Brane Brick Models in the Mirror

TL;DR

This paper uses mirror symmetry to systematically construct brane brick models for 2d (0,2) gauge theories on D1-branes probing toric Calabi-Yau 4-folds, revealing geometric dualities and conjecturing a new quadrality symmetry.

Contribution

It introduces a mirror symmetry approach to construct brane brick models from geometry and explores geometric realizations of dualities, including a conjecture of quadrality in 0d theories.

Findings

Mirror symmetry provides a systematic construction method.

Geometric transitions realize 2d (0,2) triality.

Conjecture of a quadrality symmetry in 0d theories.

Abstract

Brane brick models are Type IIA brane configurations that encode the $2d$ $\mathcal{N}=(0,2)$ gauge theories on the worldvolume of D1-branes probing toric Calabi-Yau 4-folds. We use mirror symmetry to improve our understanding of this correspondence and to provide a systematic approach for constructing brane brick models starting from geometry. The mirror configuration consists of D5-branes wrapping 4-spheres and the gauge theory is determined by how they intersect. We also explain how $2d$ $(0,2)$ triality is realized in terms of geometric transitions in the mirror geometry. Mirror symmetry leads to a geometric unification of dualities in different dimensions, where the order of duality is $n-1$ for a Calabi-Yau $n$-fold. This makes us conjecture the existence of a quadrality symmetry in $0d$. Finally, we comment on how the M-theory lift of brane brick models connects to the classification of $2d$ $(0,2)$ theories in terms of 4-manifolds.