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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 $$G'$$.

##### MSC:
 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
##### Keywords:
dimer model; McKay correspondence; derived equivalence
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