Self-dual U(1) lattice field theory with a $\theta$-term

We study U(1) gauge theories with a modified Villain action. Such theories can naturally be coupled to electric and magnetic matter, and display exact electric-magnetic duality. In their simplest formulation without a $\theta$-term, such theories are ultra-local. We extend the discussion to U(1) gauge theories with $\theta$-terms, such that $\theta$ periodicity is exact for a free theory, and show that imposing electric-magnetic duality results in a local, but not ultra-local lattice action, which is reminiscent of the L\"uscher construction of axial-symmetry preserving fermions in 4d. We discuss the coupling to electric and magnetic matter as well as to dyons. For dyonic matter the electric-magnetic duality and shifts of the $\theta$-angle by $2\pi$ together generate an SL$(2,\mathbb Z)$ duality group of transformations, just like in the continuum. We finally illustrate how the SL$(2,\mathbb Z)$ duality may be used to explore theories at finite $\theta$ without a sign problem.