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Dimer models and the special McKay correspondence. (English) Zbl 1338.14019
A 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 \(G'\).

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
14G20 Local ground fields in algebraic geometry
14E16 McKay correspondence
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