Nonlinear Self-Duality for Arbitrary Spin, Superspin, and Supersymmetry Types

We review the general formalism of duality rotations for N-extended (super)conformal gauge multiplets of arbitrary (super)spin in four dimensions, with N≥0. Self-dual models for a vector field (N=0) and for N=1 and 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 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 this review, we make use of the formalism of conformal (super)space, which is the geometric setting to describe the gauge theory of the (super)conformal group.