Exact topological quantum order in D=3 and beyond:: Branyons and brane-net condensates

We construct an exactly solvable Hamiltonian acting on a three-dimensional lattice of spin-1/2 systems that exhibits topological quantum order. The ground state is a string net and a membrane-net condensate. Excitations appear in the form of quasiparticles and fluxes, as the boundaries of strings and membranes, respectively. The degeneracy of the ground state depends on the homology of the 3-manifold. We generalize the system to D >= 4, where different topological phases may occur. The whole construction is based on certain special complexes that we call colexes.