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 , 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 of flags of closed subschemes of a smooth projective surface , associated with the operation of taking union of flags. We prove that this simplicial object has the -Segal property. For an affine complex algebraic group , we define a derived, flag analogue of the Beilinson–Drinfeld Grassmannian of -bundles on the surface , 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 and on . We sketch the construction of actions of flags analogues of the loop group and of the positive loop group on . To fixed ‘large’ flags on , we associate ‘exotic’ derived structures on the stack of -bundles on .