Dimers and cluster integrable systems

By A. Goncharov, Richard Kenyon. 2011

266 citations recorded by the source corpus.

We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph. The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus. We show that the phase space of the latter has a Lagrangian subvariety. We identify it with the space parametrizing resistor networks on the original graph.We construct several discrete quantum integrable systems.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/02ac2ea040eb76caa0b61d5e56d14d212a0fba6b

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In Geometry and Topology

Citations in the corpus

Scattering Amplitudes and the Positive Grassmannian (2012, background): “Bipartite graphs are also intimately related to dimer models, with the recent mathematical work [40] particularly closely related to our discussion.”
Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry (2013, background): “t have a central charge. More generally complex integrable systems with the total space being a compact hyperk¨ahler manifold do not have central charge. 4.2.2 Integrable systems from dimer models In [GoKen] the authors deﬁned a class of integrable systems X0 →B0 for which X0 is birationally symplec”
INTEGRABLE CLUSTER DYNAMICS OF DIRECTED NETWORKS AND PENTAGRAM MAPS (2014, unknown): “Thus, just like in [19], interior points of Δ correspond to independent integrals while integer points on the boundary of Δ correspond to Casimir functions.”
Cluster varieties from Legendrian knots (2015, methodology): “ projection into the alternating strand diagram of a bipartite graph embedded in the base surface. The ﬁlling retracts onto this graph and is an exact Lagrangian embedding of the conjugate surface of [GK]. Holonomies around the faces of the graph form a natural coordinate system on the associated ch”
Introduction to Cluster Algebras (2018, background): “The conserved quantities of the pentagrammap are sums of weights of perfect matchings, refined by a notion of homology class of a matching defined in [28].”
The twist for positroid varieties (2016, methodology): “The αf are clearly Gm-invariant, and have often been used as G V m-invariant coordinates on Gm, and as rational coordinates on Π ◦(M) induced by D [2, 11, 27].”
Introduction to Cluster Algebras. Chapters 1-3 (2016, unknown): “Urban renewal Urban renewal [33] is an operation on bipartite graphs which arises in several different contexts including statistical mechanics (spider moves in dimer models [30]), gauge theory (Seiberg duality action on brane tilings [27]), and total positivity (square moves in dual graphs of Postn”
Dimer models and cluster categories of Grassmannians (2013, background): “We note that dimer models (bipartite field theories) on surfaces with boundary have also been considered in independent work of Franco [12], and Postnikov diagrams on surfaces are also considered in [16, 23].”