  In our companion papers, we managed to derive a connection formulation of
Lorentzian General Relativity in D+1 dimensions with compact gauge group
SO(D+1) such that the connection is Poisson commuting, which implies that Loop
Quantum Gravity quantisation methods apply. We also provided the coupling to
standard matter. In this paper, we extend our methods to derive a connection
formulation of a large class of Lorentzian signature Supergravity theories, in
particular 11d SUGRA and 4d, N = 8 SUGRA, which was in fact the motivation to
consider higher dimensions. Starting from a Hamiltonian formulation in the time
gauge which yields a Spin(D) theory, a major challenge is to extend the
internal gauge group to Spin(D+1) in presence of the Rarita-Schwinger field.
This is non trivial because SUSY typically requires the Rarita-Schwinger field
to be a Majorana fermion for the Lorentzian Clifford algebra and Majorana
representations of the Clifford algebra are not available in the same spacetime
dimension for both Lorentzian and Euclidean signature. We resolve the arising
tension and provide a background independent representation of the non trivial
Dirac antibracket *-algebra for the Majorana field which significantly differs
from the analogous construction for Dirac fields already available in the
literature.
