We construct an 11D supermembrane with topological central charges induced through an irreducible winding on a G2 manifold realized from the T 7 /Z 3 2 orbifold construction. The Hamiltonian H of the theory on a T 7 target has a discrete spectrum. Within the discrete symmetries of H associated with large diffeomorphisms, the Z 2 ×Z 2 ×Z 2 group of automorphisms of the quaternionic subspaces preserving the octonionic structure is relevant. By performing the corresponding identification on the target space, the supermembrane may be formulated on a G2 manifold, preserving the discreteness of its supersymmetric spectrum. The corresponding 4D low energy effective field theory has N = 1 supersymmetry.

We introduce a new integrable equation valued on a Cayley-Dickson (C-D) algebra. In the particular case in which the algebra reduces to the complex one the new interacting term in the equation cancells and the equation becomes the known Korteweg-de Vries equation. For each C-D algebra the equation has an infinite sequence of local conserved quantities. We obtain a Bäcklund transformation in the sense of Walhquist-Estabrook for the equation for any Cayley-Dickson algebra, and relate it to a generalized Gardner equation. From it, the infinite sequence of conserved quantities follows directly. We give the explicit expression for the first few of them. From the Bäcklund transformation we get the Lax pair and the one-soliton and two-soliton solutions generalizing the known solutions for the quaternion valued KdV equation. From the Gardner equation we obtain the generalized modified KdV equation which also has an infinite sequence of conserved quantities. The new integrable equation is preserved under a subgroup of the automorphisms of the C-D algebra. In the particular case of the algebra of octonions, the equation is invariant under SU (3).

The purpose of this paper is to provide an octonionic description of the Lie group SL(2, 𝕆). The main result states that it can be obtained as a free group generated by invertible and determinant preserving transformations from 𝔥2(𝕆) onto itself. An interesting characterization is given for the generators of G2. Also, explicit isomorphisms are constructed between the Lie algebras 𝔰𝔩(2, 𝕂), for 𝕂 = ℝ, ℂ, ℍ, 𝕆, and their corresponding Lorentz algebras.

We consider a formulation of Yang-Mills theory where the gauge field is valued on an octonionic algebra and the gauge transformation is the group of automorphisms of it. We show, under mild assumptions, that the only possible gauge formulations are the usual su (2) or u(1) Yang-Mills theories. arXiv:1412.4889v2 [hep-th] 31 Dec 2015

No abstract