Nonlinear self-duality for arbitrary spin, superspin, and supersymmetry type

We review the general formalism of duality rotations for $\cal N$-extended (super)conformal gauge multiplets of arbitrary (super)spin in four dimensions, with ${\cal N} \geq 0$. Self-dual models for a vector field (${\cal N}=0$) and for ${\cal N}=1$ and ${\cal N}=2$ vector supermultiplets are naturally formulated on general (super)gravity backgrounds. For all other (super)spin values, the corresponding self-dual systems are realised on arbitrary conformally flat backgrounds. Every $\mathsf{U}(1)$ duality-invariant model is demonstrated to be self-dual with respect to a Legendre transformation. Methods are described to generate such self-dual models including superconformal ones. We show that every model for self-dual nonlinear electrodynamics admits a higher-spin extension. Throughout the review, we make use of the formalism of conformal (super)space, that is the geometric setting to describe the gauge theory of the (super)conformal group.