A natural framework for the minimal supersymmetric gauge theories (Q1111675)

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 \(\mathbb{R},\mathbb{C},\mathbb{Q}\) and \(\mathbb{O}\), the real, complex, quaternion, and octonion number systems, respectively. The purpose of this paper is to provide a basis for understanding the connection between the division algebras and the minimal supersymmetric gauge theories, using results on the connection between the Lorentz groups in \(3,4,6\) and \(10\) dimensions and the Jordan algebras \(M^n_3\) \((n=1,2,4,8)\), whose elements are in the \(3\times 3\) Hermitian matrices over \(\mathbb{R},\mathbb{C},\mathbb{Q}\) and \(\mathbb{O}\), respectively. The authors show that the Jordan algebraic framework allows the four supersymmetric gauge theories to be unified in a remarkably natural and elegant way.