Causal Space-Times on a Null Lattice

I investigate a discrete model of quantum gravity on a causal null-lattice with \SLC structure group. The description is geometric and foliates in a causal and physically transparent manner. The general observables of this model are constructed from local Lorentz symmetry considerations only. For smooth configurations, the local lattice actions reduce to the Hilbert-Palatini action, a cosmological term and the three topological terms of dimension four of Pontyagin, Euler and Nieh-Yan. Consistency conditions for a topologically hypercubic complex with null 4-simplexes are derived and a topological lattice theory that enforces these non-local constraints is constructed. The lattice integration measure is derived from an \SLC-invariant integration measure by localization of the non-local structure group. This measure is unique up to a density that depends on the local 4-volume. It can be expressed in terms of manifestly coordinate invariant geometrical quantities. The density provides an invariant regularization of the lattice integration measure that suppresses configurations with small local 4-volumes. Amplitudes conditioned on geodesic distances between local observables have a physical interpretation and may have a smooth ultraviolet limit. Numerical studies on small lattices in the unphysical strong coupling regime of large imaginary cosmological constant suggest that this model of triangulated causal manifolds is finite. Two topologically different triangulations of space-time are discussed: a single, causally connected universe and a duoverse with two causally disjoint connected components. In the duoverse, two hypercubic sublattices are causally disjoint but the local curvature depends on fields of both sublattices. This may simulate effects of dark matter in the continuum limit.