Dimers and cluster integrable systems

Alexander B. Goncharov; Richard Kenyon

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 5, page 747-813
  • ISSN: 0012-9593

Abstract

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We show that the dimer model on a bipartite graph Γ on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space Ł Γ of line bundles with connections on the graph Γ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs Γ 1 and Γ 2 areequivalentif the Newton polygons of the corresponding partition functions coincide up to translation. We define elementary transformations of bipartite surface graphs, and show that two equivalent minimal bipartite graphs are related by a sequence of elementary transformations. For each elementary transformation we define a birational Poisson isomorphism Ł Γ 1 → Ł Γ 2 providing an equivalence of the integrable systems. We show that it is a cluster Poisson transformation, as defined in [10]. We show that for any convex integral polygon N there is a non-empty finite set of minimal graphs Γ for which N is the Newton polygon of the partition function related to  Γ . Gluing the varieties Ł Γ for graphs Γ related by elementary transformations via the corresponding cluster Poisson transformations, we get a Poisson space 𝒳 N . It is a natural phase space for the integrable system. The Hamiltonians are functions on  𝒳 N , parametrized by the interior points of the Newton polygon N . We construct Casimir functions whose level sets are the symplectic leaves of  𝒳 N . The space 𝒳 N has a structure of a cluster Poisson variety. Therefore the algebra of regular functions on  𝒳 N has a non-commutative q -deformation to a * -algebra 𝒪 q ( 𝒳 N ) . We show that the Hamiltonians give rise to a commuting family of quantum Hamiltonians. Together with the quantum Casimirs, they provide a quantum integrable system. Applying the general quantization scheme [11], we get a * -representation of the * -algebra 𝒪 q ( 𝒳 N ) in a Hilbert space. The quantum Hamiltonians act by commuting unbounded selfadjoint operators. For square grid bipartite graphs on a torus we getdiscrete quantum integrable systems, where the evolution is a cluster automorphism of the * -algebra 𝒪 q ( 𝒳 N ) commuting with the quantum Hamiltonians. We show that theoctahedral recurrence, closely related to Hirota’s bilinear difference equation[20], appears this way. Any graph G on a torus 𝕋 gives rise to a bipartite graph Γ G on  𝕋 . We show that the phase space  𝒳 related to the graph Γ G has a Lagrangian subvariety ℛ , defined in each coordinate system by a system of monomial equations. We identify it with the space parametrizing resistor networks on  G . The pair ( 𝒳 , ℛ ) has a large group of cluster automorphisms. In particular, for a hexagonal grid graph we get a discrete quantum integrable system on  𝒳 whose restriction to  ℛ is essentially given by thecube recurrence[33], [4]. The set of positive real points 𝒳 N ( ℝ > 0 ) of the phase space is well defined. It is isomorphic to the moduli space of simple Harnack curves with divisors studied in [26]. The Liouville tori of the real integrable system are given by the product of ovals of the simple Harnack curves. In the sequel [17] to this paper we show that the set of complex points 𝒳 N ( ℂ ) of the phase space is birationally isomorphic to a finite cover of the Beauville complex algebraic integrable system related to the toric surface assigned to the polygon N .

How to cite

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Goncharov, Alexander B., and Kenyon, Richard. "Dimers and cluster integrable systems." Annales scientifiques de l'École Normale Supérieure 46.5 (2013): 747-813. <http://eudml.org/doc/272193>.

@article{Goncharov2013,
abstract = {We show that the dimer model on a bipartite graph $\Gamma $ on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space $\{Ł\}_\Gamma $ of line bundles with connections on the graph $\Gamma $. The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs $\Gamma _1$ and $\Gamma _2$ areequivalentif the Newton polygons of the corresponding partition functions coincide up to translation. We define elementary transformations of bipartite surface graphs, and show that two equivalent minimal bipartite graphs are related by a sequence of elementary transformations. For each elementary transformation we define a birational Poisson isomorphism $\{Ł\}_\{\Gamma _1\} \rightarrow \{Ł\}_\{\Gamma _2\}$ providing an equivalence of the integrable systems. We show that it is a cluster Poisson transformation, as defined in [10]. We show that for any convex integral polygon $N$ there is a non-empty finite set of minimal graphs $\Gamma $ for which $N$ is the Newton polygon of the partition function related to $\Gamma $. Gluing the varieties $\{Ł\}_\Gamma $ for graphs $\Gamma $ related by elementary transformations via the corresponding cluster Poisson transformations, we get a Poisson space $\{\mathcal \{X\}\}_N$. It is a natural phase space for the integrable system. The Hamiltonians are functions on $\{\mathcal \{X\}\}_N$, parametrized by the interior points of the Newton polygon $N$. We construct Casimir functions whose level sets are the symplectic leaves of $\{\mathcal \{X\}\}_N$. The space $\{\mathcal \{X\}\}_N$ has a structure of a cluster Poisson variety. Therefore the algebra of regular functions on $\{\mathcal \{X\}\}_N$ has a non-commutative $q$-deformation to a $\ast $-algebra $\{\mathcal \{O\}\}_q(\{\mathcal \{X\}\}_N)$. We show that the Hamiltonians give rise to a commuting family of quantum Hamiltonians. Together with the quantum Casimirs, they provide a quantum integrable system. Applying the general quantization scheme [11], we get a $\ast $-representation of the $\ast $-algebra $\{\mathcal \{O\}\}_q(\{\mathcal \{X\}\}_N)$ in a Hilbert space. The quantum Hamiltonians act by commuting unbounded selfadjoint operators. For square grid bipartite graphs on a torus we getdiscrete quantum integrable systems, where the evolution is a cluster automorphism of the $\ast $-algebra $\{\mathcal \{O\}\}_q(\{\mathcal \{X\}\}_N)$ commuting with the quantum Hamiltonians. We show that theoctahedral recurrence, closely related to Hirota’s bilinear difference equation[20], appears this way. Any graph $G$ on a torus $\{\mathbb \{T\}\}$ gives rise to a bipartite graph $\Gamma _G$ on $\{\mathbb \{T\}\}$. We show that the phase space $\{\mathcal \{X\}\}$ related to the graph $\Gamma _G$ has a Lagrangian subvariety $\{\mathcal \{R\}\}$, defined in each coordinate system by a system of monomial equations. We identify it with the space parametrizing resistor networks on $G$. The pair $(\{\mathcal \{X\}\}, \{\mathcal \{R\}\})$ has a large group of cluster automorphisms. In particular, for a hexagonal grid graph we get a discrete quantum integrable system on $\{\mathcal \{X\}\}$ whose restriction to $\{\mathcal \{R\}\}$ is essentially given by thecube recurrence[33], [4]. The set of positive real points $\{\mathcal \{X\}\}_N(\{\mathbb \{R\}\}_\{&gt;0\})$ of the phase space is well defined. It is isomorphic to the moduli space of simple Harnack curves with divisors studied in [26]. The Liouville tori of the real integrable system are given by the product of ovals of the simple Harnack curves. In the sequel [17] to this paper we show that the set of complex points $\{\mathcal \{X\}\}_N(\mathbb \{C\})$ of the phase space is birationally isomorphic to a finite cover of the Beauville complex algebraic integrable system related to the toric surface assigned to the polygon $N$.},
author = {Goncharov, Alexander B., Kenyon, Richard},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {integrable systems; dimers; cluster algebras},
language = {eng},
number = {5},
pages = {747-813},
publisher = {Société mathématique de France},
title = {Dimers and cluster integrable systems},
url = {http://eudml.org/doc/272193},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Goncharov, Alexander B.
AU - Kenyon, Richard
TI - Dimers and cluster integrable systems
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 5
SP - 747
EP - 813
AB - We show that the dimer model on a bipartite graph $\Gamma $ on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space ${Ł}_\Gamma $ of line bundles with connections on the graph $\Gamma $. The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs $\Gamma _1$ and $\Gamma _2$ areequivalentif the Newton polygons of the corresponding partition functions coincide up to translation. We define elementary transformations of bipartite surface graphs, and show that two equivalent minimal bipartite graphs are related by a sequence of elementary transformations. For each elementary transformation we define a birational Poisson isomorphism ${Ł}_{\Gamma _1} \rightarrow {Ł}_{\Gamma _2}$ providing an equivalence of the integrable systems. We show that it is a cluster Poisson transformation, as defined in [10]. We show that for any convex integral polygon $N$ there is a non-empty finite set of minimal graphs $\Gamma $ for which $N$ is the Newton polygon of the partition function related to $\Gamma $. Gluing the varieties ${Ł}_\Gamma $ for graphs $\Gamma $ related by elementary transformations via the corresponding cluster Poisson transformations, we get a Poisson space ${\mathcal {X}}_N$. It is a natural phase space for the integrable system. The Hamiltonians are functions on ${\mathcal {X}}_N$, parametrized by the interior points of the Newton polygon $N$. We construct Casimir functions whose level sets are the symplectic leaves of ${\mathcal {X}}_N$. The space ${\mathcal {X}}_N$ has a structure of a cluster Poisson variety. Therefore the algebra of regular functions on ${\mathcal {X}}_N$ has a non-commutative $q$-deformation to a $\ast $-algebra ${\mathcal {O}}_q({\mathcal {X}}_N)$. We show that the Hamiltonians give rise to a commuting family of quantum Hamiltonians. Together with the quantum Casimirs, they provide a quantum integrable system. Applying the general quantization scheme [11], we get a $\ast $-representation of the $\ast $-algebra ${\mathcal {O}}_q({\mathcal {X}}_N)$ in a Hilbert space. The quantum Hamiltonians act by commuting unbounded selfadjoint operators. For square grid bipartite graphs on a torus we getdiscrete quantum integrable systems, where the evolution is a cluster automorphism of the $\ast $-algebra ${\mathcal {O}}_q({\mathcal {X}}_N)$ commuting with the quantum Hamiltonians. We show that theoctahedral recurrence, closely related to Hirota’s bilinear difference equation[20], appears this way. Any graph $G$ on a torus ${\mathbb {T}}$ gives rise to a bipartite graph $\Gamma _G$ on ${\mathbb {T}}$. We show that the phase space ${\mathcal {X}}$ related to the graph $\Gamma _G$ has a Lagrangian subvariety ${\mathcal {R}}$, defined in each coordinate system by a system of monomial equations. We identify it with the space parametrizing resistor networks on $G$. The pair $({\mathcal {X}}, {\mathcal {R}})$ has a large group of cluster automorphisms. In particular, for a hexagonal grid graph we get a discrete quantum integrable system on ${\mathcal {X}}$ whose restriction to ${\mathcal {R}}$ is essentially given by thecube recurrence[33], [4]. The set of positive real points ${\mathcal {X}}_N({\mathbb {R}}_{&gt;0})$ of the phase space is well defined. It is isomorphic to the moduli space of simple Harnack curves with divisors studied in [26]. The Liouville tori of the real integrable system are given by the product of ovals of the simple Harnack curves. In the sequel [17] to this paper we show that the set of complex points ${\mathcal {X}}_N(\mathbb {C})$ of the phase space is birationally isomorphic to a finite cover of the Beauville complex algebraic integrable system related to the toric surface assigned to the polygon $N$.
LA - eng
KW - integrable systems; dimers; cluster algebras
UR - http://eudml.org/doc/272193
ER -

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