  A gauge theory for a superalgebra that includes an internal gauge (G) and
local Lorentz algebras, and that could describe the low energy particle
phenomenology is constructed. These two symmetries are connected by fermionic
supercharges. The system includes an internal gauge connection 1-form $A$, a
spin-1/2 Dirac spinor $\psi$, the Lorentz connection $\omega$, and the vielbein
$e$. The connection one-form is in the adjoint representation of G, while
$\psi$ is in the fundamental. In contrast to standard supergravity, the metric
is not a fundamental field and is in the center of the superalgebra: it is not
only invariant under the internal gauge group and under Lorentz
transformations, but is also invariant under supersymmetry. The features of
this theory that mark the difference with standard supersymmetry are: A) The
number of fermionic and bosonic states is not necessarily the same; B) There
are no superpartners with equal mass, "bosoninos", sleptons and squarks are
absent; C) Although this supersymmetry originates in a local gauge theory and
gravity is included, there is no gravitino; D) Fermions acquire mass from their
coupling to the background or from self-couplings, while bosons remain
massless. In odd dimensions, the Chern-Simons form provides an action that is
quasi-invariant under the entire superalgebra. In even dimensions, the
Yang-Mills form $ < F *F>$ is the only natural option, and the symmetry breaks
down to [G x SO(1,D-1)]. In 4D, the construction follows the Townsend - Mac
Dowell-Mansouri approach. Due to the absence of osp(4|2)-invariant traces in
four dimensions, the resulting Lagrangian is only invariant under [U(1) x
SO(3,1)], and includes a Nambu--Jona-Lasinio term. In this case, the Lagrangian
depends on a single dimensionful parameter that fixes Newton's constant, the
cosmological constant and the NJL coupling.
