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Dimer models and Calabi-Yau algebras. (English) Zbl 1237.14002
Mem. Am. Math. Soc. 1011, vii, 86 p. (2012).
This article uses techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3-dimensional Calabi-Yau algebras which are non-commutative crepant resolutions of Gorenstein affine toric threefolds. A characteristic property of an \(n\)-dimensional Calabi-Yau manifold \(X\) is that the \(n\)th power of the shift functor on \(D(X):=D^b(\mathrm{coh}(X))\), the bounded derived category of coherent sheaves on \(X\), is a Serre functor. That is, there exists a natural isomorphism \[ \text{Hom}_{D(X)}(A,B)\cong\text{Hom}_{D(X)}(B,A[n])^\ast, \;\forall A,B\in D(X). \] The idea behind Calabi-Yau algebras is to write down conditions on the algebra \(A\) such that \(D(A):=D^b(\mathrm{mod}(A))\), the bounded derived category of modules over \(A\), has the same property.
One way to study resolutions of singularities is by considering their derived categories. Of particular interest are crepant resolutions of toric Gorenstein singularities. It is conjectured by Bondal and Orlov that if \(f_1:Y_1\rightarrow X\) and \(f_2:Y_2\rightarrow X\) are crepant resolutions then there is a derived equivalence \(D(Y_1)\cong D(Y_2)\). This means that the derived category of a crepant resolution is an invariant of the singularity.
A tilting bundle \(T\) is a bundle which determines a derived equivalence \(D(Y)\cong D(A)\), where \(A=\text{End}(T)\). In this case one could consider the algebra \(A\) as a type of noncommutative crepant resolution (NCCR) of the singularity.
Given a NCCR \(A\), a (commutative) crepant resolution \(Y\) such that \(D(Y)\cong D(A)\) can be constructed as a moduli space of certain stable \(A\)-representations. This is a generalization of the McKay correspondence.
If \(X=\text{Spec}(R)\) is a Gorenstein singularity, then any crepant resolution is a Calabi-Yau variety. Therefore, if \(A\) is an NCCR it must be a Calabi-Yau algebra. The center of \(A\) must also be the coordinate ring \(R\) of the singularity. Thus any 3-dimensional Calabi-Yau algebra whose center \(R\) is the coordinate ring of a toric Gorenstein 3-fold, is potentially an NCCR of \(X=\text{Spec}(R)\).
Bocklandt proved that every graded Calabi-Yau algebra of global dimension 3 is isomorphic to a superpotential algebra. Such an algebra \(A=\mathbb C Q/(dW)\) is the quotient of the path algebra of a quiver \(Q\) by an ideal of relations \((dW)\), where the relations are generated by taking (formal) partial derivatives of a single element \(W\) called the superpotential. It encodes some informations about the syzygies. The Calabi-Yau condition is actually equivalent to saying that all the syzygies can be obtained from the superpotential.
To construct algebras from Calabi-Yau manifolds, the author uses the concept of dimer models. A dimer model is a finite bipartite tiling of a compact (oriented) Riemann surface \(Y\). Of particular interest are tilings of the 2-torus where the dimer model can be considered as a doubly periodic tiling of the plane. A dual tiling is also considered, where faces are dual to vertices and edges dual to edges. The edges of this dual tiling inherit an orientation from the bipartiteness of the dimer model. This is usually chosen so that the arrows go clockwise around a face dual to a white vertex. Therefore the dual tiling is a quiver \(Q\), with faces. The faces of this quiver encodes a superpotential \(W\), and so there is a superpotential algebra \(A=\mathbb C Q/(dW)\) associated to every dimer model. Of special importance are the perfect matchings in a dimer model used to construct a commutative ring \(R=\mathbb C[X]\) from a dimer model. Then \(R\) is the coordinate ring in three variables of an affine toric Gorenstein 3-fold \(X\). The author describes conditions on the dimer model so that its superpotential algebra is Calabi-Yau. Consistency conditions are a strong type of non-degeneracy condition, and necessary and sufficient conditions for a dimer model to be geometrically consistent are given in terms of the intersection properties of special paths called zig-zag flows on the universal cover \(\tilde Q\) of the quiver \(Q\). Geometric consistency amaounts to saying that zig-zag flows behave effectively like straight lines.
The author study some properties of zig-zag flows in a geometrically consistent dimer model. He construct, in a very explicit way, a collection of perfect matchings indexed by the 2-dimensional cones in the global zig-zag fan. It is proved that these are all the perfect matchings. Also, it is seen that each perfect matching of this form corresponds to a vertex of multiplicity one.
The concept of (noncommutative, affine, normal) toric algebras are introduced. It is hoped that they might play a similar role in noncommutative algebraic geometry to that played by toric varieties in algebraic geometry. The author shows that there is a toric algebra \(B\) naturally associated to every dimer model, and moreover, the center of this algebra is the ring \(R\) associated to the dimer model in the way described above. Therefore a given dimer model has two noncommutative algebras \(A\) and \(B\) and there is a natural algebra homomorphism \(\mathfrak h:A\rightarrow B.\) A dimer model is called algebraically consistent if this map is an isomorphism. Algebraic and geometric consistency are the two consistency conditions studied in the core of the article. One essential result is that a geometrically consistent dimer model is algebraically consistent. The proof of this is heavily dependent on the definition of perfect matchings. The main results of the article is the following theorems:
1) If a dimer model on a torus is algebraically consistent, then the algebra \(A\) obtained from it is a Calabi-Yau algebra of global dimension 3.
2) Given an algebraically consistent dimer model on a torus, the algebra \(A\) obtained from it is an NCCR of the (commutative) ring \(R\) associated to that dimer model.
3) Every Gorenstein affine toric threefold admits an NCCR, which can be obtained via a geometrically consistent dimer model.
The book is explicit; it gives all the definitions, contains clear proofs, and is an excellent text on the subject.

MSC:
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14A22 Noncommutative algebraic geometry
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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