Cluster Structures on Higher Teichmuller Spaces for Classical Groups

TL;DR

This paper demonstrates that moduli spaces of framed local systems for classical groups have cluster variety structures, enabling quantization of higher Teichmuller spaces beyond type A groups, simplifying proofs and expanding applications.

Contribution

It establishes cluster structures on moduli spaces for classical groups, extending the quantization framework of higher Teichmuller spaces beyond type A groups.

Findings

Moduli spaces $ ext{X}_{G',S}$ and $ ext{A}_{G,S}$ have cluster variety structures.

Simplifies proofs in Fock-Goncharov's earlier work.

Enables quantization of higher Teichmuller spaces for classical groups.

Abstract

Let $S$ be a surface, $G$ a simply-connected classical group, and $G'$ the associated adjoint form of the group. We show that the spaces of moduli spaces of framed local systems $\X_{G',S}$ and $\A_{G,S}$, which were constructed by Fock and Goncharov (\cite{FG1}), have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in \cite{FG1}, and also allows one to quantize higher Teichmuller space following the formalism of \cite{FG2}, \cite{FG3}, and \cite{FG5}, which was previously only possible when $G$ was of type $A$.