Unconventional supersymmetry and its breaking

We present a gauge theory for a superalgebra that includes an internal gauge (G) and local Lorentz (so(1,D−1)) algebras. These two symmetries are connected by fermionic supercharges. The field content of the system includes a (non-)abelian gauge potential A, a spin-1/2 Dirac spinor ψ, the Lorentz connection ωab, and the vielbein eμa. The connection one-form A is in the adjoint representation of G, while ψ is in the fundamental. In contrast to standard supersymmetry and 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, G, and under Lorentz transformations, SO(1,D−1), but is also invariant under supersymmetry. The distinctive features of this theory that mark the difference with standard supersymmetries are: i) the number of fermionic and bosonic states is not necessarily the same; ii) there are no superpartners with equal mass; iii) although this supersymmetry originates in a local gauge theory and gravity is included, there is no gravitino; iv) fermions acquire mass from their coupling to the background or from higher order self-couplings, while bosons remain massless. In odd dimensions, the Chern–Simons (CS) form provides an action that is (quasi-)invariant under the entire superalgebra. In even dimensions, the Yang–Mills (YM) form is the only natural option and the symmetry breaks down to G⊗SO(1,D−1). In four dimensions, the construction follows the Townsend–Mac Dowell–Mansouri approach, starting with an osp(4|2)∼usp(2,2|1) connection. Due to the absence of osp(4|2)-invariant traces in four dimensions, the resulting Lagrangian is only invariant under u(1)⊕so(3,1), which includes a Nambu–Jona-Lasinio (NJL) term. In this case, the Lagrangian depends on a single dimensionful parameter that fixes Newton's constant, the cosmological constant and the NJL coupling.