Abstract
We construct a gauge theory based in the supergroup G = SU(2, 2|2) that generalizes MacDowell-Mansouri supergravity. This is done introducing an extended notion of Hodge operator in the form of an outer automorphism of su(2, 2|2)-valued 2-form tensors. The model closely resembles a Yang-Mills theory — including the action principle, equations of motion and gauge transformations — which avoids the use of the otherwise complicated component formalism. The theory enjoys H = SO(3, 1) × ℝ × U(1) × SU(2) off-shell symmetry whilst the broken symmetries G/H, translation-type symmetries and supersymmetry, can be recovered on surface of integrability conditions of the equations of motion, for which it suffices the Rarita-Schwinger equation and torsion-like constraints to hold. Using the matter ansatz —projecting the 1 ⊗ 1/2 reducible representation into the spin-1/2 irreducible sector — we obtain (chiral) fermion models with gauge and gravity interactions.
Highlights
Besides their apparent dissimilar structure, these field theories have another important difference
In order to enforce local translation invariance of the action, it is necessary to appeal to the so-called torsion constraint, which is a consequence of the field equations and it is referred to as an on-shell symmetry
Symmetries that are realized on the surface of the field equations and which belong to certain Liegroup that contains off-shell symmetries are often referred to as “on-shell symmetries”
Summary
We present the general scheme of our model without technical details This theory will be built from a Lie (super)algebra-valued connection one-form that contains all the fields appearing in the action, as in a standard gauge theory. It turns out that Υ = 0 is an integrability condition for the equation of motion Υ = 0 and, at the same time, an indicator of whether the parameter λM generates a symmetry or not. Since [h , f] is a subset of f the gauge fields valued in the algebra elements f transform as vectors under the endomorphisms generated by the unbroken symmetry algebra h. Which is in the class of (2.11) for φ = 0 and φ = 1 Both local supersymmetry and transvection invariance are conditional symmetries of N = 1 supergravity.
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