Branes in hearts with perverse sheaves

TL;DR

This paper explores the topological properties and stability conditions of D-branes in type-IIA string theory using the derived category of coherent sheaves, focusing on their behavior in different phases of the Kähler moduli space.

Contribution

It introduces a framework for understanding D-branes as objects in the derived category with specific t-structures and grades, applied to a degenerate Calabi-Yau space with a projective curve.

Findings

Identification of regions in the Kähler moduli space corresponding to different t-structures.

Analysis of phases of topological branes in a degenerate Calabi-Yau setting.

Application of derived category techniques to string theory brane stability.

Abstract

Various topological properties of D-branes in the type--IIA theory are captured by the topologically twisted B-model, treating D-branes as objects in the bounded derived category of coherent sheaves on the compact part of the target space. The set of basic D-branes wrapped on the homology cycles of the compact space are taken to reside in the heart of t-structures of the derived category of coherent sheaves on the space at any point in the K\"ahler moduli space. The stability data entails specifying a t-structure along with a grade for sorting the branes. Considering an example of a degenerate Calabi-Yau space, obtained via geometric engineering, that retains but a projective curve as the sole non-compact part, we identify the regions in the K\"ahler moduli space of the curve that pertain to the different t-structures of the bounded derived category of coherent sheaves on the curve corresponding to the different phases of the topological branes.