2016/06/24 by Dylan G. L. Allegretti, Allegretti, Dylan G. L.
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1606.07788
openalex publication_date 2016/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.