A bosonic lattice model realizes exact chiral symmetry and its anomaly in 3+1d, with the continuum limit a compact boson theory with axion-like coupling.
Brauner,Field theories with higher-group symmetry from composite currents,JHEP04 (2021) 045, arXiv:2012.00051 [hep-th]
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Lattice chiral symmetry from bosons in 3+1d
A bosonic lattice model realizes exact chiral symmetry and its anomaly in 3+1d, with the continuum limit a compact boson theory with axion-like coupling.