In this Letter we construct Abelian extensions of the group of diffecomorphisms of a torus. We consider the Jacobian map, which is a crossed homomorphism from the group of diffeomorphisms into a toroidal gauge group. A pull-back under this map of an invariant central 2-cocycle on a gauge group turns out to be an Abelian cocycle on the group of diffeomorphisms. In the case of a circle we get an interpretation of the Virasoro-Bott cocycle as a pull-back of the Heisenberg cocycle. We also give an Abelian generalization of the Virasoro-Bott cocycle to the case of a manifold with a volume form.

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Keywords abelian extensions, group of diffeomorphisms, Virasoro-Bott group
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Journal Letters in Mathematical Physics
Billig, Y. (2003). Abelian extensions of the group of diffeomorphisms of a torus. Letters in Mathematical Physics, 64(2), 155–169. doi:10.1023/A:1025750704319