Graded quivers and B-branes at Calabi-Yau singularities

<p>A graded quiver with superpotential is a quiver whose arrows are assigned degrees <em>c</em> ∈ {0<em>,</em> 1<em>,</em> ⋯<em> , m</em>}, for some integer <em>m</em> ≥ 0, with relations generated by a superpotential of degree <em>m</em> − 1. Ordinary quivers (<em>m</em> = 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 <em>N</em> = 1 supersymmetric quiver gauge theories. It was pointed out recently that graded quivers with <em>m</em> = 2 and <em>m</em>=3 similarly describe systems of D-branes at CY 4-fold and 5-fold singularities, as 2d <em>N</em> = (0<em>,</em> 2) and 0d <em>N</em> = 1 gauge theories, respectively. In this work, we further explore the correspondence between <em>m</em>-graded quivers with superpotential, <em>Q</em>_(<em>m</em>), and CY (<em>m</em> + 2)-fold singularities, <strong>X</strong><em>_m</em>₊₂. For any <em>m</em>, the open string sector of the topological B-model on <strong>X</strong><em>_m</em>₊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 <em>m</em> ∈ 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 <em>m</em>; for instance, for one family of singularities, dubbed <em>C</em>(<em>Y</em>^1,0(ℙ<em>^m</em>)), that generalizes the conifold singularity to <em>m &gt;</em> 1, we point out the existence of a formal “duality cascade” for the corresponding graded quivers.</p>