Sheaves of categories on the moduli stack of local systems on the formal punctured disk via factorization.

Abstract: Given a DG category acted on by the category of quasi-coherent sheaves on LocSys_G(D^{\circ}), one can define a factorization Rep(G)-module category. Following ideas of Beilinson and Drinfeld, Gaitsgory conjectured that this construction loses no information: that it gives a fully faithful 2-functor QCoh(LocSys_G(D^{\circ}))-mod (DGCat) —> Rep(G)-mod^fact(DGCat). We will discuss this conjecture, its role in the local Geometric Langlands program, and we will prove a partial result in this direction. Namely, we will prove the fully faithfulness for QCoh(LocSys_G(D^{\circ}))-modules set-theoretically supported over the stack of local systems with restricted variation on the formal punctured disk.