  The heterotic string compactified on a six-torus is described by a low-energy
effective action consisting of N=4 supergravity coupled to N=4 super
Yang-Mills, a theory that was studied in detail many years ago. By explicitly
carrying out the dimensional reduction of the massless fields, we obtain the
bosonic sector of this theory. In the Abelian case the action is written with
manifest global $O(6,6+n)$ symmetry. A duality transformation that replaces the
antisymmetric tensor field by an axion brings it to a form in which the axion
and dilaton parametrize an $SL(2,R)/SO(2)$ coset, and the equations of motion
have $SL(2,R)$ symmetry. This symmetry, which combines Peccei--Quinn
translations with Montonen--Olive duality transformations, has been exploited
in several recent papers to construct black hole solutions carrying both
electric and magnetic charge. Our purpose is to explore whether, as various
authors have conjectured, an $SL(2,Z)$ subgroup could be an exact symmetry of
the full quantum string theory. If true, this would be of fundamental
importance, since this group transforms the dilaton nonlinearly and can relate
weak and strong coupling.
