A Note on Higher Dimensional Instantons and Supersymmetric Cycles

We discuss instantons in dimensions higher than four. A generalized self-dual or anti-self-dual instanton equation in n-dimensions can be defined in terms of a closed (n-4) form $\Omega$ and it was recently employed as a topological gauge fixing condition in higher dimensional generalizations of cohomological Yang-Mills theory. When $\Omega$ is a calibration which is naturally introduced on the manifold of special holomony, we argue that higher dimensional instanton may be locally characterized as a family of four dimensional instantons over a supersymmetric (n-4) cycle $\Sigma$ with respect to the calibration $\Omega$. This is an instanton configuration on the total space of the normal bundle $N(\Sigma)$ of the submanifold $\Sigma$ and regarded as a natural generalization of point-like instanton in four dimensions that plays a distinguished role in a compactification of instanton moduli space.