Abstract
We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems in Goncharov-Kenyon 2013. The dynamic on the graph is an urban renewal together with shrinking all 2-valent vertices, while it is a cluster transformation on the weight. The graph is not necessary obtained from an integral polygon. We define the Hamiltonians of a weighted graph as partition functions of all weighted perfect matchings with a common homology class, then show that they are invariant under a move on the weighted graph. This move coincides with a cluster mutation, analog to Y-seed mutation in dimer integrable systems. We construct graphs for Q-systems of type A and B and show that the Hamiltonians are conserved quantities of the systems. This reproves the results of Di Francesco-Kedem 2010 and Galashin-Pylyavskyy 2016 for the Q-systems of type A, and gives new results for that of type B. Similar to the results in Di Francesco-Kedem 2010, the conserved quantities for Q-systems of type B can also be written as partition functions of hard particles on a certain graph. For type A, we show that the conserved quantities Poisson commute under a nondegenerate Poisson bracket.
Highlights
The dimer model is an important model and has a long history in statistical mechanics [Kas63]
The first goal of this paper is to rethink the discrete dimer integrable system of [GK13] in cluster variable setting and extend it to bipartite torus graphs not necessary obtained from integral polygons
We studied a discrete dynamic on a weighted bipartite torus graph, obtained from an urban renewal on the graph and cluster mutation on the weight
Summary
The dimer model is an important model and has a long history in statistical mechanics [Kas63]. There is a move called urban renewal on the weighted graph G, which acts on its corresponding Y-seed as a Y-seed mutation This transformation is a change of coordinates on the phase space, and the Hamiltonians are invariant under the transformation. The first goal of this paper is to rethink the discrete dimer integrable system of [GK13] in cluster variable setting and extend it to bipartite torus graphs not necessary obtained from integral polygons. The second goal of the paper is to use perfect matchings on graphs to compute conserved quantities of Q-systems associated with a finite Dynkin diagram of type A and B in Section 5 and 6, respectively. Throughout the paper we denote [x]+ := [m, n] := {m, m + 1, . . . , n}
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