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Abstract

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 [1]. 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 present a novel derivation
of 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 [2]. The final section
contains some personal reminiscences of H.N.'s encounters with Peter Freund.