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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$(2,\mathbb Z)$ 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 $θ$ without a sign problem.