Dilaton-Axion Symmetry

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.