Finite symmetries of quantum character stacks

TL;DR

This paper explores the structure of categorical factorisation homology on surfaces with principal D-bundles, revealing new algebraic descriptions and connections to quantum group symmetries and moduli space quantization.

Contribution

It extends the understanding of factorisation homology to surfaces with D-bundles and provides explicit algebraic and representation-theoretic descriptions, including a quantization of moduli spaces.

Findings

Identifies factorisation homology with module categories over explicit algebras in al for surfaces with boundary.

Describes boundary conditions and point defects via equivariant representation theory.

Shows that for Dynkin diagram automorphisms, factorisation homology quantizes moduli spaces of flat twisted bundles.

Abstract

For a finite group $D$, we study categorical factorisation homology on oriented surfaces equipped with principal $D$-bundles, which `integrates' a (linear) balanced braided category $\mathcal{A}$ with $D$-action over those surfaces. For surfaces with at least one boundary component, we identify the value of factorisation homology with the category of modules over an explicit algebra in $\mathcal{A}$, extending the work of Ben-Zvi, Brochier and Jordan to surfaces with $D$-bundles. Furthermore, we show that the value of factorisation homology on annuli, boundary conditions, and point defects can be described in terms of equivariant representation theory. Our main example comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We show that in this case factorisation homology gives rise to a quantisation of the moduli space of flat twisted bundles.