Conserved quantities of Q-systems from dimer integrable systems (Q1700804)

Summary: We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems in [\textit{A. B. Goncharov} and \textit{R. Kenyon}, Ann. Sci. Éc. Norm. Supér. (4) 46, No. 5, 747--813 (2013; Zbl 1288.37025)]. 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 \textit{P. Di Francesco} and \textit{R. Kedem} [Commun. Math. Phys. 293, No. 3, 727--802 (2010; Zbl 1194.05165)] and \textit{P. Galashin} and \textit{P. Pylyavskyy} [``Quivers with subadditive labelings: classification and integrability'', Preprint, \url{arXiv:1606.04878}] for the Q-systems of type A, and gives new results for that of type B. Similar to the results in [Di Francesco and Kedem, loc. cit.], 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.