A $3D$ Field-Theoretic Model: Discrete Duality Symmetry

We demonstrate the discrete duality symmetry between the Abelian 1-form and 2-form basic gauge fields in the context of a three $(2 + 1)$-dimensional ($3D$) combined system of the field-theoretic model of the free Abelian 1-from and 2-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. The classical gauge-fixed Lagrangian density of this theory is generalized to its quantum counterpart as the BRST and co-BRST invariant Lagrangian density. We show clearly the existence of the off-shell nilpotent (co-)BRST symmetry transformations and establish their intimate connection through a set of underlying discrete duality symmetry transformations in our $3D$ BRST-quantized theory. We provide the mathematical basis for the existence of the discrete duality symmetry transformations in our theory through the Hodge duality operator (that is defined on the $3D$ flat Minkowskian spacetime manifold). We briefly mention a bosonic symmetry transformation which is constructed from the anticommutator of the above off-shell nilpotent (co-)BRST symmetry transformations. We lay emphasis on the algebraic structures of the existing continuous and discrete duality symmetry transformations for our $3D$ BRST-quantized theory (where they are treated as operators). We also comment on the appearance of a pseudo-scalar field (with negative kinetic term). This field happens to be one of the possible candidates for the phantom field of the cosmological models.