Tame twistings and $\Theta$-data

The goal of this paper is to assign an intrinsic meaning to the space of quantum parameters $\operatorname{Par}_G$ appearing in the geometric Langlands program of Beilinson-Drinfeld. We introduce tame twistings, a variant of twisted differential operators (TDOs) for which regularity of twisted $\mathscr D$-modules is well-defined. Our main result is that for a proper curve $X$, $\operatorname{Par}_G$ is precisely the moduli space of factorization tame twistings on the affine Grassmannian.