  Taking the simple examples of an Abelian 1-form gauge theory in two (1 +
1)-dimensions, a 2-form gauge theory in four (3 + 1)-dimensions and a 3-form
gauge theory in six (5 + 1)-dimensions of spacetime, we establish that such
gauge theories respect, in addition to the gauge symmetry transformations that
are generated by the first-class constraints of the theory, additional
continuous symmetry transformations. We christen the latter symmetry
transformations as the dual-gauge transformations. We generalize the above
gauge and dual-gauge transformations to obtain the proper (anti-)BRST and
(anti-)dual-BRST transformations for the Abelian 3-form gauge theory within the
framework of BRST formalism. We concisely mention such symmetries for the 2D
free Abelian 1-form and 4D free Abelian 2-form gauge theories and briefly
discuss their topological aspects in our present endeavor. We conjecture that
any arbitrary Abelian p-form gauge theory would respect the above cited
additional symmetry in D = 2p (p = 1, 2, 3,...) dimensions of spacetime. By
exploiting the above inputs, we establish that the Abelian 3-form gauge theory,
in six (5 + 1)-dimensions of spacetime, is a perfect model for the Hodge theory
whose discrete and continuous symmetry transformations provide the physical
realizations of all aspects of the de Rham cohomological operators of
differential geometry. As far as the physical utility of the above nilpotent
symmetries is concerned, we demonstrate that the 2D Abelian 1-form gauge theory
is a perfect model of a new class of topological theory and 4D Abelian 2-form
as well as 6D Abelian 3-form gauge theories are the field theoretic models for
the quasi-topological field theory.
