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

Abstract

© 2021 American Physical Society. We propose a bosonic Uκ(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=∫d3x(KIJ/4π)AIdAJ. 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 areanomalous (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.