Quantum character varieties and braided module categories

Understand

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points.

• These are categorical invariants $\int_S\mathcal A$ of a surface $S$, determined by the choice of a braided tensor category $\mathcal A$, and computed via factorization homology.
• We identify the algebraic data governing marked points and boundary components with the notion of a {\em braided module category} for $\mathcal A$, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called {\em quantum moment maps}.
• We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction.