By exploiting the Stueckelberg approach, we obtain a gauge theory for the two (1+1)-dimensional (2D) Proca theory and demonstrate that this theory is endowed with, in addition to the usual Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetries, the on-shell nilpotent (anti-)co-BRST symmetries, under which, the total gauge-fixing term remains invariant. The anticommutator of the BRST and co-BRST (as well as anti-BRST and anti-co-BRST) symmetries define a unique bosonic symmetry in the theory, under which, the ghost part of the Lagrangian density remains invariant. To establish connections of the above symmetries with the Hodge theory, we invoke a pseudo-scalar field in the theory. Ultimately, we demonstrate that the full theory provides a field theoretic example for the Hodge theory where the continuous symmetry transformations provide a physical realization of the de Rham cohomological operators and discrete symmetries of the theory lead to the physical realization of the Hodge duality operation of differential geometry. We also mention the physical implications and utility of our present investigation.

We exploit the standard tools and techniques of the augmented version of the Bonora-Tonin superfield formalism to derive the off-shell nilpotent and absolutely anticommuting (anti-)Becchi-Rouet-Stora-Tyutin (BRST) and (anti-)co-BRST symmetry transformations for the BRSTinvariant Lagrangian density of a self-dual bosonic system. In the derivation of the full set of the above symmetry transformations, we invoke the (dual-)horizontality conditions, and (anti-)BRST-and (anti-)co-BRST-invariant restrictions on the superfields that are defined on the (2, 2)-dimensional supermanifold. The latter is parameterized by the bosonic variable x μ (μ = 0, 1) and a pair of Grassmannian variables θ andθ (with θ 2 =θ 2 = 0 and θθ +θθ = 0). The dynamics of this system is such that, instead of the full (2, 2)-dimensional superspace coordinates (x μ , θ,θ), we require only the specific (1, 2)-dimensional super-subspace variables (t, θ,θ) for its description. This is a novel observation in the context of the superfield approach to the BRST formalism. The application of the dual-horizontality condition, in the derivation of a set of proper (anti-)co-BRST symmetries, is also one of the new ingredients of our present endeavor where we have exploited the augmented version of the superfield approach to the BRST formalism.

We exploit the strength of the superspace (SUSP) unitary operator to obtain the results of the application of the horizontality condition (HC) within the framework of augmented version of superfield formalism that is applied to the interacting systems of Abelian 1-form gauge theories where the U(1) Abelian 1-form gauge field couples to the Dirac and complex scalar fields in the physical four (3 + 1)-dimensions of spacetime. These interacting theories are generalized onto a (4, 2)-dimensional supermanifold that is parametrized by the four (3 + 1)-dimensional (4D) spacetime variables and a pair of Grassmannian variables. To derive the (anti-)BRST symmetries for the matter fields, we impose the gauge invariant restrictions (GIRs) on the superfields defined on the (4, 2)-dimensional supermanifold. We discuss various outcomes that emerge out from our knowledge of the SUSP unitary operator and its hermitian conjugate. The latter operator is derived without imposing any operation of hermitian conjugation on the parameters and fields of our theory from outside. This is an interesting observation in our present investigation.

Abstract:We consider the toy model of a rigid rotor as an example of the Hodge theory within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism and show that the internal symmetries of this theory lead to the derivation of canonical brackets amongst the creation and annihilation operators of the dynamical variables where the definition of the canonical conjugate momenta is not required. We invoke only the spin-statistics theorem, normal ordering and basic concepts of continuous symmetries (and their generators) to derive the canonical brackets for the model of a one (0 + 1)-dimensional (1D) rigid rotor without using the definition of the canonical conjugate momenta anywhere. Our present method of derivation of the basic brackets is conjectured to be true for a class of theories that provide a set of tractable physical examples for the Hodge theory.

We exploit the standard techniques of the supervariable approach to derive the nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for a toy model of the Hodge theory (i.e. a rigid rotor) and provide the geometrical meaning and interpretation to them. Furthermore, we also derive the nilpotent (anti-)co-BRST symmetry transformations for this theory within the framework of the above supervariable approach. We capture the (anti-)BRST and (anti-)co-BRST invariance of the Lagrangian of our present theory within the framework of augmented supervariable formalism. We also express the (anti-)BRST and (anti-)co-BRST charges in terms of the supervariables (obtained after the application of the (dual-)horizontality conditions and (anti-)BRST and (anti-)co-BRST invariant restrictions) to provide the geometrical interpretations for their nilpotency and anticommutativity properties. The application of the dual-horizontality condition and ensuing proper (i.e. nilpotent and absolutely anticommuting) fermionic (anti-)co-BRST symmetries are completely novel results in our present investigation.