  In four-dimensional gauge theory there exists a well-known correspondence
between instantons and holomorphic curves, and a similar correspondence exists
between certain octonionic instantons and triholomorphic curves. We prove that
this latter correspondence stems from the dynamics of various dimensional
reductions of ten-dimensional supersymmetric Yang-Mills theory. More precisely
we show that the dimensional reduction of the (5+1)-dimensional supersymmetric
sigma model with hyperkaehler (but otherwise arbitrary) target X to a
four-dimensional hyperkaehler manifold M is a topological sigma model
localising on the space of triholomorphic maps M -> X (or hyperinstantons).
When X is the moduli space M_K of instantons on a four-dimensional hyperkaehler
manifold K, this theory has an interpretation in terms of supersymmetric gauge
theory. In this case, the topological sigma model can be understood as an
adiabatic limit of the dimensional reduction of ten-dimensional supersymmetric
Yang-Mills on the eight-dimensional manifold M x K of holonomy Sp(1) x Sp(1) in
Spin(7), which is a cohomological theory localising on the moduli space of
octonionic instantons.
