A Note on Higher Dimensional Instantons and Sypersymmetric 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 Ω and it was recently employed as a topological gauge fixing condition in higher dimensional generalizations of cohomological Yang-Mills theory. When Ω 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 Σ with respect to the calibration Ω. This is an instanton configuration on the total space of the normal bundle N(Σ) of the submanifold Σ 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. ∗ Talk presented at the workshop on “Gauge Theory and Integrable Models” (YITP, Kyoto), January 26-29, 1999.