Lattice realization of compact $U(1)$ Chern-Simons theory with exact 1-symmetries

We propose a bosonic $U(1)$ rotor model on a three dimensional space-time lattice. With the inclusion of a Maxwell term, we show that the low-energy semi-classical behavior of our model is consistent with the Chern-Simons field theory, $S= \int \text{d}^3 x\ \frac{K_{IJ}}{4\pi} A_I \text{d} A_J$ at low energies. We require that the action be periodic in the lattice variables, which enforces the quantization of the $K$-matrix as a symmetric integer matrix with even diagonals. We also show that our lattice model has the exact $1$-symmetries. In particular, some of those 1-symmetries are anomalous (i.e. non-on-site) in the expected way. The anomaly can be probed via the breaking of those $1$-symmetries by the boundaries of space-time.