Taking the simple examples of an Abelian 1-form gauge theory in two (1+1)-dimensions, a 2-form gauge theory in four (3+1)-dimensions and a 3-form gauge theory in six (5+1)-dimensions of space–time, we establish that such gauge theories respect, in addition to the gauge symmetry transformations that are generated by the first-class constraints of the theory, additional continuous symmetry transformations. We christen the latter symmetry transformations as the dual-gauge transformations. We generalize the above gauge and dual-gauge transformations to obtain the proper (anti-)BRST and (anti-)dual-BRST transformations for the Abelian 3-form gauge theory within the framework of BRST formalism. We concisely mention such symmetries for the 2D free Abelian 1-form and 4D free Abelian 2-form gauge theories and briefly discuss their topological aspects in our present endeavor. We conjecture that any arbitrary Abelian p-form gauge theory would respect the above cited additional symmetry in D = 2p(p = 1, 2, 3, …) dimensions of space–time. By exploiting the above inputs, we establish that the Abelian 3-form gauge theory, in six (5+1)-dimensions of space–time, is a perfect model for the Hodge theory whose discrete and continuous symmetry transformations provide the physical realizations of all aspects of the de Rham cohomological operators of differential geometry. As far as the physical utility of the above nilpotent symmetries is concerned, we demonstrate that the 2D Abelian 1-form gauge theory is a perfect model of a new class of topological theory and 4D Abelian 2-form as well as 6D Abelian 3-form gauge theories are the field theoretic models for the quasi-topological field theory.

We clearly and consistently supersymmetrize the celebrated horizontality condition to derive the off-shell nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for the supersymmetric system of a free spinning relativistic particle within the framework of superfield approach to BRST formalism. For the precise determination of the proper (anti-)BRST symmetry transformations for all the bosonic and fermionic dynamical variables of our system, we consider the present theory on a (1, 2)-dimensional supermanifold parameterized by an even (bosonic) variable (τ ) and a pair of odd (fermionic) variables θ andθ (with θ 2 =θ 2 = 0, θθ +θθ = 0) of the Grassmann algebra. One of the most important and novel features of our present investigation is the derivation of (anti-)BRST invariant Curci-Ferrari type restriction which turns out to be responsible for the absolute anticommutativity of the (anti-)BRST transformations and existence of the coupled (but equivalent) Lagrangians for the present theory of a supersymmetric system. These observations are completely new results for this model. 11.30.Ph; 02.20.+b Keywords: Supersymmetry, free spinning particle, superfield formalism, (anti-)BRST symmetries, Curci-Ferrari type restriction, supersymmetric horizontality condition † The BT superfield formalism, applied to a given (non-)Abelian gauge theory, utilizes only the HC. In a set of papers (see, e.g. [8-10]), we have generalized the HC in a consistent fashion by incorporating some other appropriate restrictions for the derivation of the (anti-)BRST symmetry transformations for the gauge as well as matter fields of a given gauge/reparametrization invariant theory.

h i g h l i g h t s• A novel method has been proposed for the derivation of N = 2 SUSY transformations. • General N = 2 SUSY quantum mechanical (QM) model with a general superpotential, is considered. • The above SUSY QM model is generalized onto a (1, 2)-dimensional supermanifold.• SUSY invariant restrictions are imposed on the (anti-)chiral supervariables. • Geometrical meaning of the nilpotency property is provided.Nilpotency property a b s t r a c t Using the supersymmetric (SUSY) invariant restrictions on the (anti-)chiral supervariables, we derive the off-shell nilpotent symmetries of the general one (0 + 1)-dimensional N = 2 SUSY quantum mechanical (QM) model which is considered on a (1, 2)-dimensional supermanifold (parametrized by a bosonic variable t and a pair of Grassmannian variables θ andθ with θ 2 =θ 2 = 0, θθ +θ θ = 0). We provide the geometrical meanings to the two SUSY transformations of our present theory which are valid for any arbitrary type of superpotential. We express the conserved charges and Lagrangian of the theory in terms of the supervariables (that are obtained after the application of SUSY invariant restrictions) and provide the geometrical interpretation for the nilpotency property and SUSY invariance of the Lagrangian for the general N = 2 * 559 SUSY quantum theory. We also comment on the mathematical interpretation of the above symmetry transformations.

Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism, we demonstrate the existence of the novel off-shell nilpotent (anti-)dual-BRST symmetries in the context of a six (5 + 1)-dimensional (6D) free Abelian 3-form gauge theory. Under these local and continuous symmetry transformations, the total gauge-fixing term of the Lagrangian density remains invariant. This observation should be contrasted with the offshell nilpotent (anti-)BRST symmetry transformations, under which, the total kinetic term of the theory remains invariant. The anticommutator of the above nilpotent (anti-)BRST and (anti-)dual-BRST transformations leads to the derivation of a bosonic symmetry in the theory. There exists a discrete symmetry transformation in the theory which provides a thread of connection between the nilpotent (anti-)BRST and (anti-)dual-BRST transformations. This theory is endowed with a ghost-scale symmetry, too. We discuss the algebra of these symmetry transformations and show that the structure of the algebra is reminiscent of the algebra of de Rham cohomological operators of differential geometry.

We derive the on-shell as well as off-shell nilpotent supersymmetric (SUSY) symmetry transformations for the N = 2 SUSY quantum mechanical model of a one (0+1)-dimensional (1D) free SUSY particle by exploiting the SUSY invariant restrictions (SUSYIRs) on the (anti-)chiral supervariables of the SUSY theory that is defined on a (1, 2)-dimensional supermanifold (parametrized by a bosonic variable t and a pair of Grassmannian variables θ andθ with θ 2 =θ 2 = 0, θθ +θθ = 0). Within the framework of our novel approach, we express the Lagrangian and conserved SUSY charges in terms of the (anti-)chiral supervariables to demonstrate the SUSY invariance of the Lagrangian as well as the nilpotency of the SUSY conserved charges in a simple manner. Our approach has the potential to be generalized to the description of other N = 2 SUSY quantum mechanical systems with physically interesting potential functions. To corroborate the above assertion, we apply our method to derive the N = 2 continuous and nilpotent SUSY transformations for one of the simplest interacting SUSY system of a 1D harmonic oscillator. PACS numbers: 11.30.Pb; Keywords: N = 2 SUSY quantum mechanics; N = 2 SUSY free particle and harmonic oscillator; on-shell and off-shell nilpotent symmetries; SUSY invariant restrictions; N = 2 SUSY algebra and its cohomological interpretation