Weyl Groups and Cluster Structures of Families of Log Calabi–Yau Surfaces

Abstract

Abstract Given a generic Looijenga pair $(Y,D)$ together with a toric model $\rho :(Y,D)\rightarrow (\overline {Y},\overline {D})$, one can construct a seed $\textbf {s}$ such that the corresponding $\mathcal {X}$-cluster variety $\mathcal {X}_{\textbf {s}}$ can be viewed as the universal family of the log Calabi–Yau surface $U=Y\setminus D$. In cases where $(Y,D)$ is positive and $\mathcal {X}_{\textbf {s}}$ is not acyclic, we describe the action of the Weyl group of $(Y,D)$ on the scattering diagram $\mathfrak {D}_{\textbf {s}}$. Moreover, we show that there is a Weyl group element $\textbf {w}$ of order $2$ that either agrees with or approximates the Donaldson–Thomas transformation $\textrm {DT}_{\mathcal {X}_{\textbf {s}}}$ of $\mathcal {X}_{\textbf {s}}$. As a corollary, $\textrm {DT}_{\mathcal {X}_{\textbf {s}}}$ is cluster. In positive non-acyclic cases, we also apply the folding technique as developed in [20] and construct a maximally folded new seed $\overline {\textbf {s}}$ from $\textbf {s}$. The $\mathcal {X}$-cluster variety $\mathcal {X}_{\overline {\textbf {s}}}$ is a locally closed subvariety of $\mathcal {X}_{\textbf {s}}$ and corresponds to the maximally degenerate subfamily in the universal family. We show that the action of the special Weyl group element $\textbf {w}$ on $\mathfrak {D}_{\textbf {s}}$ descends to $\mathfrak {D}_{\overline {\textbf {s}}}$ and permutes distinct subfans in $\mathfrak {D}_{\overline {\textbf {s}}}$, generalizing the well-known case of the Markov quiver.