Brane brick models in the mirror

TLDR: In this article, the authors use mirror symmetry to improve their understanding of the correspondence between 2D (0, 2) triality and 4-manifolds and provide a systematic approach for constructing brane brick models starting from geometry.

Abstract: Brane brick models are Type IIA brane configurations that encode the 2d $$ \mathcal{N}=\left(0,2\right) $$ 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.