Moduli spaces of curves and representation theory (original) (raw)

We establish a canonical isomorphism between the second cohomology of the Lie algebra of regular differential operators on ℂx of degree ≦1, and the second singular cohomology of the moduli spacehat F_{g - 1} 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 p, 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 ℂx and related topological spaces. The basic tool is a canonical map fromhat F_{g - 1} to the infinite-dimensional Grassmannian of subspaces of H, which is the orbit of the subspace H - of holomorphic functions on ℂx vanishing at ∞, under the group Aut H. As an application, we give a Lie-algebraic proof of the Mumford formula: λ n =(6 n 2-6 n+1)λ1, 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.