Twisted differential K-characters and D-branes

TL;DR

This paper develops a detailed mathematical framework using twisted differential K-theory to describe D-branes in type II superstring backgrounds, providing a refined classification of D-brane charges and world-volumes beyond previous cohomological methods.

Contribution

It introduces new models of differential twisted K-theory that depend on the twisting cocycle, clarifies gauge theories on D-branes, and extends the classification to backgrounds with non-vanishing B-fields.

Findings

Complete characterization of D-brane world-volumes and charges within K-theory.

Construction of models of twisted differential K-theory depending on the twisting cocycle.

Extension of the classification of D-branes to backgrounds with non-zero B-field.

Abstract

We analyse in detail the language of partially non-abelian Deligne cohomology and of twisted differential K-theory, in order to describe the geometry of type II superstring backgrounds with D-branes. This description will also provide the opportunity to show some mathematical results of independent interest. In particular, we begin classifying the possible gauge theories on a D-brane or on a stack of D-branes using the intrinsic tool of long exact sequences. Afterwards, we recall how to construct two relevant models of differential twisted K-theory, paying particular attention to the dependence on the twisting cocycle within its cohomology class. In this way we will be able to define twisted K-homology and twisted Cheeger-Simons K-characters in the category of simply-connected manifolds, eliminating any unnatural dependence on the cocycle. The ambiguity left for non simply-connected manifolds will naturally correspond to the ambiguity in the gauge theory, following the previous classification. This picture will allow for a complete characterization of D-brane world-volumes, the Wess-Zumino action and topological D-brane charges within the K-theoretical framework, that can be compared step by step to the old cohomological classification. This has already been done for backgrounds with vanishing B-field; here we remove this hypothesis.