SL(2, R) invariance of non-linear electrodynamics coupled to an axion and a dilaton

The most general Lagrangian for non-linear electrodynamics coupled to an axion a and a dilaton φ with SL(2, R) invariant equations of motion is −12 (∇φ)2 −12e2φ (∇a)2 +14aFμν ★ Fμν + Linv(gμν, e−12φ Fϱσ) where Linv(gμν, Fϱσ) is a Lagrangian whose equations of motion are invariant under electric-magnetic duality rotations. In particular there is a unique generalization of Born-Infeld theory admitting SL(2, R) invariant equations of motion.