Crepant resolutions and brane tilings II: Tilting bundles

TL;DR

This paper explores the construction of tilting bundles on crepant resolutions of singular 3-Calabi-Yau varieties derived from brane tilings, confirming key conjectures in the field.

Contribution

It provides an explicit toric description of tilting bundles on crepant resolutions, proving conjectures by Hanany, Herzog, Vegh, and Aspinwall.

Findings

Explicit toric description of tilting bundles

Proof of Hanany, Herzog, and Vegh's conjecture

Verification of Aspinwall's conjecture

Abstract

Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety. Using the brane tiling, we can also construct all crepant resolutions of the above variety. We give an explicit toric description of tilting bundles on these crepant resolutions. This result proves the conjecture of Hanany, Herzog and Vegh and a version of the conjecture of Aspinwall.