A geometric model for syzygies over 2-Calabi-Yau tilted algebras

In this article, we consider the class of 2-Calabi-Yau tilted algebras that are defined by a quiver with potential whose dual graph is a tree. These algebras are wild in general. For every such algebra $B$, we construct a polygon $\mathcal{S}$ with a checkerboard pattern in its interior that gives rise to a category Diag$(\mathcal{S})$. The indecomposable objects of Diag$(\mathcal{S})$ are the 2-diagonals in $\mathcal{S}$, and its morphisms are given by certain pivoting moves between the 2-diagonals. We conjecture that the category Diag$(\mathcal{S})$ is equivalent to the stable syzygy category over the algebra $B$, such that the rotation of the polygon corresponds to the shift functor on the syzygies. In particular, the number of indecomposable syzygies is finite and the projective resolutions are periodic. We prove the conjecture in the special case where every chordless cycle in the quiver is of length three. As a consequence, we obtain an explicit description of the projective resolutions. Moreover, we show that the syzygy category is equivalent to the 2-cluster category of type $\mathbb{A}$, and we introduce a new derived invariant for the algebra $B$ that can be read off easily from the quiver.