Producing a Poisson cluster variety using dimer models II

Last time, we discussed how the coordinate ring of the moduli space of line bundles on a graph provided the right object to give gauge invariant functions on bipartite graphs. We then discussed how to glue together such moduli spaces to get a global modified partition function on the graph and how to associate a Newton polygon to it. In this lecture, we will explore how to recover a family of graphs whose moduli spaces of line bundles are all related by a cluster mutation. We will also define a Poisson structure of these varieties and discuss the construction of special Casimir elements with respect to the Poisson structure. Finally, time permitting, we will discuss how this leads to a set of Hamiltonians corresponding to the interior points of the Newton polygon.

Date published: Wednesday, November 16, 2022