Self-dual U(1) lattice field theory with a θ-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 θ-term, such theories are ultra-local. We extend the discussion to U(1) gauge theories with θ-terms, such that θ 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 Lüscher 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 θ-angle by 2π together generate an SL(2, ℤ) duality group of transformations, just like in the continuum. We finally illustrate how the SL(2, ℤ) duality may be used to explore theories at finite θ without a sign problem.