Constraints, conserved charges and extended BRST algebra for a 3D field-theoretic example for Hodge theory

We perform the constraint analysis of a three (2 + 1)-dimensional (3D) field-theoretic example for Hodge theory (i) at the classical level within the ambit of Lagrangian formulation, and (ii) at the quantum level within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We derive the conserved charges corresponding to the six continuous symmetries of our present theory. These six continuous summery transformations are the nilpotent (anti-)BRST and (anti-)co-BRST symmetries, a unique bosonic symmetry and the ghost-scale symmetry. It turns out that the Noether conserved (anti-)BRST charges are found to be non-nilpotent even though they are derived from the off-shell nilpotent versions of the continuous and infinitesimal (anti-)BRST symmetry transformations. We obtain the nilpotent versions of the (anti-)BRST charges from the non-nilpotent Noether (anti-)BRST charges and discuss the physicality criteria w.r.t. the latter to demonstrate that the operator forms of the first-class constraints (of the classical gauge theory) annihilate the physical states at the quantum level. This observation is consistent with Dirac's quantization conditions for the systems that are endowed with the constraints. We lay emphasis on the existence of a single (anti-)BRST invariant Curci-Ferrari (CF) type restriction in our theory and derive it from various theoretical angles. • The classical theory of the 3D combined system of the Abelian 1-form and 2-form theories is endowed with the first-class constraints. • The classical gauge symmetry has been elevated to their quantum counterparts (anti-)BRST symmetries. • In the BRST quantized theory, there are six continuous symmetries. • Noether conserved charges have been computed corresponding to the above continuous symmetries. • The extended BRST algebra has been obtained among the appropriate conserved charges and its relationship with the Hodge algebra has been established.