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Moduli spaces of local systems and higher Teichmüller theory

Vladimir Fock; Alexander Goncharov — 2006

Publications Mathématiques de l'IHÉS

Let G be a split semisimple algebraic group over with trivial center. Let S be a compact oriented surface, with or without boundary. We define representations of the fundamental group of S to G(), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related...

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock; Alexander B. Goncharov — 2009

Annales scientifiques de l'École Normale Supérieure

A cluster ensemble is a pair $(𝒳, 𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $Γ$ . The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p : 𝒜 \to 𝒳$ . The space $𝒜$ is equipped with a closed $2$ -form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role...

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