Nilpotent (anti-)co-BRST symmetries and their consequences in a 4D BRST-quantized field-theoretic system

We demonstrate the existence of the infinitesimal, continuous and off-shell nilpotent (anti-)co-BRST symmetry transformations for the coupled (but equivalent) Lagrangian densities in the case of a four (3 + 1)-dimensional (4D) combined field-theoretic system of the free Abelian 3-form and 1-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. Using the standard theoretical tricks of the Noether theorem, we derive the Noether (anti-)co-BRST currents and corresponding conserved (anti-)co-BRST charges. We establish that the latter are not nilpotent of order two due to the presence of the non-trivial Curci-Ferrari (CF) type restrictions on our theory (which have been deduced, in our present endeavor, from two different theoretical angles). We derive the off-shell nilpotent versions of the conserved (anti-)co-BRST charges and discuss the physicality criteria w.r.t. them. We demonstrate that the physical states (existing in the total quantum Hilbert space of states) are those that are annihilated by the operator forms of the dual versions of the first-class constraints on our theory. We comment very briefly on the annihilation of the physical states by the operator forms of the first-class constraints due to the physicality criteria w.r.t. the nilpotent versions of the (anti-)BRST charges because our 4D theory also respects the off-shell nilpotent (anti-)BRST symmetries. The presence of the discrete duality symmetry transformations is responsible for a deep connection amongst the first-class constraints and their dual counterparts.