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Holomorphic quantization and unitary representations of the Teichmüller group. (English) Zbl 0846.58030

Brylinski, Jean-Luc (ed.) et al., Lie theory and geometry: in honor of Bertram Kostant on the occasion of his 65th birthday. Invited papers, some originated at a symposium held at MIT, Cambridge, MA, USA in May 1993. Boston, MA: Birkhäuser. Prog. Math. 123, 21-64 (1994).
The authors present two approaches to the integrable connection of Witten and its monodromy. The first is a continuation of Brylinski’s work on nonabelian theta functions. It is based on the description of the quantum bundle in terms of loop groups and on the Sugawara construction, which realizes the Virasoro Lie algebra as quadratic elements in some completion of the enveloping algebra of a loop algebra.
The second method is based on a new approach to the construction and study of holomorphic line bundles with hermitian metrics by cohomological means. This provides us with a cohomological method to construct holomorphic line bundles with hermitian metrics and gives also an alternate approach to the theory of the Quillen metric on determinant line bundles, which is usually based on the spectral analysis of the Laplace operator.
For the entire collection see [Zbl 0807.00014].

MSC:

53D50 Geometric quantization
22E67 Loop groups and related constructions, group-theoretic treatment
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
32L05 Holomorphic bundles and generalizations
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