  Group theory indicates the existence of a $SO(8) X SO(7) \subset SO(16)$
invariant self-duality equation for a 3-form in 16 dimensions. It is a signal
for interesting topological field theories, especially on 8-dimensional
manifolds with holonomy group smaller than or equal to Spin(7), with a gauge
symmetry that is SO(8) or SO(7). Dimensional reduction also provides new
supersymmetric theories in 4 and lower dimensions, as well as a model with
gravitational interactions in 8 dimensions, which relies on the octonionic
gravitational self-duality equation.
