Moduli spaces of curves and representation theory |
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Authors: | E Arbarello C De Concini V G Kac C Procesi |
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Institution: | (1) Dipartimento di Matematica, Universita Degli Studi di Roma La Sapienza , Rome, Italy;(2) Dipartimento di Matematica, Universita di Roma II Tor Vergata , Rome, Italy;(3) Department of Mathematics, M.I.T., 02139 Cambridge, MA, USA |
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Abstract: | We establish a canonical isomorphism between the second cohomology of the Lie algebra of regular differential operators on x of degree 1, and the second singular cohomology of the moduli space
of quintuples (C, p, z, L, ]), whereC is a smooth genusg Riemann surface,p a point onC, z a local parameter atp, L a degreeg–1 line bundle onC, and ] 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 x and related topological spaces. The basic tool is a canonical map from
to the infinite-dimensional Grassmannian of subspaces ofH, which is the orbit of the subspaceH
– of holomorphic functions on x vanishing at , under the group AutH. As an application, we give a Lie-algebraic proof of the Mumford formula:
n
=(6n
2–6n+1) 1, where
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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