I will present the recent work (collaboration with K.Kremnizer, Oxford) on constructing factorization spaces over Beilinson-Drinfeld Grassmannians in differential geometry, where the underlying manifold can be of arbitrary dimension, not necessarily a Riemann surface as in the usual theory of vertex operator algebras. Time permitting, I will also present the definition of quasi-coherent and coherent sheaves in differential geometry (also collaboration with K.Kremnizer), that is necessary for linearizing factorization spaces and obtaining chiral algebras.