Self-dual U(1) lattice field theory with a ?-term

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