  We demonstrate a few striking similarities and some glaring differences
between (i) the free four (3 + 1)-dimensional (4D) Abelian 2-form gauge theory,
and (ii) the anomalous two (1 + 1)-dimensional (2D) Abelian 1-form gauge
theory, within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We
demonstrate that the Lagrangian densities of the above two theories transform
in a similar fashion under a set of symmetry transformations even though they
are endowed with a drastically different variety of constraint structures.
Taking the help of our understanding of the 4D Abelian 2-form gauge theory, we
prove that the gauge invariant version of the anomalous 2D Abelian 1-form gauge
theory is a new field-theoretic model for the Hodge theory where all the de
Rham cohomological operators of differential geometry find their physical
realizations in the language of proper symmetry transformations. The
corresponding conserved charges obey an algebra that is reminiscent of the
algebra of the cohomological operators. We briefly comment on the consistency
of the 2D anomalous 1-form gauge theory in the language of restrictions on the
harmonic state of the (anti-) BRST and (anti-) co-BRST invariant version of the
above 2D theory.
