Cluster Duality and Non-Holomorphic Spectral Curves

Explore cluster duality and its geometric interpretations in this 49-minute lecture by Vladimir Fock from Strasbourg University, presented at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the correspondence between tropical points of cluster A-varieties and canonical basis functions on X-varieties, extending the concept of duality between integers and the multiplicative group. Examine the geometric interpretations of tropical points for local systems on curves, including colored graphs on surfaces using affine Weyl group generators, Lagrangian coverings in cotangent bundles representing integer homology classes, and connections to the space of local systems with affine group values. Discover how these concepts relate to measured laminations and the "cells" of local system spaces. Follow the lecture's progression through topics such as bipartite graphs, metroids, configuration spaces, surfaces, tropical limits, and the geometric answers provided by this framework.