  We present a novel formulation of the instanton equations in 8-dimensional
Yang-Mills theory. This formulation reveals these equations as the last member
of a series of gauge-theoretical equations associated with the real division
algebras, including flatness in dimension 2 and (anti-)self-duality in 4. Using
this formulation we prove that (in flat space) these equations can be
understood in terms of moment maps on the space of connections and the moduli
space of solutions is obtained via a generalised symplectic quotient: a Kaehler
quotient in dimension 2, a hyperkaehler quotient in dimension 4 and an
octonionic Kaehler quotient in dimension 8. One can extend these equations to
curved space: whereas the 2-dimensional equations make sense on any surface,
and the 4-dimensional equations make sense on an arbitrary oriented manifold,
the 8-dimensional equations only make sense for manifolds whose holonomy is
contained in Spin(7). The interpretation of the equations in terms of moment
maps further constraints the manifolds: the surface must be orientable, the
4-manifold must be hyperkaehler and the 8-manifold must be flat.
