It is postulated that quantum gravity is a sum over causal structures coupled to matter via scale evolution. Quantized causal structures can be described by studying simple matrix models where matrices are replaced by an algebra of quantum mechanical
observables. In particular, previous studies constructed quantum gravity models by quantizing the moduli of Laplace, weight and defining-function operators on Fefferman-Graham ambient spaces. The algebra of these operators underlies conformal geometries. We extend those results to include fermions by taking an osp(1|2) Dirac square root of these algebras. The theory is a simple, Grassmann, two-matrix model. Its quantum action is a Chern-Simons theory whose differential is a first-quantized, quantum mechanical BRST operator. The theory is a basic ingredient for building fundamental theories of physical observables.
We present the BRST cohomologies of a class of constraint (super) Lie algebras as detour complexes. By giving physical interpretations to the components of detour complexes as gauge invariances, Bianchi identities and equations of motion we obtain a
large class of new gauge theories. The pivotal new machinery is a treatment of the ghost Hilbert space designed to manifest the detour structure. Along with general results, we give details for three of these theories which correspond to gauge invariant spinning particle models of totally symmetric, antisymmetric and Kahler antisymmetric forms. In particular, we give details of our recent announcement of a (p,q)-form Kahler electromagnetism. We also discuss how our results generalize to other special geometries.
