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

Abstract

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.

arXiv comment4 pages + appendices

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Cited by in corpus (9)

• Modified Villain formulation of abelian Chern-Simons theory
• Emergent higher-symmetry protected topological orders in the confined phase of $U(1)$ gauge theory
• Exactly solvable models for U(1) symmetry-enriched topological phases
• 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems
• Deconfined criticality and bosonization duality in easy-plane Chern-Simons two-dimensional antiferromagnets
• Lattice Chern-Simons-Maxwell Theory and its Chirality
• Lattice BF Theory, Dumbbells, and Composite Fermions
• Bosonization duality in 2+1 dimensions and critical current correlation functions in Chern-Simons $U(1)\times U(1)$ Abelian Higgs model
• Abelian Chern-Simons Gauge Theory on The Lattice