Lattice Clifford fractons and their Chern-Simons-like theory

We use Dirac matrix representations of the Clifford algebra to build fracton models on lattice and their effective Chern-Simons-like theory. As an example we fractons in odd $D$ spatial dimensions $(D+1)$ The model possesses anti-symmetric $K$ resembling that hierarchical quantum Hall states. gauge charges are conserved sub-dimensional manifolds which ensures fractonic behavior. construction extends any built from commuting projectors with tensor products spin-$1/2$ degrees freedom at sites.