Abstract

It was established long ago that SO(2) electric-magnetic duality is an {\em off-shell} symmetry of the free Maxwell theory, i.e., that it leaves invariant the action and not just the equations of motion. We review here that analysis and extend it to the Maxwell field coupled to scalar fields defined on the SL(2,R)/SO(2) coset space, showing that SL(2,R) is in that case an {\em off-shell} symmetry. We also show how the result can be generalized to many Maxwell fields and Sp(2n,R) duality symmetry - or a subgroup of it, recovering in particular the case of maximal supergravity in four dimensions with E7,7 symmetry. We finally indicate further possible extensions to twisted self-duality equations for p-forms, including Chern-Simons terms and Pauli couplings, as well as linearized gravity, which will be treated in depth elsewhere.