  Based on the superconformal algebra we construct a dual operator that
introduces a grading among bosonic generators independent of the boson/fermion
grading of the superalgebra. This dual operator allows us to construct an
action that is gauge invariant under the grading even bosonic generators. We
provide a self-dual notion based on the dual operator. We use the definition of
the dual operator to contruct a model with gauge invariance $SO(1,3)\times
SU(N) \times U(1) \subset SU(2,2|N)$. The choice of a graded dual operator
allows us to overcome technical difficulties of $U(N)$ unified theories based
on the superconformal group. The gravity action reproduces the Einstein-Hilbert
action in certain sector of the theory. The definition of the dual operator
allows us to include fermionic matter in the gauge connection in a geometric
manner. We give a summary with possible phenomenological parameters that are
included in the model.
