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.

我们构建了一个基于超群G = SU(2 , 2 | 2)的规范理论，它概括了 MacDowell-Mansouri 超引力。这是通过以su (2 , 2 | 2)-valued 2-form 张量的外部自同构的形式引入霍奇算子的扩展概念来完成的。该模型非常类似于 Yang-Mills 理论——包括作用原理、运动方程和规范变换——避免使用其他复杂的组件形式主义。该理论享有H = SO(3, 1) × ℝ × U(1) × SU(2) 壳外对称性，而破坏对称性G/H，平移型对称性和超对称性，可以在运动方程的可积条件的表面上恢复，对于它，它足以满足 Rarita-Schwinger 方程和类扭转约束。使用物质ansatz——将 1 ⊗ 1/2 可约表示投影到自旋 1/2 不可约扇区——我们获得了具有规范和重力相互作用的（手性）费米子模型。