Derived Categories and Zero-Brane Stability

TL;DR

This paper establishes a connection between topological field theories, D-branes, and derived categories, showing how 0-branes become unstable in Calabi-Yau moduli space and how derived categories behave under birational transformations.

Contribution

It introduces a class of topological field theories linking D-branes to the bounded derived category of coherent sheaves and analyzes 0-brane stability and derived category invariance under birational changes.

Findings

D-branes form the bounded derived category of coherent sheaves.

Any 0-brane on a Calabi-Yau threefold can become unstable.

Derived categories can be invariant under birational transformations.

Abstract

We define a particular class of topological field theories associated to open strings and prove the resulting D-branes and open strings form the bounded derived category of coherent sheaves. This derivation is a variant of some ideas proposed recently by Douglas. We then argue that any 0-brane on any Calabi-Yau threefold must become unstable along some path in the Kahler moduli space. As a byproduct of this analysis we see how the derived category can be invariant under a birational transformation between Calabi-Yaus.