Abstract
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.
Highlights
An important aspect of many quantum field theories is the fact that they can be augmented with topological θ-terms which can cause dramatic changes in the low-energy physics
The corresponding physical phenomena are nonperturbative in nature such that genuinely non-perturbative approaches are necessary to study these systems. Such terms are well known in the context of non-abelian gauge theories, as they are the source of the famous strong CP problem
As we summarize in Appendix A, Eq (17) defines an exterior derivative operator on link fields, i.e., Fp =p
Summary
An important aspect of many quantum field theories is the fact that they can be augmented with topological θ-terms which can cause dramatic changes in the low-energy physics. The constraint is imposed by the presence of ZN lattice symmetries, which are mapped into the conservation of the magnetic flux modulo 2πN in the corresponding (2 + 1)d effective quantum field theory. The lattice discretization of the theory has to be such that it admits a solution of the complex action problem if the theory is to be useful for simulations This can sometimes be achieved by a transformation to dual variables which are worldlines for matter fields and worldsheets for the gauge degrees of freedom. This approach will allow us to couple magnetic matter, and implement an electric-magnetic duality even in the presence of electric and magnetic matter It serves as a natural way to define θ-terms in both 2d and 4d lattice gauge theories.
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