A flag version of Beilinson-Drinfeld Grassmannian for surfaces

In this paper we define and study a generalization of the Belinson-Drinfeld Grassmannian to the case where the curve is replaced by a smooth projective surface $X$, and the trivialization data are given with respect to a flag of closed subschemes. In order to do this, we first establish some general formal gluing results for moduli of almost perfect complexes, perfect complexes and torsors. We then construct a simplicial object of flags of closed subschemes of a smooth projective surface $X$, naturally associated to the operation of taking union of flags. We prove that this simplicial object has the $2$-Segal property. For an affine complex algebraic group $G$, we finally define a flag analog $\mathcal{G}r_X$ of the Beilinson-Drinfeld Grassmannian of $G$-bundles on the surface $X$, and show that most of the properties of the Beilinson-Drinfeld Grassmannian for curves can be extended to our flag generalization. In particular, we prove a factorization formula, the existence of a canonical flat connection, we construct actions of the loop group and of the positive loop group on $\mathcal{G}r_X$, and define a fusion product on sheaves on $\mathcal{G}r_X$.