B-branes and supersymmetric quivers in 2d

Abstract

We study 2d \( \mathcal{N} \) = (0, 2) supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY₄) singularities. On general grounds, the holomorphic sector of these theories — matter content and (classical) superpotential interactions — should be fully captured by the topological B-model on the CY₄. By studying a number of examples, we confirm this expectation and flesh out the dictionary between B-brane category and supersymmetric quiver: the matter content of the supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext groups of coherent sheaves), while the superpotential interactions are encoded in the A_∞ algebra satisfied by the morphisms. This provides us with a derivation of the supersymmetric quiver directly from the CY₄ geometry. We also suggest a relation between triality of \( \mathcal{N} \) = (0,2) gauge theories and certain mutations of exceptional collections of sheaves. 0d \( \mathcal{N} \) = 1 supersymmetric quivers, corresponding to D-instantons probing CY₅ singularities, can be discussed similarly.