We present N\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal{N} $$\end{document} = 2 superconformal U(1) duality-invariant models for an Abelian vector multiplet coupled to conformal supergravity. In a Minkowski background, such a nonlinear theory is expected to describe (the planar part of) the low-energy effective action for the N\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal{N} $$\end{document} = 4 SU(N) super-Yang-Mills (SYM) theory on its Coulomb branch where (i) the gauge group SU(N) is spontaneously broken to SU(N − 1) × U(1); and (ii) the dynamics is captured by a single N\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal{N} $$\end{document} = 2 vector multiplet associated with the U(1) factor of the unbroken group. Additionally, a local U(1) duality-invariant action generating the N\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal{N} $$\end{document} = 2 super-Weyl anomaly is proposed. By providing a new derivation of the recently constructed U(1) duality-invariant N\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal{N} $$\end{document} = 1 superconformal electrodynamics, we introduce its SL(2, ℝ) duality-invariant coupling to the dilaton-axion multiplet.
