Brane Tilings and Exceptional Collections

TL;DR

This paper establishes a translation between brane tilings and exceptional collections, linking gauge theories on D3-branes with geometric data of Calabi-Yau singularities, thus enhancing understanding of their interrelation.

Contribution

It provides a systematic dictionary connecting brane tilings and exceptional collections, enabling translation between gauge theory representations and geometric structures.

Findings

Computed exceptional collections from brane tilings.

Derived periodic quivers from exceptional collections.

Provided new insights into quiver theory construction.

Abstract

Both brane tilings and exceptional collections are useful tools for describing the low energy gauge theory on a stack of D3-branes probing a Calabi-Yau singularity. We provide a dictionary that translates between these two heretofore unconnected languages. Given a brane tiling, we compute an exceptional collection of line bundles associated to the base of the non-compact Calabi-Yau threefold. Given an exceptional collection, we derive the periodic quiver of the gauge theory which is the graph theoretic dual of the brane tiling. Our results give new insight to the construction of quiver theories and their relation to geometry.