2d Supersymmetric Gauge Theories, D-branes and Trialities

TL;DR

This paper reviews how brane configurations in string theory realize 2d supersymmetric gauge theories, introduces Brane Brick Models for encoding these theories, and discusses their role in understanding triality and related geometries.

Contribution

It introduces Brane Brick Models as a new class of Type IIA brane configurations that encode 2d (0,2) gauge theories and connect them to probed geometries, advancing the understanding of brane realizations of triality.

Findings

Brane Brick Models fully encode 2d (0,2) quiver gauge theories.

These models provide a brane realization of triality.

They streamline the connection between gauge theories and toric Calabi-Yau 4-folds.

Abstract

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