We propose a bosonic ${\mathrm{U}}^{\kappa }(1)$ 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, $S=\int d^{3}x(K_{IJ}/4\pi )A_{I}d{A}_J$. 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.

中文翻译：

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CompactUk（1）Chern-Simons理论作为具有精确离散1对称性的局部Bosonic晶格模型

我们建议 ${\mathrm{ü}}^{\kappa }（\mathrm{1个}）$三维时空晶格上的转子模型。通过包括麦克斯韦项，我们表明可以通过半经典方法可靠地获得我们模型的低能特性。这些性质与Chern-Simons场论的性质相同，$\mathrm{小号}=\int d^{3}X（{ķ}_{\mathrm{一世}Ĵ}/4\pi ）{\mathrm{一种}}_{\mathrm{一世}}d{\mathrm{一种}}_{Ĵ}$。我们要求每个链接上的晶格变量必须紧凑（即，取圆上的值），从而强制量化$ķ$矩阵是具有偶数对角线的对称整数矩阵。我们的晶格模型还具有精确的1对称性，这在Chern-Simons场论中引起了1形式的对称性。特别地，按照预期的方式，这些1对称中的某些是异常的（即非现场的）。可以通过边界破坏那些晶格1对称性来探测异常。