# Conserved quantities of Q-systems from dimer integrable systems

## TL;DR

This paper investigates a discrete dynamic on bipartite graphs related to dimer integrable systems, demonstrating invariance of Hamiltonians and constructing conserved quantities for Q-systems of types A and B.

## Contribution

It introduces a novel dynamic on weighted bipartite graphs not derived from polygons, showing invariance of Hamiltonians and constructing conserved quantities for Q-systems.

## Key findings

- Hamiltonians are invariant under graph moves.
- Constructed graphs for Q-systems of types A and B.
- Conserved quantities Poisson commute in type A.

## 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 show that all Hamiltonians, partition functions of all weighted perfect matchings with a common homology class, 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. The conserved quantities can be written as partition functions of hard particles on a certain graph. For type A, they Poisson commute under a nondegenerate Poisson bracket.