  We formulate $U(1)$ $bda$ Chern-Simons theory, which is also called BF
theory, on a lattice, adapting a method proposed by Kantor and Susskind for the
groups $\mathbb{R}$ and $\mathbb{Z}_N$. Our method applies to any finite or
infinite abelian group. We study the discrete symmetries and use the model to
provide a rigorous treatment of the composite fermion theory of the fractional
quantum Hall effect (FQHE), with no ambiguities relating to intersecting
Wilson/'t Hooft lines. We derive Jain's fractions, and one can also calculate
corrections to the mean field solution within this framework. We also
generalize the formalism to higher form gauge models in arbitrary dimension,
and suggest a possible non-Abelian extension.
