  In a previous work, we have been able to settle Jackiw's et al. chiral gauge
theory for Dirac fermions in graphene in an N=1 supersymmetric framework, using
a tau3-QED prescription, defined by means of a single pair of gauge charged
superfields, but without preserving a global phase symmetry associated to the
electric charge. In the present work, we propose another N=1-generalisation
which indeed preserves this symmetry, namely, a straightforward extension built
upon a set of two pairs of (chiral) gauge-charged superfields plus an extra
pair of electrically neutral superfields. We then further proceed to establish,
via a dimensional reduction procedure, an N=2 extension, allowing for the
identification of non-perturbative features, as we put forward Bogomol'nyi
equations and obtain vortex-like solutions saturating a topologically
non-trivial bound. Remarkably, the bosonic projection of the N=2 functional
space onto the saturated regime analysed herewith reveals to be free from extra
scalar degrees of freedom that would otherwise demand a phenomenological
interpretation. The investigation of Jackiw's model within an N=2 complex
superspace is also motivated by the assumption that an R-parity-like symmetry
could provide a route to incorporate the global phase-fermion number invariance
as an external-like symmetry of the theory, thus associating the electric
charge in graphene to the complex covariance (super-)space for the N=2-D=3
setup. We prove such a hypothesis to be realisable, as we build up the model
endowed with all the symmetries required to further extend Jackiw's chiral
gauge theory.
