Topics on the geometry of D-brane charges and Ramond-Ramond fields - Part II: Low-degree RR fields

We discuss the geometry of the Ramond-Ramond fields of low degree via the language of abelian p-gerbes, working in type II superstring theory with vanishing H-flux. A Dp-brane is a source for the Ramond-Ramond field strength G_(p+2), which can be thought of as the curvature of a connection on a p-gerbe with U(1)-band. The highest degree forms of the connection are the local Ramond-Ramond potentials C_(p+1). This picture is clear for p >= 0, a 0-gerbe being a line bundle. Actually, one can encounter also lower degree Ramond-Ramond field strengths: G_1 in IIB superstring theory, and G_0 in IIA superstring theory. Although the geometrical nature of these objects is simple in itself, it is interesting to explicitly clarify the analogy with the geometry of the higher degree fields and potentials, introducing the notions of (-1)-gerbes with connection and (-2)-gerbes. We define these gerbes and we show that they are better described by a variant of ordinary Cech cohomology.