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Modified 2D Proca theory: off-shell nilpotent symmetries and their mathematical as well as physical implications

We derive the off-shell nilpotent Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST, (anti-)co-BRST, a bosonic and the ghost-scale symmetry transformations for a couple of equivalent Lagrangian densities of the two (1 + 1)-dimensional (2D) Stueckelberg-modified version of Proca theory which also incorporates a pseudo-scalar field. We also discuss algebraically suitable discrete symmetry transformations of the theory. Finally, we demonstrate the relevance of the above continuous and discrete symmetries in the context of differential geometry and establish that our present massive 2D theory is a tractable field theoretic model for the Hodge theory. One of the key novel observations of our present investigation is the appearance of Curci-Ferrari (CF) type restrictions even in the case of our present massive 2D Abelian 1-form gauge theory. We also point out the mathematical as well as physical implications of the above pseudo-scalar field in our present theory.