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An explicit construction of a pre-quantum line bundle for the moduli space offlat G-bundles over a Riemann surface is given, where G is any non-simplyconnected compact simple Lie group. This work helps to explain a curiouscoincidence previously observed between Toledano-Laredo's work classifyingcentral extensions of loop groups LG and the author's previous work on theobstruction to pre-quantization of the moduli space of flat G-bundles.

Central extensions of loop groups and obstruction to pre-quantization