Moduli spaces of curves and representation theory

E. Arbarello, C. De Concini, V. G. Kac, C. Procesi

Research output: Contribution to journalArticlepeer-review

144 Citations (Scopus)

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 {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 ℂx 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 ℂx vanishing at ∞, under the group Aut H. As an application, we give a Lie-algebraic proof of the Mumford formula: λn=(6 n2-6 n+1)λ1, 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.

Original languageEnglish
Pages (from-to)1-36
Number of pages36
JournalCommunications in Mathematical Physics
Volume117
Issue number1
DOIs
Publication statusPublished - Mar 1988
Externally publishedYes

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