  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.

High Energy Physics - Theory;Commutative Algebra;Algebraic Geometry;Combinatorics