A note on higher dimensional instantons and supersymmetric cycles

We discuss instantons in dimensions higher than four. A generalized self-du al or anti-self-dual instanton equation in n-dimensions can be defined in t erms of a closed (n - 4) form Ohm and it was recently employed as a topolog ical gauge fixing condition in higher dimensional generalizations of cohomo logical Yang-Mills theory. When Ohm is a calibration which is naturally int roduced on the manifold of special holonomy, we argue that higher dimension al instanton may be locally characterized as a family of four dimensional i nstantons over a supersymmetric (n - 4) cycle Sigma with respect to the cal ibration Ohm. This is an instanton configuration on the total space of the normal bundle N(Sigma) of the submanifold Sigma and regarded as a natural g eneralization of point-like instanton in four dimensions that plays a disti nguished role in a compactification of instanton moduli space.