Apparaît dans la collection : Combinatorics and Arithmetic for Physics : Special Days

Cluster duality is a correspondence between tropical points of a cluster A-variety and a canonical basis of functions on the corresponding X-variety. (It is a gen- eralization of duality between integers and the multiplicative group.) In the talk we will suggest related geometric interpretations of the tropical points of the A- variety for local system of the curve. On on hand it can be considered as a class of graphs on the surfaces colored by generators of the affine Weyl group. This is a generalization of the notion of a measured laminaiton. On the other hand it can be interpreted as a class class of Lagrangian coverings in the cotangent bundle to the curve representing integer classes of homology. Finally they are related to the ”cells” of the space of local system on the curve with values in the affine group. (Joint with A.Thomas and V.Tatitscheff)