Graded quivers and B-branes at Calabi-Yau singularities

A graded quiver with superpotential is a quiver whose arrows are assigned degrees c {0, 1, , m}, for some integer m 0, with relations generated by a superpotential of degree m - 1. Ordinary quivers (m = 1) often describe the open string sector of D-brane systems; in particular, they capture the physics of D3-branes at local Calabi-Yau (CY) 3-fold singularities in type IIB string theory, in the guise of 4d N = 1 supersymmetric quiver gauge theories. It was pointed out recently that graded quivers with m = 2 and m=3 similarly describe systems of D-branes at CY 4-fold and 5-fold singularities, as 2d N = (0, 2) and 0d N = 1 gauge theories, respectively. In this work, we further explore the correspondence between m-graded quivers with superpotential, Q((m)), and CY (m + 2)-fold singularities, Xm+2. For any m, the open string sector of the topological B-model on Xm+2 can be described in terms of a graded quiver. We illustrate this correspondence explicitly with a few infinite families of toric singularities indexed by m N, for which we derive toric graded quivers associated to the geometry, using several complementary perspectives. Many interesting aspects of supersymmetric quiver gauge theories can be formally extended to any m; for instance, for one family of singularities, dubbed C(Y-1,Y-0(P-m)), that generalizes the conifold singularity to m > 1, we point out the existence of a formal duality cascade for the corresponding graded quivers.