Abelian gauge theories on the lattice: $\theta$-terms and compact gauge theory with(out) monopoles.
We discuss a particular lattice discretization of the abelian gauge theories on the lattice in arbitrary dimension. The construction is based on gauging the center symmetry of a non-compact abelian gauge theory, which results in the Villain type action. We show that this construction has several benefits over the conventional $U(1)$ lattice gauge theory construction, such as electric magnetic duality, natural coupling of the theory to magnetically charged matter in four dimensions, complete control over the monopoles and their charges in three dimensions and a natural $\theta$-term in two dimensions. Moreover we show that for bosonic matter our formulation can be mapped to a worldline/worldsheet representation where the complex action problem is solved. We illustrate our construction by explicit dualizations of the $CP(N-1)$ and the gauge Higgs model in $2d$ with a $\theta$ term, as well as the gauge Higgs model in $3d$ with constrained monopole charges. These models are of importance in low dimensional antiferromagnets. Further we perform a natural construction of the $\theta$-term in four dimensional gauge theories, and demonstrate the Witten effect which endows magnetic matter with a fractional electric charge. We extend this discussion to $PSU(N)=SU(N)/\mathbb Z_N$ non-abelian gauge theories and the construction of the discrete $\theta$-terms on a cubic lattice.
Publisher URL: http://arxiv.org/abs/1901.02637
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