Compact Uk(1) Chern-Simons Theory as a Local Bosonic Lattice Model with Exact Discrete 1-Symmetries

We propose a bosonic U-k(1) rotor model on a three dimensional spacetime lattice. With the inclusion of a Maxwell term, we show that the low-energy properties of our model can be obtained reliably via a semiclassical approach. Those properties are the same as that of the Chern-Simons field theory, S = integral d(3)x(K-IJ/4 pi)A(I)dA(J). We require the lattice variables on each link to be compact (i.e., take values on circles), which enforces the quantization of the K matrix as a symmetric integer matrix with even diagonals. Our lattice model also has exact 1-symmetries, which gives rise to the 1-form symmetry in the Chern-Simons field theory. 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 lattice 1-symmetries by the boundaries.

Automated summary

In this study, a bosonic U-k(1) rotor model with a Maxwell term on a three-dimensional spacetime lattice is proposed. By using a semiclassical approach, the low-energy properties of the model are shown to be equivalent to the Chern-Simons field theory. The lattice variables on each link are required to be compact, enforcing the quantization of the K matrix as a symmetric integer matrix with even diagonals. Additionally, the lattice model exhibits exact 1-symmetries, some of which are anomalous and can be probed by breaking the symmetries at the boundaries.