Quantum geometry of moduli spaces of local systems

Abstract: Let $G$ be a split semi-simple algebraic group over $\mathbb{Q}$. We introduce a natural cluster structure on moduli spaces of $G$-local systems over surfaces with marked points. As a consequence, the moduli spaces of $G$-local systems admit natural Poisson structures, and can be further quantized. We will study the principal series representations of such quantum spaces. It will recover many classical topics, such as the $q$-deformed Toda systems, quantum groups, and the modular functor conjecture for such representations. This talk will mainly be based on joint work with A.B. Goncharov.

algebraic geometrycombinatoricsdifferential geometrygeometric topologyquantum algebrarepresentation theorysymplectic geometry

Audience: researchers in the topic

Legendrians, Cluster algebras, and Mirror symmetry

Series comments: Schedule
School: January 4–8, 2021
Conference: January 11–15, 2021