Abelian gauge theories on the lattice: θ-Terms and compact gauge theory with(out) monopoles

We discuss a particular lattice discretization of abelian gauge theories in arbitrary dimensions. The construction is based on gauging the center symmetry of a non-compact abelian gauge theory, which results in a 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 θ-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 θ term, as well as the gauge Higgs model in 3d with constrained monopole charges. These models are of importance in low dimensional anti-ferromagnets. Further we perform a natural construction of the θ-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)/ZN non-abelian gauge theories and the construction of discrete θ-terms on a cubic lattice.