D-brane couplings and Generalised Geometry

The goal of this paper is to re-examine D-brane Ramond-Ramond field couplings in the presence of a B-field. We will argue that the generalised geometry induced on the world volume by the B-field results in an important but subtle change on the coupling. In order to explain this, we use the language of differential K-theory. The expression determining the coupling is then seen to be a consequence of the Riemann-Roch theorem. Our key assertion is that the appropriate Riemann-Roch theorem changes in the presence of the B-field. In particular, the A-hat forms appearing in the theorem are now constructed using the torsionful Levi-Civita connection associated to the generalised geometry. As we shall see, the resulting expression not only agrees with recently discovered local couplings on the D-brane worldvolume involving RR fields and derivatives of the B-field, but also makes the coupling manifestly T-duality invariant.