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Nonlinear Self-Duality in Even Dimensions

We show that the Born-Infeld theory with n complex abelian gauge fields written in an auxiliary field formulation has a U(n,n) duality group. We conjecture the form of the Lagrangian obtained by eliminating the auxiliary fields and then introduce a new reality structure leading to a Born-Infeld theory with n real fields and an Sp(2n,R) duality symmetry. The real and complex constructions are extended to arbitrary even dimensions. The maximal noncompact duality group is U(n,n) for complex fields. For real fields the duality group is Sp(2n,R) if half of the dimension of space-time is even and O(n,n) if it is odd. We also discuss duality under the maximal compact subgroup, which is the self-duality group of the theory obtained by fixing the expectation value of a scalar field. Supersymmetric versions of self-dual theories in four dimensions are also discussed.