The worldvolume theories of D1-branes probing toric Calabi-Yau 4-folds give rise to a rich family of 2d (0,2) quiver gauge theories. Remarkably, both the probed Calabi-Yau geometry and the associated Type IIA brane configuration can be encoded in a single combinatorial object: a tessellation of the 3-torus known as a brane brick model. This talk gives a pedagogical introduction to brane brick models, showing how they capture both the Lagrangian of the corresponding 2d (0,2) gauge theory and the geometry of the toric Calabi-Yau 4-fold. We then review how Gadde-Gukov-Putrov triality relates distinct brane brick models that describe the same Calabi-Yau geometry. From the perspective of Calabi-Yau mirror symmetry, we further explain how brane brick models and triality connect to brane tilings and Seiberg duality, linking gauge theories and gauge theory phenomena across different spacetime dimensions. More recently, brane brick models have been classified for smooth toric Fano 3-folds defined by reflexive polytopes. We illustrate how a family of birational transformations relating these Fano 3-folds can be identified with mass deformations of the corresponding brane brick models and their 2d (0,2) theories.
https://arxiv.org/abs/2510.05517
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https://arxiv.org/abs/1609.01723
https://arxiv.org/abs/1602.01834
https://arxiv.org/abs/1510.01744
https://arxiv.org/abs/1506.03818