Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) approach, we discuss mainly the fermionic (i.e., off-shell nilpotent) (anti-)BRST, (anti-)co-BRST, and some discrete dual symmetries of the appropriate Lagrangian densities for a two (1+1)-dimensional (2D) modified Proca (i.e., a massive Abelian 1-form) theory without any interaction with matter fields. One of the novel observations of our present investigation is the existence of some kinds of restrictions in the case of our present Stückelberg-modified version of the 2D Proca theory which is not like the standard Curci-Ferrari (CF) condition of a non-Abelian 1-form gauge theory. Some kinds of similarities and a few differences between them have been pointed out in our present investigation. To establish the sanctity of the above off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetries, we derive them by using our newly proposed (anti-)chiral superfield formalism where a few specific and appropriate sets of invariant quantities play a decisive role. We express the (anti-)BRST and (anti-)co-BRST conserved charges in terms of the superfields that are obtained after the applications of (anti-)BRST and (anti-)co-BRST invariant restrictions and prove their off-shell nilpotency and absolute anticommutativity properties, too. Finally, we make some comments on (i) the novelty of our restrictions/obstructions and (ii) the physics behind the negative kinetic term associated with the pseudoscalar field of our present theory.

We exploit the potential and power of the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST invariant restrictions on the (anti-)chiral supervariables to derive the proper nilpotent (anti-)BRST symmetries for the reparameterization invariant one (0+1)dimensional (1D) toy models of a free relativistic particle as well as a free spinning (i.e. supersymmetric) relativistic particle within the framework of (anti-)chiral supervariable approach to BRST formalism. Despite the (anti-)chiral super expansions of the (anti-)chiral supervariables, we observe that the (anti-)BRST charges, for the above toy models, turn out to be absolutely anticommuting in nature. This is one of the novel observations of our present endeavor. For this proof, we utilize the beauty and strength of Curci-Ferrari (CF)-type restriction in the context of a spinning relativistic particle but no such restriction is required in the case of a free scalar relativistic particle. We have also captured the nilpotency property of the conserved charges as well as the (anti-)BRST invariance of the appropriate Lagrangian(s) of our present toy models within the framework of (anti-)chiral supervariable approach.PACS numbers: 11.30.Ph; 02.20.+b Keywords: (Anti-)chiral supervariable approach; a free scalar relativistic particle; a free massless spinning relativistic particle; off-shell nilpotent (anti-)BRST symmetries; nilpotent (anti-)BRST charges; absolute anticommutativity of the (anti-)BRST charges; chiral and anti-chiral super expansion(s); (anti-)BRST invariant restrictions; Curci-Ferrari type restriction; coupled Lagrangians * To be precise, we shall utilize the augmented version of the supervariable approach to BRST formalism where we shall consider only the (anti-)chiral super expansions of the supervariables. We christen our approach as " the supervariable approach" because we are dealing with "variables" and not the "fields" in our present endeavor (where we are concerned with only the 1D toy models of relativistic particles).

We derive the off-shell nilpotent (fermionic) (anti-)BRST symmetry transformations by exploiting the (anti-)chiral superfield approach (ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism for the interacting Abelian 1-form gauge theories where there is a coupling between the U(1) Abelian 1-form gauge field and Dirac as well as complex scalar fields. We exploit the (anti-)BRST invariant restrictions on the (anti-)chiral superfields to derive the fermionic symmetries of our present D-dimensional Abelian 1-form gauge theories. The novel observation of our present investigation is the derivation of the absolute anticommutativity of the nilpotent (anti-)BRST charges despite the fact that our ordinary D-dimensional interacting Abelian 1-form gauge theories are generalized onto the (D, 1)-dimensional (anti-)chiral super submanifolds (of the general (D, 2)-dimensional supermanifold) where only the (anti-)chiral super expansions of the (anti-)chiral superfields have been taken into account. We also discuss the nilpotency of the (anti-)BRST charges and (anti-)BRST invariance of the Lagrangian densities of our present interacting Abelian 1-form gauge theories within the framework of ACSA to BRST formalism. PACS numbers: 11.15.-q, 11.30.Pb Keywords:Interacting U(1) Abelian 1-form gauge theories; Dirac and complex scalar fields; ACSA to BRST formalism; chiral and anti-chiral superfields; (anti-)BRST invariant restrictions; conserved charges; nilpotency and absolute anticommutativity. * In the case of N = 2 SUSY quantum mechanical models, the anticommutator of two distinct SUSY transformations on a variable leads to the time derivative on that specific variable.

We exploit the power and potential of the (anti-)chiral superfield approach (ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism to derive the nilpotent (anti-) BRST symmetry transformations for any arbitrary D-dimensional interacting non-Abelian 1-form gauge theory where there is an SU(N) gauge invariant coupling between the gauge field and the Dirac fields. We derive the conserved and nilpotent (anti-)BRST charges and establish their nilpotency and absolute anticommutativity properties within the framework of ACSA to BRST formalism. The clinching proof of the absolute anticommutativity property of the conserved and nilpotent (anti-)BRST charges is a novel result in view of the fact that we consider, in our present endeavor, only the (anti-)chiral super expansions of the superfields that are defined on the (D, 1)-dimensional super-submanifolds of the general (D, 2)-dimensional supermanifold on which our D-dimensional ordinary interacting non-Abelian 1-form gauge theory is generalized. To corroborate the novelty of the above result, we apply the ACSA to an N = 2 supersymmetric (SUSY) quantum mechanical (QM) model of a harmonic oscillator and show that the nilpotent and conserved N = 2 super charges of this system do not absolutely anticommute.Keywords: (Anti-)chiral superfield approach; interacting non-Abelian 1-form gauge theory with Dirac fields; (anti-)BRST symmetries; (anti-)BRST charges; N = 2 SUSY harmonic oscillator; N = 2 SUSY symmetries; N = 2 SUSY charges, (anti-)chiral supervariable approach; nilpotency property; absolute anticommutativity property * The beauty of the (anti-)chiral superfield/supervariable approach is the observation that we obtain the (anti-)BRST symmetries for all the fields/variables of the theory from the (anti-)BRST (i.e. quantum gauge) invariant restrictions on the (anti-)chiral superfileds/supervariables.

We apply the geometrical supervariable approach to derive the appropriate quantum Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetries for the toy model of a free scalar relativistic particle by exploiting the classical reparameterization symmetry of this theory. The supervariable approach leads to the derivation of an (anti-)BRST invariant Curci-Ferrari (CF)-type restriction which is the hallmark of a quantum theory (discussed within the framework of BRST formalism). We derive the conserved and offshell nilpotent (anti-)BRST charges and prove their absolute anticommutativity property by using the virtues of CF-type restriction of our present theory. We establish the sanctity of the existence of CF-type restriction (i) by considering the (anti-)BRST symmetries for the coupled (but equivalent) Lagrangians, and (ii) by proving the symmetry invariance of the Lagrangians within the framework of supervariable approach. We capture the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges within the framework of (anti-)chiral supervariable approach (ACSA) to BRST formalism. One of the novel observations of our present endeavor is the derivation of CF-type restriction by using the modified Bonora-Tonin (BT) supervariable approach (while deriving the (anti-)BRST symmetries for the target spacetime and/or momenta variables) and by symmetry considerations of the Lagrangians of the theory. The rest of the (anti-)BRST symmetries for the other variables of our theory are derived by using the ACSA to BRST formalism.