A hidden supersymmetry of supersymmetric mechanics on the sphere S6

Abstract

In the supersymmetric classical or quantal mechanics of particle motion on the sphere S 6, an SO(7)-invariant theory, the existence of a totally antisymmetric tensor e ijk (defined by the product law of octonions) invariant under the G 2 sub-group of SO(7) allows the construction of a Killing–Yano tensor and the supercharge Q′ of a hidden supersymmetry of the theory. The canonical bracket { Q′, Q′} (or quantally the corresponding anticommutator) yields not the Hamiltonian of the theory but instead an operator determined by the quadratic Casimir operator of G 2. The situation is compared with the theory of a spin- 1 2 particle moving in the field of a Dirac monopole. Use of the tensor e ijk to break the SO(7)-invariance of the original theory of motion on S 6 to G 2 invariance is discussed.