SUPERSYMMETRIC YANG-MILLS, OCTONIONIC INSTANTONS AND TRIHOLOMORPHIC CURVES

In four-dimensional gauge theory there exists a well-known corresponde nce between instantons and holomorphic curves, and a similar correspon dence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynami cs of various dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory. More precisely we show that the dimensional reduct ion of the (5+1)-dimensional supersymmetric sigma model with hyper-Kah ler (but otherwise arbitrary) target X to a four-dimensional hyper-Kah ler manifold M is a topological sigma model localising on the space of triholomorphic maps M --> X (or hyperinstantons), When X is the modul i space M-K of instantons on a four-dimensional hyper-Kahler 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 su persymmetric Yang-Mills on the eight-dimensional manifold M x K of hol onomy Sp(1) x Sp(1) subset of Spin(7), which is a cohomological theory localising on the moduli space of octonionic instantons. (C) 1998 Els evier Science B.V.