D-brane stability, geometric engineering, and monodromy in the Derived Category

TL;DR

This paper explores the stability and monodromy of topological B-type D-branes on Calabi-Yau threefolds using derived categories, proposing a classification of massless D-branes and testing it through Fourier-Mukai functors and pi-stability.

Contribution

It introduces a conjectural classification of massless D-branes in the derived category framework and verifies it via composition formulas and stability analysis.

Findings

Established a composition formula for Fourier-Mukai functors.

Rederived the stable spectrum of N=2 SU(2) Seiberg-Witten theory.

Linked monodromies to massless D-branes in the moduli space.

Abstract

We discuss aspects of topological B-type D-branes in the framework of the derived category of coherent sheaves on a Calabi-Yau 3-fold X. We analyze the link between massless D-branes and monodromies in the CFT moduli space. A classification of all massless D-branes at any point in the moduli space is conjectured, together with an associated monodromy. We test the conjectures in two independent ways. First we establish a composition formula for certain Fourier-Mukai functors, which is a consequence of the triangulated structure of D(X). Secondly, using pi-stability we rederive the stable soliton spectrum of the pure N=2 supersymmetric SU(2) Seiberg-Witten theory. In this approach, the simplicity of the spectrum rests on Grothendieck's theorem concerning vector bundles over P^1.