On syzygy categories of dimer tree algebras

A dimer tree algebra A is the Jacobian algebra of a quiver Q (without loops and 2-cycles) with a canonical potential such that (i) every arrow lies in a chordless oriented cycle; and (ii) the dual graph is a tree. The category of non-projective syzygies over A is equivalent to the stable category of (maximal) Cohen Macaulay modules as well as to the singularity category of A. It is a triangulated 3-Calabi-Yau category.
We give a geometric model for the syzygy category for dimer tree algebras in terms of 2-diagonals in a polygon with checkerboard pattern.
This checkerboard pattern can be extended to a dimer model on the disk.