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Lattice BF Theory, Dumbbells, and Composite Fermions

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.