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首页> 外文期刊>Annales scientifiques de l'Ecole normale superieure >CLUSTER ENSEMBLES, QUANTIZATION AND THE DILOGARITHM
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CLUSTER ENSEMBLES, QUANTIZATION AND THE DILOGARITHM

机译：簇团，量化和对数

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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 Gamma. The space A is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism p : A -> 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 in the cluster ensemble structure. We define a non-commutative q-deformation of the X-space. When q is a root of unity the algebra of functions on the q-deformed X-space has a large center, which includes the algebra of functions on the original X-space. The main example is provided by the pair of moduli spaces assigned in [6] to a topological surface S with a finite set of points at the boundary and a split semisimple algebraic group G. It is an algebraic-geometric avatar of higher Teichmuller theory on S related to G. We suggest that there exists a duality between the A and X spaces. In particular, we conjecture that the tropical points of one of the spaces parametrise a basis in the space of functions on the Langlands dual space. We provide some evidence for the duality conjectures in the finite type case.
机译：集群合奏是一对（X，A）正空间（即配备正图集的变体），具有对称群Gamma的作用。空间A与簇代数的光谱密切相关[12]。这两个空间的相态为p：A->X。空间A配备有可能是简并的闭合2形式，并且空间X具有泊松结构。映射p与这些结构兼容。双对数及其动机和量子化身在团簇集成结构中起着核心作用。我们定义了X空间的非交换q变形。当q为单位根时，q变形的X空间上的函数代数的中心较大，其中包括原始X空间上的函数代数。主要示例由在[6]中分配给在边界上具有有限点集和分裂半简单代数群G的拓扑表面S的一对模空间提供。这是更高的Teichmuller理论的代数几何化身。 S与G有关。我们建议A和X空间之间存在对偶性。特别是，我们推测一个空间的热带点参数化了Langlands对偶空间上函数空间的基础。我们为有限类型情况下的对偶猜想提供了一些证据。

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