A flag version of Beilinson–Drinfeld Grassmannian for surfaces

Abstract We define and study a generalization of the Beilinson–Drinfeld Grassmannian to the case where the curve is replaced by a smooth projective surface $X$ , and the trivialization data are given on loci associated with a nonlinear flag of closed subschemes. We first establish some general formal gluing results for moduli of (almost) perfect complexes and torsors. We construct a simplicial object $\operatorname {\underline {\mathsf{Fl}}}_X$ of flags of closed subschemes of a smooth projective surface $X$ , associated with 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 define a derived, flag analogue $\mathcal{G}r_X$ 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: we prove a factorization formula, the existence of a canonical flat connection and define a chiral product on suitable sheaves on $\operatorname {\underline {\mathsf{Fl}}}_X$ and on $\mathcal{G}r_X$ . We sketch the construction of actions of flags analogues of the loop group and of the positive loop group on $\mathcal{G}r_X$ . To fixed ‘large’ flags on $X$ , we associate ‘exotic’ derived structures on the stack of $G$ -bundles on $X$ .