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Starting from a recently proposed linear formulation in terms of auxiliaryfields, we study $n$-field generalizations of Born and Born-Infeld theories. Inthis description the Lagrangian is quadratic in the vector field strengths andthe symmetry properties (including the characteristic self-duality) of thecorresponding non-linear theory are manifest as on-shell duality symmetries anddepend on the choice of the (homogeneous) manifold spanned by the auxiliaryscalar fields and the symplectic frame. By suitably choosing these definingproperties of the quadratic Lagrangian, we are able to reproduce some knownmulti-field Born-Infeld theories and to derive new non-linear models, such asthe $n$-field Born theory. We also discuss non-Abelian generalizations of thesetheories obtained by choosing the vector fields in the adjoint representationof an off-shell compact global symmetry group $K$ and replacing them bynon-Abelian, $K$-covariant field strengths, thus promoting $K$ to a gaugegroup.

On Multifield Born and Born-Infeld Theories and their non-Abelian Generalizations