  We demonstrate that the two (1 + 1)-dimensional (2D) free 1-form Abelian
gauge theory provides an interesting field theoretical model for the Hodge
theory. The physical symmetries of the theory correspond to all the basic
mathematical ingredients that are required in the definition of the de Rham
cohomological operators of differential geometry. The conserved charges,
corresponding to the above continuous symmetry transformations, constitute an
algebra that is reminiscent of the algebra obeyed by the de Rham cohomological
operators. The topological features of the above theory are discussed in terms
of the BRST and co-BRST operators. The super de Rham cohomological operators
are exploited in the derivation of the nilpotent (anti-)BRST, (anti-)co-BRST
symmetry transformations and the equations of motion for all the fields of the
theory, within the framework of the superfield formulation. The derivation of
the equations of motion, by exploiting the super Laplacian operator, is a
completely new result in the framework of the superfield approach to BRST
formalism. In an Appendix, the interacting 2D Abelian gauge theory (where there
is a coupling between the U(1) gauge field and the Dirac fields) is also shown
to provide a tractable field theoretical model for the Hodge theory.
