N $$ \mathcal{N} $$ = 2 extended MacDowell-Mansouri supergravity

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