We present new models of non-linear electromagnetism which satisfy the Noether-Gaillard-Zumino current conservation and are, therefore, self-dual. The new models differ from the Born-Infeld-type models in that they deform the Maxwell theory starting
with terms like $lambda (partial F)^{4}$. We provide a recursive algorithm to find all higher order terms in the action of the form $lambda^{n} partial ^{4n} F^{2n+2} $, which are necessary for the U(1) duality current conservation. We use one of these models to find a self-dual completion of the $lambda (partial F)^{4}$ correction to the open string action. We discuss the implication of these findings for the issue of UV finiteness of ${cal N}=8$ supergravity.
We find the most general supersymmetric solutions of ungauged N=1,d=4 supergravity coupled to an arbitrary number of vector and chiral supermultiplets, which turn out to be essentially pp-waves and strings. We also introduce magnetic 1-forms and thei
r supersymmetry transformations and 2-forms associated to the isometries of the scalar manifold and their supersymmetry transformations. Only the latter can couple to BPS objects (strings), in agreement with our results.
