Weyl groups and cluster structures of families of log Calabi-Yau surfaces

Abstract

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

Cited by in corpus (1)

• Stable maps to Looijenga pairs