B-branes and supersymmetric quivers in 2d

Understand

We study 2d $\mathcal{N}=(0,2)$ supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY$_4$) 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 CY$_4$.
• 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 $A_\infty$ algebra satisfied by the morphisms.
• This provides us with a derivation of the supersymmetric quiver directly from the CY$_4$ geometry.