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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 themodel has the rich phase diagram thanks to the presence of both electricallyand magnetically charged particles. We discuss the topological nature of theoblique confinement phase of the model at $\theta=\pi$, and show how itsappearance can be consistent with the anomaly constraint. We also construct the$SL(2,\mathbb{Z})$ self-dual theory out of the Cardy-Rabinovici model bygauging a part of its one-form symmetry. This self-duality has a mixed 't Hooftanomaly 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 usmore hints to explore possible dynamics of non-Abelian gauge theories withnonzero $\theta$ angles.

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