## Abstract

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 {Mathematical expression} of quintuples (C, p, z, L, [φ{symbol}]), 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{symbol}] 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 {Mathematical expression} to the infinite-dimensional Grassmannian of subspaces of H, which is the orbit of the subspace H_{-} of holomorphic functions on ℂ^{x} 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)λ_{1}, 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.

Original language | English |
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Pages (from-to) | 1-36 |

Number of pages | 36 |

Journal | Communications in Mathematical Physics |

Volume | 117 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1988 |

Externally published | Yes |