Gauge-invariant couplings of noncommutative branes to Ramond-Ramond backgrounds

We derive the couplings of noncommutative D-branes to spatially varying Ram ond-Ramond fields, extending our earlier results in hep-th/0009101. These c ouplings are expressed in terms of *(n) products of operators involving ope n Wilson lines. Equivalence of the noncommutative to the commutative coupli ngs implies interesting identities as well as an expression for the Seiberg -Witten map that was previously conjectured. We generalise our couplings to include transverse scalars, thereby obtaining a Seiberg-Witten map relatin g commutative and noncommutative descriptions of these scalars. RR coupling s for unstable non-BPS branes are also proposed.