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