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

We introduce and initiate the study of a general class of $2d$ $\mathcal{N}=(0,2)$ quiver gauge theories, defined in terms of certain 2-dimensional CW complexes on oriented 3-manifolds. We refer to this class of theories as BFT$_2$$\mbox{'}$s. They are natural generalizations of Brane Brick Models, which capture the gauge theories on D1-branes probing toric Calabi-Yau 4-folds. The dynamics and triality of the gauge theories translate into simple transformations of the underlying CW complexes. We introduce various combinatorial tools for analyzing these theories and investigate their connections to toric Calabi-Yau manifolds, which arise as their master and moduli spaces. Invariance of the moduli space is indeed a powerful criterion for identifying theories in the same triality class. We also investigate the reducibility of these theories.