Quantum parameters of the geometric Langlands theory

Fix a smooth, complete algebraic curve $X$ over an algebraically closed field $k$ of characteristic zero. To a reductive group $G$ over $k$, we associate an algebraic stack $\operatorname{Par}_G$ of quantum parameters for the geometric Langlands theory. Then we construct a family of (quasi-)twistings parametrized by $\operatorname{Par}_G$, whose module categories give rise to twisted $\mathcal D$-modules on $\operatorname{Bun}_G$ as well as quasi-coherent sheaves on the DG stack $\operatorname{LocSys}_G$.