Abstract
Let $S$ be a surface, $G$ a simply connected classical group, and $G^{\prime }$ the associated adjoint form of the group. We show that the moduli spaces of framed local systems ${\mathcal{X}}_{G^{\prime },S}$ and ${\mathcal{A}}_{G,S}$, which were constructed by Fock and Goncharov [‘Moduli spaces of local systems and higher Teichmuller theory’, Publ. Math. Inst. Hautes Études Sci.103 (2006), 1–212], have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in that paper, and also allows one to quantize higher Teichmuller space, which was previously only possible when $G$ was of type $A$.
Highlights
Let S be a topological surface with nonempty boundary, let G be a connected semisimple group, and let G be the adjoint form of this group
We are interested in the space LG,S, the moduli space of G-local systems on the surface S, or, equivalently, the space of representations of π1(S) into G
We show that (AG,S, XG,S) forms a cluster ensemble
Summary
Let S be a topological surface with nonempty boundary, let G be a connected semisimple group, and let G be the adjoint form of this group. In each of these sections, the most important subsections are the first and the last two, consisting of the construction of the seed for the cluster algebra structure on Confm AG, as well the sequences of mutations that realize the S3 symmetries on each triangle and the flip of a triangulation. We show how the cluster algebra structure is related to reduced words in the Weyl group; we show how the cactus sequence of mutations plays a role in constructing the seeds and computing some of the S3 symmetries; we explain how folding the seed for SL2n gives the seed for Sp2n, how Langlands duality relates the seeds for Sp2n and Spi n2n+1, and how unfolding the seed for Spi n2n+1 gives the seed for Spin2n+2.
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