Topological aspects of $4$D Abelian lattice gauge theories with the $\theta$ parameter

Abstract

We study a four-dimensional $U(1)$ gauge theory with the $\theta$ angle, which was originally proposed by Cardy and Rabinovici. It is known that the model has the rich phase diagram thanks to the presence of both electrically and magnetically charged particles. We discuss the topological nature of the oblique confinement phase of the model at $\theta =\pi$, and show how its appearance can be consistent with the anomaly constraint. We also construct the $SL(2,\mathbb{Z})$ self-dual theory out of the Cardy-Rabinovici model by gauging a part of its one-form symmetry. This self-duality has a mixed 't Hooft anomaly with gravity, and its implications on the phase diagram is uncovered. As the model shares the same global symmetry and 't Hooft anomaly with those of $SU(N)$ Yang-Mills theory, studying its topological aspects would provide us more hints to explore possible dynamics of non-Abelian gauge theories with nonzero $\theta$ angles.

33 pages, 3 figures (v2) refs added, some discussions added

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• Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality
• Obstructions to Gapped Phases from Non-Invertible Symmetries
• Investigating a (3+1)D Topological $\theta$-Term in the Hamiltonian Formulation of Lattice Gauge Theories for Quantum and Classical Simulations
• Theory of oblique topological insulators
• Higher Berry Phase of Fermions and Index Theorem
• Ising model as a $U(1)$ Lattice Gauge Theory with a $\theta$-term
• Study of gapped phases of 4d gauge theories using temporal gauging of the ${\mathbb{Z}}_N$ 1-form symmetry
• New conformal field theories by gauging electric and magnetic currents
• Intertwined order of generalized global symmetries
• Anomaly-induced edge currents in hydrodynamics with parity anomaly
• Generalized symmetry enriched criticality in (3+1)d