Orbifold Reduction and 2d (0,2) Gauge Theories

We introduce Orbifold Reduction, a new method for generating $2d$ $(0,2)$ gauge theories associated to D1-branes probing singular toric Calabi-Yau 4-folds starting from $4d$ $\mathcal{N}=1$ gauge theories on D3-branes probing toric Calabi-Yau 3-folds. The new procedure generalizes dimensional reduction and orbifolding. In terms of T-dual configurations, it generates brane brick models starting from brane tilings. Orbifold reduction provides an agile approach for generating $2d$ $(0,2)$ theories with a brane realization. We present three practical applications of the new algorithm: the connection between $4d$ Seiberg duality and $2d$ triality, a combinatorial method for generating theories related by triality and a $2d$ $(0,2)$ generalization of the Klebanov-Witten mass deformation.