  We present an Sp(2n,R) duality invariant Born-Infeld U(1)^2n gauge theory
with scalar fields. To implement this duality we had to introduce complex gauge
fields and as a result the rank of the duality group is only half as large as
that of the corresponding Maxwell gauge theory with the same number of gauge
fields. The latter is self-dual under Sp(4n,R), the largest allowed duality
group. A special case appears for n=1 when one can also write an SL(2,R)
duality invariant Born-Infeld theory with a real gauge field. We also describe
the supersymmetric version of the above construction.
