D-Branes on Toric Calabi-Yau Varieties

TL;DR

This paper studies B-type D-branes on noncompact toric Calabi-Yau spaces, presenting a method to identify tilting line bundles that reveal the invariance of the derived category across different phases in the Kähler moduli space.

Contribution

It introduces a general program to find tilting line bundles for D-branes on toric Calabi-Yau varieties, showing their stability and invariance across phases in the moduli space.

Findings

The tilting set often remains fixed across phases.

The derived category is invariant across all phases.

Certain line bundles in the tilting set stay stable throughout the moduli space.

Abstract

We analyze B-type D-branes on noncompact toric Calabi--Yau spaces. A general program is presented to find a set of tilting line bundles that yields the associated quiver and its relations. In many cases, this set remains fixed as one moves between phases in the K\"ahler moduli space. This gives a particularly simple picture of how the derived category remains invariant across all phases. The combinatorial problems involving local cohomology used to determine the tilting set are also related to questions of Pi-stability as one moves between phases. As a result, in some cases precisely those line bundles in the tilting set remain stable over the whole moduli space in some sense.