Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Journée ALPE -- Generalized Character Varieties and Quantization via Factorization Homology

Factorization homology is a local-to-global invariant which "integrates" disk algebras in symmetric monoidal higher categories over manifolds. In this talk I will focus on a particular instance of factorization homology on surfaces where the input algebraic data is a braided monoidal category. If one takes the representation category of a quantum group as an input, it was shown by Ben-Zvi, Brochier and Jordan (BZBJ) that categorical factorization homology provides a quantization of the Fock-Rosly Poisson structure on the classical G-character variety. I will discuss two applications of the factorization homology approach for quantizing (generalized) character varieties. First, I will explain how to compute categorical factorization homology on surfaces with principal D-bundles decorations, for D a finite group. The main example comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We will see that in this case factorization homology gives rise to a quantization of Out(G)-twisted character varieties (This is based on joint work with Lukas Müller). In a second part we will consider surfaces that are decorated with marked points. It was shown by BZBJ that the algebraic data governing marked points are braided module categories and I will discuss an example related to the theory of dynamical quantum groups. https://damienespadon.wixsite.com/dersympapp/alpe