Abstract
We update and detail the formulation of the duality-invariant systems of $N$ interacting Abelian gauge fields with $N$ auxiliary bispinor fields added. In this setting, the self-duality amounts to $U(N)$ invariance of the nonlinear interaction of the auxiliary fields. The $U(N)$ self-dual Lagrangians arise after solving the nonlinear equations of motion for the auxiliary fields. We also elaborate on a new extended version of the bispinor field formulation involving some additional scalar auxiliary fields and study $U(N)$ invariant interactions with derivatives of the auxiliary bispinor fields. Such interactions generate higher-derivative $U(N)$ self-dual theories.
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