Comparing two approaches to the K-theory classification of D-branes

TL;DR

This paper compares two main K-theory-based methods for classifying D-brane charges in type II superstring theory, establishing a link between them and discussing their equivalence with rational coefficients.

Contribution

It explicitly connects the Gysin map approach with the Atiyah-Hirzebruch spectral sequence in D-brane charge classification, clarifying their relationship.

Findings

The Gysin map yields a representative element of the spectral sequence class.

The two methods are fully equivalent when using rational coefficients.

An explicit link between the two approaches is established.

Abstract

We consider the two main classification methods of D-brane charges via K-theory, in type II superstring theory with vanishing B-field: the Gysin map approach and the one based on the Atiyah-Hirzebruch spectral sequence. Then, we find out an explicit link between these two approaches: the Gysin map provides a representative element of the equivalence class obtained via the spectral sequence. We also briefly discuss the case of rational coefficients, characterized by a complete equivalence between the two classification methods.