The geometry of cluster varieties from surfaces
arXiv:1606.07788
Abstract
Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented surfaces with boundary. The main original contribution of this thesis is to develop the properties of a particular kind of cluster variety called the symplectic double. We show that the symplectic double is birational to a certain moduli space of local systems associated to a doubled surface. We define a version of the notion of measured lamination on such a surface and prove that the space of all such laminations is a tropicalization of the symplectic double. We describe a canonical map from this space of laminations into the algebra of rational functions on the symplectic double. The second main contribution of this thesis is a proof of Fock and Goncharov's duality conjectures for quantum cluster varieties associated to a disk with finitely many marked points on its boundary. These duality conjectures identify a canonical set of elements in the quantized algebra of functions on a cluster variety satisfying a number of special properties.
173 pages. Ph.D. thesis. Version 2: Updated to match arXiv:1410.3035, arXiv:1509.01342, and arXiv:1605.01599