Supersymmetric Yang-Mills theories:not quite the usual perspective

In this paper, we take up an old thread of development concerning the characterization of supersymmetric theories without any use of anticommuting variables that goes back to one of the authors' very early work (Nicolai H 1980 Nucl. Phys. B 176 419). Our special focus here will be on the formulation of supersymmetric Yang-Mills theories, extending previous results beyond D = 4 dimensions. This perspective is likely to provide new insights into these theories, and in particular the maximally extended N = 4 theory. As a new result we re-derive the admissible dimensions for interacting (pure) super-Yang-Mills theories to exist. This article is dedicated to the memory of Peter Freund, amongst many other things an early contributor to supersymmetry, and an author of one of the very first papers on superconformal gauge theories (Crispim Romao J et al 1977 Nucl. Phys. B 122 170). The final section contains some personal reminiscences of H N's encounters with Peter Freund.