Graded quivers, generalized dimer models and toric geometry

Abstract

Abstract The open string sector of the topological B-model on CY (m + 2)-folds is de- scribed 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 world- volume of D(5 − 2m)-branes for m = 0, …, 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.