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We introduce and initiate the study of a general class of $2d$$\mathcal{N}=(0,2)$ quiver gauge theories, defined in terms of certain2-dimensional CW complexes on oriented 3-manifolds. We refer to this class oftheories as BFT$_2$$\mbox{'}$s. They are natural generalizations of Brane BrickModels, which capture the gauge theories on D1-branes probing toric Calabi-Yau4-folds. The dynamics and triality of the gauge theories translate into simpletransformations of the underlying CW complexes. We introduce variouscombinatorial tools for analyzing these theories and investigate theirconnections to toric Calabi-Yau manifolds, which arise as their master andmoduli spaces. Invariance of the moduli space is indeed a powerful criterionfor identifying theories in the same triality class. We also investigate thereducibility of these theories.

BFT$_2$: a General Class of $2d$ $\mathcal{N}=(0,2)$ Theories, 3-Manifolds and Toric Geometry