Abstract
In this thesis we extend the factorization homology method for quantizing G-character varieties due to D. Ben-Zvi, A. Brochier and D. Jordan to surfaces with certain decorations to obtain functorial quantizations of twisted- and dynamical character varieties. For the former, we will consider surfaces decorated with D-bundles, for D a finite group, and local coefficients given by balanced braided tensor categories with a D-action. Our main example comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We show that in this case factorization homology gives rise to a quantization of the moduli space of flat twisted G-bundles (this part is based on joint work with L. Müller). In a second part, we consider surfaces with marked points and local coefficients coming from the theory of dynamical quantum groups: local coefficients for the bulk are quantum group representations and the point defects are governed by dynamical twists coming from solutions to the quantum dynamical Yang–Baxter equation. We show that factorization homology gives rise to a deformation quantization of a Fock-Rosly type Poisson bracket defined in terms of classical dynamical r-matrices. These Poisson structures have previously appeared in the context of Chern-Simons theory coupled to dynamical sources as studied by E. Buffenoir and P. Roche.