Gauge theory and the division algebras

Abstract

We present a novel formulation of the instanton equations in eight-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 Kähler quotient in dimension 2, a hyperkähler quotient in dimension 4 and an octonionic Kähler quotient in dimension 8. One can extend these equations to curved space: whereas the two-dimensional equations make sense on any surface, and the four-dimensional equations make sense on an arbitrary oriented manifold, the eight-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 oriented, the 4-manifold must be hyperkähler and the 8-manifold must be flat.