K-homology in algebraic geometry and D-branes

TL;DR

This paper explores the use of K-homology and K-theory in algebraic geometry to classify D-branes, establishing a mathematical framework linking D-brane charges to K-homology.

Contribution

It demonstrates how D-branes can be classified by K-homology using algebraic geometry tools, extending topological K-theory methods to a holomorphic setting.

Findings

D-branes wrapping subvarieties are classified by relative K-groups.

D-brane charges are properly described by K-homology.

The duality between relative K-groups and K-homology is established.

Abstract

In this article, we study how the Grothendieck group of coherent sheaves can be used to describe D-branes. We show how global bound state construction in topological $K$-theory can be adapted to our context, showing that D-branes wrapping a subvariety are holomorphically classified by a relative $K$-group. By taking the duality between the relative $K$-groups and the $K$-homologies, we show that D-brane charge of type IIB superstrings is properly classified by the $K$-homology.