Graded Quivers, Generalized Dimer Models and Toric Geometry

Abstract

The open string sector of the topological B-model model on CY $(m+2)$-folds is described by $m$-graded quivers with superpotentials. This correspondence extends to general $m$ the well known connection between CY $(m+2)$-folds and gauge theories on the worldvolume of D$(5-2m)$-branes for $m=0,\dots ,3$. We introduce $m$-dimers, which fully encode the $m$-graded quivers and their superpotentials, in the case in which the CY $(m+2)$-folds are toric. Generalizing the well known $m=1,2$ cases, $m$-dimers significantly simplify the connection between geometry and $m$-graded quivers. A key result of this paper is the generalization of the concept of perfect matching, which plays a central role in this map, to arbitrary $m$. We also introduce a simplified algorithm for the computation of perfect matchings, which generalizes the Kasteleyn matrix approach to any $m$. We illustrate these new tools with a few infinite families of CY singularities.

arXiv comment54 pages, 6 figures

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• Toroidal and Elliptic Quiver BPS Algebras and Beyond
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• BFT${}_2$: a General Class of $2d$ $\mathcal{N}=(0,2)$ Theories, 3-Manifolds and Toric Geometry
• Charging solid partitions
• Calabi-Yau Products: Graded Quivers for General Toric Calabi-Yaus
• On the Classification of Duality Webs for Graded Quivers
• Some Open Questions in Quiver Gauge Theory
• Charge functions for odd dimensional partitions