# Graded quivers and B-branes at Calabi-Yau singularities

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## Abstract

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 \( \mathcal{N} \) = 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 \( \mathcal{N} \) = (0*,* 2) and 0d \( \mathcal{N} \) = 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, **X**_{m+2}. For any *m*, the open string sector of the topological B-model on **X**_{m+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*^{1,0}(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.

## Keywords

D-branes Supersymmetry and Duality Topological Strings## Notes

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