2d Cardy-Rabinovici model with the modified Villain lattice: exact dualities and symmetries

A bstract The Cardy-Rabinovici model is a toy model of the lattice U(1) gauge theories to study various oblique confinement states associated with the nonzero θ angles. We reformulate the 2d version of this model using the modified Villain lattice formalism, and we establish the exact θ periodicity for the Witten effect and the strong-weak duality at the finite lattice spacings. We then study the phase structure of this model based on the duality, symmetry and anomaly, and the perturbative renormalization group.