A flag version of Beilinson–Drinfeld Grassmannian for surfaces

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 trivializa- tion 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 com- plexes and torsors. We construct a simplicial object FlX 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 GrX 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 general- ization: we prove a factorization formula, the existence of a canonical flat connection and define a chiral product on suitable sheaves on FlX and on GrX . We sketch the construction of actions of flags analogues of the loop group and of the positive loop group on GrX . To fixed ‘large’ flags on X, we associate ‘exotic’ derived structures on the stack of G-bundles on X.