Abstract
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.
Highlights
Supersymmetry (SUSY), a symmetry that unifies spacetime transformations and internal gauge symmetries, combining bosons and fermions, presents a curious paradox: On the one hand, there is a wide consensus among theoretical –and even experimental– physicists that this unification must exist and be reflected in the particle spectrum of the standard model [1, 2]
In SUSY there is no approximate degeneracy to be explained by the symmetry; instead, there seems to be a need to explain the complete absence of a symmetry for which there are compelling mathematical arguments
Including s = 1/2 fermions in the superconnection requires the introduction of a metric structure, and the closure of the SUSY algebra requires the Lorentz group, which brings in the spin connection ωab
Summary
Supersymmetry (SUSY), a symmetry that unifies spacetime transformations and internal gauge symmetries, combining bosons and fermions, presents a curious paradox: On the one hand, there is a wide consensus among theoretical –and even experimental– physicists that this unification must exist and be reflected in the particle spectrum of the standard model [1, 2]. The second point of departure from the standard global (rigid) SUSY is that we assume the fundamental fields in an adjoint representation, as parts of a connection one-form for a superalgebra, and not in a vector representation of the supergroup In this approach, bosons and fermions are parts of the same connection, an idea that has been exploited since the mid-70s to construct supergravities [11], and was extended to construct Chern-Simons (CS) gauge theories with local off-shell SUSY including gravity for all odd dimensions [12, 13, 14, 15]. This allows reading the supersymmetry transformations as defined on the tangent space, which for all practical purposes can be taken as Minkowski spacetime This approach gives rise to a scenario where, as in the Standard Model, bosons are interaction carriers described by massless connection fields in the adjoint representation of the gauge algebra, while fermions are vectors under the gauge group (sections in the gauge bundle) and their currents are sources for the bosonic fields. These two ingredients make the incorporation of gravity practically unavoidable
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