origin of the various supersymmetric theories is desirable. In particular, we will consider the four minimal supersymmetric gauge theories, and show that these theories can be 'unified' in the sense that they can all be put on an equal footing by using the four division algebras.
Recently, Schwarz [1] has suggested that there may be a correspondence between the minimal supersymmetric gauge theories [2] and the division algebras [3]. This speculation stems from the observation that the minimal supersymmetric gauge theories exist only in spacetime dimensions of 3, 4, 6, and 10. In these dimensions the number of propagating Bose and Fermi degrees of freedom is one for D = 3, two for D = 4, four for D = 6, and eight for D = 10. These dimensions are the dimensions of the four division algebras R, C, Q, and O, the real, complex, quaternion, and octonion number system, respectively. The purpose of this Letter is to provide a basis for understanding the connection between the division algebras and the minimal supersymmetric gauge theories, using results of recent work by ourselves [4] on the connection between the Lorentz groups in 3, 4, 6, and 10 dimensions and the Jordan algebras [5].
The minimal supersymmetric gauge theory consists of the gauge field A~ and the spinor 2 a, where a labels the generators of a semi-simple gauge group. The action for these fields, for D spacetime dimensions is [ 1,2]