Dimer models and the special McKay correspondence.

*(English)*Zbl 1338.14019A dimer model is a bicolored graph on a real 2-torus giving a polygon division of the torus. Associated with a dimer model is its characteristic polynomial defined combinatorially in terms of perfect matchings, and a quiver with relations. The Newton polygon of the characteristic polynomial is called the characteristic polygon. If a dimer model is non-degenerate and the stability parameter \(\theta\) is generic, then the moduli space \(\mathcal{M}_\theta\) of stable representations of the corresponding quiver with dimension vector \((1,\dots,1)\) is a smooth toric Calabi-Yau 3-fold. Corresponding to each vertex \(v\) of the quiver (or a face of the dimer model) is a tautological line bundle \(\mathcal{L}_v\) on \(\mathcal{M}_\theta\), and the edges give rise to the universal morphism from the path algebra of the quiver with relations \(\mathbb{C}\Gamma\to \text{End}(\bigoplus_v \mathcal{L}_v)\). If the tautological bundle \(\bigoplus_v \mathcal{L}_v\) is a tilting object (Condition T), and the universal morphism above is an isomorphism (Condition E), then it is known that the bounded derived categories of coherent sheaves on \(\mathcal{M}_\theta\), and finitely generated \(\mathbb{C}\Gamma\)-modules are naturally equivalent. The notion of consistency condition on a dimer model ensures the Calabi-Yau property of \(\mathbb{C}\Gamma\).

The paper under review studies the behavior of a consistent dimer model under the removal of a corner from the characteristic polygon. It provides an explicit algorithm to remove some of the edges from the dimer model \(G\) and produce another consistent dimer model \(G'\) with a corner of the associated characteristic polynomial is removed. This result refines a previously studied operation of removing a triangle from the characteristic polygon, and provides a more constructive proof the fact that for any lattice polygon \(\Delta\), there is a consistent dimer model whose characteristic polygon coincides with \(\Delta\). The other main result of the paper under review is the following: let \(G\) be a consistent dimer model. Then for any generic stability parameter \(\theta\), the tautological bundle \(\bigoplus_v \mathcal{L}_v\) satisfies the conditions T and E. Furthermore, for certain choices of \(\theta\), one can find some \(\theta'\) such that the conditions T and E hold for \(\mathcal{M}_\theta\) if and only if they hold for \(\mathcal{M}'_{\theta'}\), the corresponding moduli space associated with the dimer model \(G'\).

The paper under review studies the behavior of a consistent dimer model under the removal of a corner from the characteristic polygon. It provides an explicit algorithm to remove some of the edges from the dimer model \(G\) and produce another consistent dimer model \(G'\) with a corner of the associated characteristic polynomial is removed. This result refines a previously studied operation of removing a triangle from the characteristic polygon, and provides a more constructive proof the fact that for any lattice polygon \(\Delta\), there is a consistent dimer model whose characteristic polygon coincides with \(\Delta\). The other main result of the paper under review is the following: let \(G\) be a consistent dimer model. Then for any generic stability parameter \(\theta\), the tautological bundle \(\bigoplus_v \mathcal{L}_v\) satisfies the conditions T and E. Furthermore, for certain choices of \(\theta\), one can find some \(\theta'\) such that the conditions T and E hold for \(\mathcal{M}_\theta\) if and only if they hold for \(\mathcal{M}'_{\theta'}\), the corresponding moduli space associated with the dimer model \(G'\).

Reviewer: Amin Gholampour (College Park)