Narrow Results

In this paper, we demonstrate the existence of a set of novel discrete symmetry transformations in the case of an interacting ${𝒩}=2$ supersymmetric quantum mechanical model of a system of an electron moving on a sphere in the background of a magnetic monopole and establish its interpretation in the language of differential geometry. These discrete symmetries are, over and above, the usual three continuous symmetries of the theory which together provide the physical realizations of the de Rham cohomological operators of differential geometry. We derive the nilpotent ${𝒩}=2$ SUSY transformations by exploiting our idea of supervariable approach and provide geometrical meaning to these transformations in the language of Grassmannian translational generators on a $(1,2)$-dimensional supermanifold on which our ${𝒩}=2$ SUSY quantum mechanical model is generalized. We express the conserved supercharges and the invariance of the Lagrangian in terms of the supervariables (obtained after the imposition of the SUSY invariant restrictions) and provide the geometrical meaning to (i) the nilpotency property of the ${𝒩}=2$ supercharges, and (ii) the SUSY invariance of the Lagrangian of our ${𝒩}=2$ SUSY theory.

We discuss a set of novel discrete symmetry transformations of the ${𝒩}=4$ supersymmetric quantum mechanical model of a charged particle moving on a sphere in the background of Dirac magnetic monopole. The usual five continuous symmetries (and their conserved Noether charges) and two discrete symmetries together provide the physical realizations of the de Rham cohomological operators of differential geometry. We have also exploited the supervariable approach to derive the nilpotent ${𝒩}=4$ SUSY transformations and provided the geometrical interpretation in the language of translational generators along the Grassmannian directions ${𝜃}^{\alpha }$ and ${\overline{𝜃}}^{\alpha }$ onto $(1,4)$-dimensional supermanifold.

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 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 ${𝒩}=2$ supercharges of this system do not absolutely anticommute.

We derive the nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations for the system of a toy model of Hodge theory (i.e. a rigid rotor) by exploiting the (anti-)BRST and (anti-)co-BRST invariant restrictions on the (anti-)chiral supervariables that are defined on the appropriately chosen $(1,1)$-dimensional super-submanifolds of the general$(1,2)$-dimensional supermanifold on which our system of a one $(0+1)$-dimensional (1D) toy model of Hodge theory is considered within the framework of the augmented version of the (anti-)chiral supervariable approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism. The general $(1,2)$-dimensional supermanifold is parametrized by the superspace coordinates $(t,𝜃,\overline{𝜃})$, where $t$ is the bosonic evolution parameter and $(𝜃,\overline{𝜃})$ are the Grassmannian variables which obey the standard fermionic relationships: ${𝜃}^2={\overline{𝜃}}^2=0$, $𝜃\overline{𝜃}+\overline{𝜃}𝜃=0$. We provide the geometrical interpretations for the symmetry invariance and nilpotency property. Furthermore, in our present endeavor, we establish the property of absolute anticommutativity of the conserved fermionic charges which is a completely novel and surprising observation in our present endeavor where we have considered only the (anti-)chiral supervariables. To corroborate the novelty of the above observation, we apply this ACSA to an ${𝒩}=2$ SUSY quantum mechanical (QM) system of a free particle and show that the ${𝒩}=2$ SUSY conserved and nilpotent charges do not absolutely anticommute.