$$ \mathcal{N} $$ = 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.

Gauging the superconformal group with a graded dual operator

“…The simplest example is the Georgy-Glashow SU(5) model [9,10]. An SU (5) model with softly broken supersymmetry was also proposed in [11] and, in order to solve the proton decay and the doublet-triplet splitting problems of the Georgy-Glashow SU(5) model, the flipped SU(5) model was proposed in [12]. Some years later, it was introduced the supersymmetric flipped SU (5) model that produce hierarchical neutrino masses [13][14][15][16].…”

“…We formulate the model purely in terms of a SU(2, 2|N ) connection. Following the procedure of [4,5,17], we gauge and break some of the SU(2, 2|N ) symmetries explicitly in the action. The generators of the ordinary conformal group, J a , K a , J ab and D, together with internal generators T I and Z of SU(N ) and U(1) respectively, form the bosonic algebra.…”

“…The symmetries generated by the S-grading odd generators, and the supersymmetries are going to be on-shell symmetries. These on-shell symmetries were discussed in [5], where for a supergravity model with…”

“…The gauge symmetries associated to the S-grading even generators are going to be genuine gauge symmetries, while the transformations associated to the S-grading odd generators are going to be conditional symmetries [5]. Under a gauge transformation of the gauge potential, δA = D A G, where…”

“…Such operator can be found naturally in the superconformal algebra. These algebras have been used to construct related theories of matter [4] and extended MacDowell-Mansouri supergravities [5].…”