CLUSTER ENSEMBLES, QUANTIZATION AND THE DILOGARITHM

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.