We establish a canonical isomorphism between the second coho- mology of the Lie algebra of regular differential operators on (Cx of degree ^ 1, and the second singular cohomology of the moduli space &g-γ of quintuples (C, p, z, L, (_φ~ where C is a smooth genus g Riemann surface, p a point on C, z a local parameter at /?, L a degree g— 1 line bundle on C, and (φ) a class of local trivializations of L at p which differ by a non-zero factor. The construction uses an interplay between various infinite-dimensional manifolds based on the topological space H of germs of holomorphic functions in a neighborhood of 0 in (C x and related topological spaces. The basic tool is a canonical map from #0_! to the infinite-dimensional Grassmannian of subspaces off/, which is the orbit of the subspace H_ of holomorphic functions on (Cx vanishing at oo, under the group AutH. As an application, we give a Lie-algebraic proof of the Mumford formula: λn = (6n2 — 6n + ί)λu where λn is the determinant line bundle of the vector bundle on the moduli space of curves of genus g, whose fiber over C is the space of differentials of degree n on C.