A lattice U(1) Chern-Simons theory via lattice Deligne-Beilinson cohomology

A bstract We define Deligne-Beilinson (DB) cohomology on a cubic lattice and use it to formulate and analyze lattice U(1) Chern-Simons theory at even levels. The continuum DB cohomology provides a refined mathematical framework for continuum U(1) connections constructed in a patchwise manner. The lattice DB cohomology we construct retains many essential properties of the continuum DB cohomology and naturally incorporates a notion of self-linking number. The lattice U(1) Chern-Simons action formulated using the lattice DB cohomology is expressed as a simple quadratic form via the star product, which naturally exhibits level quantization. Framed Wilson lines respecting staggered symmetry are defined in a gauge-invariant manner, and their expectation values are shown to be given by the self-linking number, as follows from completing the square. Using the lattice Hodge decomposition, we explicitly characterize the DB cohomology on a three-dimensional cubic toroidal lattice and write the gauge-fixed path integral for the lattice Chern-Simons theory explicitly as a finite-dimensional integral. To regulate divergences in the lattice Chern-Simons path integral arising from the staggered symmetry, we introduce a small Maxwell term. Once a Wilson line or an external field is fixed, the error introduced by this Maxwell regularization can be bounded at linear order in the small Maxwell coupling.