2-calibrations on R2n

The notion of calibration was defined by Harvey and Lawson in 1982 and since then this notion proved to be important in the fields of minimal submanifolds and special holonomy in dimension 7 and 8. Calibrated submanifolds are natural examples of homologically volume minimizing submanifolds and in dimensions 7 and 8 the associative, coassociative and Cayley calibrations possess a rich geometry.The notion of the self duality of 2-forms is a central notion in classical Yang-Mills theory in dimension 4 and there are several generalizations of this notion to higher even dimensions which are essentially equivalent and which play a role in higher dimensional gauge theories.This two notions, calibrations and self duality were rather unrelated yet. In this work, the relationship between the notions of 2-calibrations and strong self-duality of 2-forms is investigated. It is shown that if the notion of calibration is suitably modified, then the two notions essentially coincide. Moreover, inspired by the notion of normed triality, a notion of normed duality is defined and it is shown that an anti-symmetric normed duality coincides with the notion of modified 2-calibration.In the theory of calibrations, most of the important and interesting examples are given by forms on Euclidean spaces with constant coefficients and these are shown to satisfy also our modified condition. We proved that on R^2n any modified calibration 2-field (and as a consequence the corresponding strong self-dual 2-form field) has necessarily constant coefficients.