Graded quivers and B-branes at Calabi-Yau singularities

A graded quiver with superpotential is a quiver whose arrows are assigned degreesandnbsp;candnbsp;andisin; {0,andnbsp;1,andnbsp;⋯andnbsp;, m}, for some integerandnbsp;mandnbsp;andge; 0, with relations generated by a superpotential of degreeandnbsp;mandnbsp;andminus; 1. Ordinary quivers (mandnbsp;= 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 4dandnbsp;Nandnbsp;= 1 supersymmetric quiver gauge theories. It was pointed out recently that graded quivers withandnbsp;mandnbsp;= 2 andandnbsp;m=3 similarly describe systems of D-branes at CY 4-fold and 5-fold singularities, as 2dandnbsp;Nandnbsp;= (0,andnbsp;2) and 0dandnbsp;Nandnbsp;= 1 gauge theories, respectively. In this work, we further explore the correspondence betweenandnbsp;m-graded quivers with superpotential,andnbsp;Q(m), and CY (mandnbsp;+ 2)-fold singularities,andnbsp;Xm+2. For anyandnbsp;m, the open string sector of the topological B-model onandnbsp;Xm+2andnbsp;can be described in terms of a graded quiver. We illustrate this correspondence explicitly with a few infinite families of toric singularities indexed byandnbsp;mandnbsp;andisin; ℕ, for which we derive andldquo;toricandrdquo; graded quivers associated to the geometry, using several complementary perspectives. Many interesting aspects of supersymmetric quiver gauge theories can be formally extended to anyandnbsp;m; for instance, for one family of singularities, dubbedandnbsp;C(Y1,0(ℙm)), that generalizes the conifold singularity toandnbsp;m andgt;andnbsp;1, we point out the existence of a formal andldquo;duality cascadeandrdquo; for the corresponding graded quivers.