2d Supersymmetric Gauge Theories, D-branes and Trialities

Engineering quantum field theories in String Theory in terms of branes is a powerful approach for understanding their dynamics. We review recent progress in the realization of $2d$ $\mathcal{N}=(0,2)$ gauge theories in terms of branes. We discuss Brane Brick Models, a new class of Type IIA brane configurations which are T-dual to D1-branes over singular toric Calabi-Yau 4-folds. They fully encode the infinite class of $2d$ $\mathcal{N}=(0,2)$ quiver gauge theories on the worldvolume of the D1-branes and significantly streamline their connection to the probed geometries. As an application, we explain how these constructions provide a brane realization of triality. We also comment on the realization of $2d$ $\mathcal{N}=(0,1)$ theories via Spin(7) orientifolds. This note is based on the author's talk at the Nankai Symposium on Mathematical Dialogues celebrating the 110$^{th}$ anniversary of the birth of Prof. S.-S. Chern