# Graded Quivers, Generalized Dimer Models and Toric Geometry

## TL;DR

This paper introduces m-dimers as a new combinatorial tool to encode m-graded quivers and superpotentials for toric Calabi-Yau (CY) (m+2)-folds, generalizing known cases and simplifying the geometric-quiver correspondence.

## Contribution

It generalizes the concept of perfect matchings and the Kasteleyn matrix to arbitrary m, providing a unified framework for toric CY (m+2)-folds and their associated gauge theories.

## Key findings

- Introduced m-dimers to encode m-graded quivers and superpotentials.
- Generalized perfect matchings and Kasteleyn matrix to any m.
- Applied new tools to infinite families of CY singularities.

## 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 $m=0,\ldots, 3$. 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.