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

TL;DR

This paper explores the geometric structure of low-degree Ramond-Ramond fields in type II superstring theory using abelian p-gerbes, clarifying their relation to higher and lower degree fields and introducing new gerbe concepts.

Contribution

It introduces the notions of (-1)- and (-2)-gerbes with connection and shows they are effectively described by a variant of Cech cohomology, extending the geometric framework.

Findings

Clarifies the geometry of low-degree RR fields using p-gerbes.

Defines and characterizes (-1)- and (-2)-gerbes with connection.

Shows these gerbes are described by a variant of Cech cohomology.

Abstract

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.