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A bstract We introduce and initiate the study of a general class of 2 d $$ \mathcal{N} $$ 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 ’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. 

A bstract We initiate the geometric engineering of 2d $$ \mathcal{N} $$ N = (0 , 1) gauge theories on D1-branes probing singularities. To do so, we introduce a new class of backgrounds obtained as quotients of Calabi-Yau 4-folds by a combination of an anti-holomorphic involution leading to a Spin(7) cone and worldsheet parity. We refer to such constructions as Spin(7) orientifolds . Spin(7) orientifolds explicitly realize the perspective on 2d $$ \mathcal{N} $$ N = (0 , 1) theories as real slices of $$ \mathcal{N} $$ N = (0 , 2) ones. Remarkably, this projection is geometrically realized as Joyce’s construction of Spin(7) manifolds via quotients of Calabi-Yau 4-folds by anti-holomorphic involutions. We illustrate this construction in numerous examples with both orbifold and non-orbifold parent singularities, discuss the role of the choice of vector structure in the orientifold quotient, and study partial resolutions. 

A bstract We present a new, geometric perspective on the recently proposed triality of 2d $$ \mathcal{N} $$ N = (0 , 1) gauge theories, based on its engineering in terms of D1-branes probing Spin(7) orientifolds. In this context, triality translates into the fact that multiple gauge theories correspond to the same underlying orientifold. We show how Spin(7) orientifolds based on a particular involution, which we call the universal involution, give rise to precisely the original version of $$ \mathcal{N} $$ N = (0 , 1) triality. Interestingly, our work also shows that the space of possibilities is significantly richer. Indeed, general Spin(7) orientifolds extend triality to theories that can be regarded as consisting of coupled $$ \mathcal{N} $$ N = (0 , 2) and (0 , 1) sectors. The geometric construction of 2d gauge theories in terms of D1-branes at singularities therefore leads to extensions of triality that interpolate between the pure $$ \mathcal{N} $$ N = (0 , 2) and (0 , 1) cases.