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.
Highlights
Many supersymmetric quantum field theories can be engineered on systems of branes in string theory
We study 2d N = (0, 2) supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY4) singularities
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, while the superpotential interactions are encoded in the A∞ algebra satisfied by the morphisms
Summary
Many supersymmetric quantum field theories can be engineered on systems of branes in string theory. There are massless open strings connecting the fractional branes among themselves, which realize bifundamental (or adjoint) matter fields XIJ In this way, the low-energy open string sector at the singularity is described by a supersymmetric quiver gauge theory : to each fractional brane EI , we associate a node in the quiver, denoted by eI. D1-branes on X4 lead to the richer structure of 2d N = (0, 2) quiver gauge theories Those quivers have two distinct types of arrows, corresponding to (0, 2) chiral multiplets XIJ and (0, 2) fermi multiplets ΛIJ , respectively. The 0d N = 1 matrix models have two kinds of holomorphic “superpotentials”, distinct from the 2d superpotentials, denoted by F (X) and H(X) [49].6 These interactions terms can be recovered from the fractional branes by considering the product structure between Ext groups. Where the sum is over all pairs of quiver paths p : eI → · · · → eJ and p : eJ → · · · → eI based at fermi multiplets ΛIJ such that the closed path p + p coincides with P
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