1 M ay 2 00 9 Dimer models and the special McKay correspondence Akira

Dimer models are introduced by string theorists (see e.g. [11, 12, 13, 16, 17, 18]) to study supersymmetric quiver gauge theories in four dimensions. A dimer model is a bicolored graph on a 2-torus encoding the information of a quiver with relations. If a dimer model is non-degenerate, then the moduli space Mθ of stable representations of the quiver with dimension vector (1, . . . , 1) with respect to a generic stability parameter θ in the sense of King [23] is a smooth toric Calabi-Yau 3-fold [20]. The convex hull of one-dimensional cones of the fan describing this toric manifold is a lattice polygon described in purely combinatorial way using perfect matchings. Although the structure of the fan is not determined by this lattice polygon, any fan structure gives an equivalent derived categories of coherent sheaves [1, 3]. In this paper, we study the behavior of the dimer model under the operation of removing a vertex from the lattice polygon and taking the convex hull of the rest. This generalizes the work of Gulotta [15] where he studies the operation of removing a triangle from the lattice polygon, and the special McKay correspondence plays an essential role in this generalization. The main result in this paper is the following: