A Note on Higher Dimensional Instantons and Supersymmetric Cycles

We discuss instantons in dimensions higher than four. A generalized self-dualor anti-self-dual instanton equation in n-dimensions can be defined in terms ofa closed (n-4) form $\Omega$ and it was recently employed as a topologicalgauge fixing condition in higher dimensional generalizations of cohomologicalYang-Mills theory. When $\Omega$ is a calibration which is naturally introducedon the manifold of special holomony, we argue that higher dimensional instantonmay be locally characterized as a family of four dimensional instantons over asupersymmetric (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 naturalgeneralization of point-like instanton in four dimensions that plays adistinguished role in a compactification of instanton moduli space.Comment: 14 pages, latex, Talk presented at the workshop on Gauge Theory and Integrable Models (YITP, Kyoto), January 26-29, 1999, the title correcte