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Superconformal duality-invariant models and $\mathcal{N} = 4$ SYM effective action

We present $\mathcal{N}=2$ superconformal $\mathsf{U}(1)$ 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 $\mathcal{N}=4$ $\mathsf{SU}(N)$ super-Yang-Mills (SYM) theory on its Coulomb branch where (i) the gauge group $\mathsf{SU}(N)$ is spontaneously broken to $\mathsf{SU}(N-1) \times \mathsf{U}(1)$; and (ii) the dynamics is captured by a single $\mathcal{N}=2$ vector multiplet associated with the $\mathsf{U}(1)$ factor of the unbroken group. Additionally, a local $\mathsf{U}(1)$ duality-invariant action generating the $\mathcal{N}=2$ super-Weyl anomaly is proposed. By providing a new derivation of the recently constructed $\mathsf{U}(1)$ duality-invariant $\mathcal{N}=1$ superconformal electrodynamics, we introduce its $\mathsf{SL}(2,{\mathbb R})$ duality-invariant coupling to the dilaton-axion multiplet.