Lattice Chern-Simons-Maxwell Theory and its Chirality

We define and solve the $\text{U(1)}$ Chern-Simons-Maxwell theory on spacetime lattice, with an emphasis on the chirality of the theory. Realizing Chern-Simons theory on lattice has been a problem of interest for decades, and over the years it has gradually become clear that there are two key points: 1) Some non-topological term, such as a Maxwell term, is necessary -- this is true even in the continuum, but more manifestly on the lattice; 2) the $\text{U(1)}$ gauge field should be implemented in the Villainized form to retain its topological properties. Putting the two ideas together seriously, we show all interesting properties of a chiral Chern-Simons theory are reproduced in an explicitly regularized manner on the lattice. These include the bosonic and fermionic level quantization, the bulk and chiral edge spectrum, the Wilson loop flux attachment (with point-split framing or geometric framing depending on the Maxwell coupling), the Wilson loop spin, the ground state degeneracy, and, most non-trivially, the chiral gravitational anomaly.