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We establish a canonical isomorphism between the second cohomology of the Lie algebra of regular differential operators on (C x 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 (C x vanishing at oo, under the group AutH. As an application, we give a Lie-algebraic proof of the Mumford formula: λ n = (6n 2 -6n + ί)λ u where λ n 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.

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