An Approach to Cluster Structures on Moduli of Local Systems for General Groups

Abstract

Let $S$ be a surface, $G$ a simply-connected classical group, and $G'$ the associated adjoint form of the group. In \cite{FG1}, it was shown that the moduli spaces of framed local systems $\X_{G',S}$ and $\A_{G,S}$ have the structure of cluster varieties, and thus together form a cluster ensemble, when $G$ had type $A$. This was extended to classical groups in \cite{Le}. In this paper we give an algorithm for constructing the cluster structure for general reductive groups $G$. The algorithm can be carried out under some mild hypotheses, which we explain, and which we believe hold in general. We show that these hypotheses hold when $G$ has type $G_2$, and therefore we are able to construct the cluster structure in this case. We also illustrate our approach by rederiving the cluster structure for $G$ of type $A$. Our goals are to give some heuristics for the approach taken in \cite{Le}, point out the difficulties that arise for more general groups, and to record some useful calculations. Forthcoming work by Goncharov and Shen gives a different approach to constructing the cluster structure on $\X_{G',S}$ and $\A_{G,S}$. We hope that some of the ideas here complement their more comprehensive work.