Continuous and discrete symmetries in a 4D field-theoretic system: Symmetry operators and their algebraic structures

Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism, we show the existence of i) a couple of off-shell nilpotent (i.e., fermionic) BRST and co-BRST symmetry transformations, and ii) a full set of non-nilpotent (i.e., bosonic) symmetry transformations for an appropriate Lagrangian density that describes the combined system of the free Abelian 3-form and 1-form gauge theories in the physical four (3 + 1)-dimensions of the flat Minkowskian spacetime. This combined BRST-quantized field-theoretic system is essential for the existence of the off-shell nilpotent co-BRST and non-nilpotent bosonic symmetry transformations in the theory. We concentrate on the full algebraic structures of the above continuous symmetry transformation operators along with a couple of very useful discrete duality symmetry transformation operators existing in our four (3 + 1)-dimensional (4D) field-theoretic model. We establish the relevance of the algebraic structures, respected by the above discrete and continuous symmetry operators, to the algebraic structures that are obeyed by the de Rham cohomological operators of differential geometry. One of the highlights of our present letter is the observation that there are no “exotic” fields with the negative kinetic terms in our present 4D field-theoretic example for Hodge theory.