Abstract
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 space\(\hat F_{g - 1} \) of quintuples (C, p, z, L, [ϕ]), whereC is a smooth genusg Riemann surface,p a point onC, z a local parameter atp, L a degreeg−1 line bundle onC, and [ϕ] a class of local trivializations ofL atp which differ by a non-zero factor. The construction uses an interplay between various infinite-dimensional manifolds based on the topological spaceH of germs of holomorphic functions in a neighborhood of 0 in ℂx and related topological spaces. The basic tool is a canonical map from\(\hat F_{g - 1} \) to the infinite-dimensional Grassmannian of subspaces ofH, which is the orbit of the subspaceH − of holomorphic functions on ℂx vanishing at ∞, under the group AutH. As an application, we give a Lie-algebraic proof of the Mumford formula: λ n =(6n 2−6n+1)λ1, where λ n is the determinant line bundle of the vector bundle on the moduli space of curves of genusg, whose fiber overC is the space of differentials of degreen onC.
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Communicated by A. Jaffe
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Arbarello, E., De Concini, C., Kac, V.G. et al. Moduli spaces of curves and representation theory. Commun.Math. Phys. 117, 1–36 (1988). https://doi.org/10.1007/BF01228409
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DOI: https://doi.org/10.1007/BF01228409