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Cluster ensembles, quantization and the dilogarithm

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  • Vladimir Fock, Denmark
  • Alexander B. Goncharov, Denmark
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 . 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 [7] 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 Teichmüller 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.
Original languageEnglish
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Volume42
Issue6
Pages (from-to)865-930
Number of pages66
ISSN0012-9593
Publication statusPublished - 2009

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