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We present a new, geometric perspective on the recently proposed triality of2d $\mathcal{N}=(0,1)$ gauge theories, based on its engineering in terms ofD1-branes probing Spin(7) orientifolds. In this context, triality translatesinto the fact that multiple gauge theories correspond to the same underlyingorientifold. We show how Spin(7) orientifolds based on a particular involution,which we call the universal involution, give rise to precisely the originalversion of $\mathcal{N}=(0,1)$ triality. Interestingly, our work also showsthat the space of possibilities is significantly richer. Indeed, generalSpin(7) orientifolds extend triality to theories that can be regarded asconsisting of coupled $\mathcal{N}=(0,2)$ and $(0,1)$ sectors. The geometricconstruction of 2d gauge theories in terms of D1-branes at singularitiestherefore leads to extensions of triality that interpolate between the pure$\mathcal{N}=(0,2)$ and $(0,1)$ cases.

Spin(7) Orientifolds and 2d $\mathcal{N}= (0,1)$ Triality